Create the page "Degree (angle)" on this wiki! See also the search results found.

• Conic section
  ...described in [[Cartesian coordinates]] by a [[polynomial]] [[equation]] of degree two or less. ...cutting plane makes an [[angle]] with an [[axis]] that is larger than the angle between the slant height and the axis.
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• Geometry
  ...he intersection two [[ray]]s at their endpoints. It is measure in either [[degree]]s or [[radian]]s, though the less-common [[Système international|metric]]
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• 2002 AMC 12A Problems
  Find the degree measure of an angle whose complement is 25% of its supplement. ...a right triangle with <math>\angle ACB</math> as its right angle, <math>m\angle ABC = 60^{\circ}</math>, and <math>AB = 10</math>. Let <math>P</math> be ra
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• 2001 AMC 12 Problems
  ...1,4)</math>, and <math>E=(0,4)</math>. What is the probability that <math>\angle APB</math> is obtuse? A polynomial of degree four with leading coefficient 1 and integer coefficients has two real zeros
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• 2002 AMC 12B Problems
  Let <math>v, w, x, y, </math> and <math>z</math> be the degree measures of the five angles of a pentagon. Suppose that <math>v < w < x < y Let <math>\triangle XOY</math> be a right-angled triangle with <math>\angle XOY = 90^{\circ}</math>. Let <math>M</math> and <math>N</math> be the midpo
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• 2006 AMC 12A Problems/Problem 22
  ...probability of choosing a point is <math>1 / 2</math>, or if the total arc degree measures add up to <math>\frac{1}{2} \cdot 360^{\circ} = 180^{\circ}</math> ...an [[isosceles triangle]]: <math>\angle AOP = 15^{\circ}</math> and <math>\angle OAP = OAB - 60^{\circ} = \frac{180^{\circ}-30^{\circ}}{2} - 60^{\circ} = 15
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• 2006 AMC 10A Problems/Problem 19
  How many non-[[similar]] triangles have angles whose degree measures are distinct positive integers in [[arithmetic progression]]? ...c{180}{3} = 60</math> degrees. The minimum possible value for the smallest angle is <math>1</math> and the highest possible is <math>59</math> (since the nu
  2 KB (259 words) - 03:10, 22 June 2023
• 2005 AIME II Problems/Problem 12
  ...E<BF </math> and <math> E </math> between <math> A </math> and <math> F, m\angle EOF =45^\circ, </math> and <math> EF=400. </math> Given that <math> BF=p+q\ ...math>). Then <math>\tan \angle EOG = \frac{x}{450}</math>, and <math>\tan \angle FOG = \frac{y}{450}</math>.
  13 KB (2,080 words) - 21:20, 11 December 2022
• 2004 AIME II Problems/Problem 1
  ...angle]] which contains the entire chord is <math>60 \cdot 2 = 120</math> [[degree]]s, so the area of the [[sector]] is <math>\frac{1}{3}r^2\pi</math>; the re
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• 1984 AIME Problems
  ...circ</math> and <math>180^\circ</math> in the complex plane. Determine the degree measure of <math>\theta</math>. ...{ cm}^2</math>. These two faces meet each other at a <math>30^\circ</math> angle. Find the volume of the tetrahedron in <math>\mbox{cm}^3</math>.
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• 2001 AIME I Problems
  ...</math> degrees and <math>45</math> degrees, respectively. The bisector of angle <math>A</math> intersects <math>\overline{BC}</math> at <math>T</math>, and ...where <math>d < 120.</math> The length of the chord of a <math>3d</math>-degree arc is <math>- m + \sqrt {n}</math> centimeters, where <math>m</math> and <
  7 KB (1,212 words) - 22:16, 17 December 2023
• 1983 AIME Problems/Problem 14
  ...must add up to <math>180^{\circ}</math>. By the [[Law of Cosines]], <math>\angle APB=\cos^{-1}\left(\frac{{-11}}{24}\right)</math>. Also, angles <math>QPA</ ...). In other words, since <math>\angle TP'N = 90^\circ</math>, then <math> \angle TPN = 90^\circ</math>.
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• 2001 AIME I Problems/Problem 13
  ...where <math>d < 120.</math> The length of the chord of a <math>3d</math>-degree arc is <math>- m + \sqrt {n}</math> centimeters, where <math>m</math> and < ...of the chords, <math>AD</math> and <math>BC</math>, of the <math>2d</math>-degree arcs can be represented as <math>x + 20</math>, as given in the problem.
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• 2000 AIME II Problems/Problem 15
  ...in(1)=\frac{1}{60}</math> in degrees, convert to radians and use the small angle approximation <math>\sin(x)=x</math>. Hence, <math>S=\sin(1)</math>, and our angle is <math>\boxed{001}</math>.
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• 1959 IMO Problems/Problem 4
  ...> is <math>ab</math>, since <math>a</math> and <math>b</math> form a right angle. However, twice the area of the triangle is also the product of <math>c</m We denote the angle between <math>b</math> and <math>c</math> as <math>\alpha</math>. The prob
  6 KB (939 words) - 17:31, 15 July 2023
• Right angle
  ...ne]]s. The [[measure of an angle | measure]] of every right angle is 90 [[degree (geometry) | degrees]] or <math>\frac \pi 2</math> [[radian]]s. When drawin * [[Angle]]
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• Supplementary
  ...ementary if they sum to <math>\displaystyle \pi</math> [[radian]]s (180 [[degree (geometry) | degrees]]).
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• Angle
  ...ex]] of the angle, and the rays themselves are called the [[side]]s of the angle. ...If there is no ambiguity, this notation can be shortened to simply <math>\angle B</math>.
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• Radian
  ...measurement for [[angle|angles]]. In a circle, the measure of a [[central angle]] in radians is the ratio of the length of the [[intercepted arc]] to the l ...the whole circumference of the circle. Thus, radians can be converted to [[Degree (geometry)|degrees]]: <math>2\pi\; rad=360^\circ\implies\pi \;rad=180^\circ
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• Degree
  * [[Degree (geometry)]], the degrees in an [[angle]]. * [[Degree (graph theory)]], the degree of a [[vertex]] in a [[graph (graph theory) | graph]].
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• Hexagon
  ...ternal [[angle]] of a [[Regular polygon | regular]] hexagon measures 120 [[degree (geometry) | degrees]], so the sum of the angles is <math>720^{\circ}</math
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• Straight angle
  ...ting in opposite directions. The measure of every straight angle is 180 [[degree (geometry) | degrees]] or <math>\pi</math> [[radian]]s. * [[Angle]]
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• Obtuse angle
  An '''obtuse angle''' is an [[angle]] with a [[measure]] greater than 90 [[degree (geometry) | degrees]] (<math>\frac\pi2</math> [[radian]]s) but less than 1 Any triangle with an obtuse angle is considered an [[Obtuse Triangle]]
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• Acute angle
  ...ngle''' is an [[angle]] with [[measure]] greater than 0 but less than 90 [[degree (geometry) | degrees]] (<math>\frac\pi2</math> [[radian]]s). Examples:http://www.google.com/search?q=acute+angle&hl=en&client=safari&rls=en&prmd=imvns&tbm=isch&tbo=u&source=univ&sa=X&ei=vp
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• Measure
  ...'measure''' can mean the amount of [[degree (geometry) | degree]]s in an [[angle]], the [[length]] of a [[line segment]], or a [[function]] that assigns a [ ...h>\angle ABC\cong\angle DEF</math>, then <math>\mbox{m}\angle ABC=\mbox{m}\angle DEF</math>.
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• Degree (geometry)
  The '''degree''' of an [[angle]] is a [[measure]]ment of the angle, representing <math>\frac1{360}</math> of a full [[rotation]]. Degrees are indicated by the degree symbol; one degree would be written <math>1^{\circ}</math>.
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• Point
  ...angle]] can be denoted by either one or three points. If there is only one angle with this point as its [[vertex]], only one point is needed; if there are m ...y either two or three points. If the arc has a [[measure]] less than 180 [[degree]]s, only two points are needed; if the arc has a measure equal to or greate
  3 KB (461 words) - 18:59, 22 April 2008
• Mock AIME 6 Pre 2005/Problems
  <math>13</math>. In <math>\triangle ABC</math>, <math>\angle A=30^{\circ}</math>, <math>AC=28\sqrt{3}</math>, and <math>\displaystyle AB <math>14</math>. Let <math>\displaystyle f</math> be a polynomial of degree <math>\displaystyle 2005</math> with leading coefficient <math>\displaystyl
  7 KB (1,110 words) - 05:15, 31 December 2006
• 2007 AIME II Problems/Problem 11
  ...r circle (twice), while the <math>180 - 2(30) = 120^{\circ}</math> central angle in the smaller circle indicates to take <math>\frac{120}{360} = \frac 13</m
  2 KB (407 words) - 16:31, 29 February 2020
• 2007 AMC 12A Problems
  ...angle <math>ADC</math> measures 140 degrees. What is the degree measure of angle <math>BAD</math>? ...10 to 3, and so on, ending back at 12. What is the degree measure of the [[angle]] at each [[vertex]] in the star polygon?
  11 KB (1,750 words) - 13:35, 15 April 2022
• 2007 AMC 12B Problems/Problem 3
  ...e AOB = 140^{\circ}</math>, as shown. What is the degree measure of <math>\angle ABC</math>? ...\angle ABO=20^o</math> and <math>\angle OBC=30^o</math>. Therefore, <math>\angle ABC=50^o</math>, or <math>\mathrm{(D)}</math>.
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• 2007 AMC 12B Problems/Problem 7
  ...gle A = \angle B = 90^{\circ}</math>. What is the degree measure of <math>\angle E</math>? ...also 5, triangle <math>CDE</math> is [[equilateral triangle|equilateral]]. Angle <math>E</math> is therefore <math>90+60=150 \Rightarrow \mathrm {(E)}</math
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• Central angle
  ...entral angle subtends is by definition equal to the measure of the central angle, and is known as the [[arc segment]]'s [[angular distance]]. * [[Degree]]
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• Octagon
  ...ternal [[angle]] of a [[Regular polygon | regular]] octagon measures 135 [[degree (geometry) | degrees]], so the sum of the angles is <math>1080^{\circ}</mat
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• 2007 AMC 12A Problems/Problem 6
  ...angle <math>ADC</math> measures 140 degrees. What is the degree measure of angle <math>BAD</math>? We angle chase and find out that:
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• 2007 AMC 12A Problems/Problem 8
  ...10 to 3, and so on, ending back at 12. What is the degree measure of the [[angle]] at each [[vertex]] in the star polygon? ...of the [[right angle]]). Thus, the angle at each vertex is an [[inscribed angle]] subtending <math>60</math> degrees, making the answer <math>\frac 1260 =
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• 2005 AMC 12A Problems/Problem 15
  ...ered pair for <math>D (x,y)</math>, and notice how <math>E</math> is a 180 degree rotation of <math>D</math>, using the rotation matrix formula we get <math> ...sion of <math>DC</math> hits the circle <math>F</math>. Notice that <math>\angle DCB = 90^\circ</math>. This implies that <math>DC=CF</math>. WOLG, let <mat
  14 KB (1,970 words) - 17:02, 18 August 2023
• 2006 AIME II Problems/Problem 1
  ...[right angle]]s, and <math>\angle B, \angle C, \angle E,</math> and <math>\angle F</math> are [[congruent]]. The area of the hexagonal region is <math>2116( Because <math>\angle
  2 KB (336 words) - 03:19, 12 January 2024
• 2007 AMC 12B Problems
  ...e AOB = 140^{\circ}</math>, as shown. What is the degree measure of <math>\angle ABC</math>? ...gle A = \angle B = 90^{\circ}</math>. What is the degree measure of <math>\angle E</math>?
  12 KB (1,814 words) - 12:58, 19 February 2020
• 2002 AMC 12A Problems/Problem 4
  Find the degree measure of an angle whose complement is 25% of its supplement. ...e <math>90^{\circ}</math> as <math>\frac{3}{4}</math> of the supplementary angle.
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• 1999 AHSME Problems
  ...mum number of points of intersection of the graphs of two different fourth degree polynomial functions <math>y = p(x)</math> and <math>y = q(x)</math>, each ...4 \cos B = 6</math> and <math>4 \sin B + 3 \cos A = 1</math>. Then <math>\angle C</math> in degrees is
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• Reflex angle
  A '''reflex angle''' is an angle whose measure is greater than 180 degrees and less than 360 degrees. |style = "background:#f2f2f2; text-align:center;" | '''[[Degree (geometry)|Degree]]s'''&nbsp;&nbsp;
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• 2007 AMC 10A Problems
  ...angle <math>ADC</math> measures 140 degrees. What is the degree measure of angle <math>BAD</math>? ...des has length <math>4</math>, and each two consecutive sides form a right angle. Suppose that <math>\overline{AG}</math> and <math>\overline{CH}</math> mee
  13 KB (2,058 words) - 17:54, 29 March 2024
• 1966 AHSME Problems
  ...tered at <math>O</math>. <math>C</math> is a point on the circle such that angle <math>BOC</math> is <math>60^\circ</math>. If the diameter of the circle is ...h>BAC</math> are in the ratio <math>2: 3: 4</math>. <math>BD</math> is the angle-bisector drawn to the shortest side <math>AC</math>, dividing it into segme
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• Three Greek problems of antiquity
  ...then show that <math>x</math> must be an [[algebraic number]] and that the degree of it's [[minimal polynomial]] must be a power of <math>2</math>. ==Trisection of the General Angle==
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• 2007 AMC 12B Problems/Problem 14
  .../math> to <math>\overline{BC}</math> be <math>D</math>. Also, since <math>\angle{PQD}=60</math>, let the foot of the altitude from <math>P</math> to <math>\
  3 KB (401 words) - 22:58, 8 May 2023
• 2008 AMC 10B Problems/Problem 24
  ...<math>m\angle BCD = 170^\circ</math>. What is the degree measure of <math>\angle BAD</math>? ...ath>AYB</math> equal to <math>60</math> degrees. We can also find out that angle <math>CYB</math> equals <math>120</math> degrees.
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• 2008 AMC 10A Problems/Problem 19
  ...RS,\, \angle P'RQ'</math> are complementary, it follows that the arc has a degree measure of <math>90^{\circ}</math> and length <math>\frac14</math> of the [
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• 2008 iTest Problems
  Let <math>\triangle XOY</math> be a right-angled triangle with <math>\angle XOY=90^\circ</math>. Let <math>M</math> and <math>N</math> be the midpoints ...C</math> in the sand such that <math>\angle ACB=90^\circ</math> and <math>\angle CBA=15^\circ</math>. He then picks a point at random from within the triang
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• 2002 AMC 12A Problems/Problem 7
  ...that in a circle with radius <math> r </math>, a subtended arc opposite an angle of <math>\theta</math> degrees has length <math>\frac{\theta}{360} \cdot 2\ ...er based on the choices. Firstly, if a higher angle measure equals a lower angle measure, that means that the ratio of areas must be lower, cutting D and E.
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• 2009 AMC 10A Problems/Problem 17
  ...s congruent to <math>DBA</math> and angle <math>ADC</math> is also a right angle. By the transitive property: ...angle has a degree measure of <math>180</math>, angle <math>BEA</math> and angle <math>DBA</math> are similar.
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• 2001 AMC 12 Problems/Problem 24
  ...math>2\cdot BD=CD</math> and <math>\angle DAB=15^\circ</math>. Find <math>\angle ACB.</math> ...<math>\triangle{ABH}</math> we get <math>BH=kt</math>. Using the 60-degree angle in <math>\triangle{ADH}</math>, we obtain <math>DH=\frac{\sqrt{3}}{3}kt</ma
  8 KB (1,316 words) - 22:48, 7 March 2024
• 2009 AMC 10B Problems
  ...>\angle A = \frac 52 \angle B</math>. What is the degree measure of <math>\angle D</math>? ...h>, <math>AB=1</math>, and <math>BC=2</math>. The angle bisector of <math>\angle A</math> intersects side <math>\overline{BC}</math> at <math>D</math>. What
  15 KB (2,262 words) - 00:53, 18 June 2021
• 2009 AMC 10B Problems/Problem 9
  ...>\angle A = \frac 52 \angle B</math>. What is the degree measure of <math>\angle D</math>? <math>\triangle ABC</math> is isosceles, hence <math>\angle ACB = \angle CAB</math>.
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• 2009 AMC 10B Problems/Problem 24
  ...let <math>x</math> be the angle measure in degrees of the larger interior angle of the trapezoid. What is <math>x</math>? ...\circ}2 = 80^\circ</math>, hence the degree measure of the larger interior angle is <math>180^\circ - 80^\circ = \boxed{100^\circ \Longrightarrow A}</math>.
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• 1989 AJHSME Problems/Problem 10
  What is the number of [[degree|degrees]] in the smaller [[angle]] between the hour hand and the minute hand on a clock that reads seven o'c ...clock. A circle is <math>360^\circ</math>, so the measure of the smaller angle is
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• 2003 AMC 12B Problems/Problem 25
  ...m amount of freedom we have to place the third point is a <math>120</math> degree arc(if the first two are the same point), with a <math>\frac{1}{3}</math> p ...degrees, the probability changes linearly (every degree it moves, adds one degree to where the third could be).
  4 KB (635 words) - 11:46, 1 September 2022
• 2010 AMC 12A Problems/Problem 17
  ...ngle{EFA}</math> are congruent by SAS, so the side opposite the 120 degree angle is also the same (since the triangles are congruent). Thus <math>\triangle{
  4 KB (690 words) - 10:13, 14 October 2022
• 2010 AMC 10B Problems
  ...ath> is a diameter and <math>C</math> is a point on the circle with <math>\angle COB = 50^\circ</math>. What is the degree measure of <math>\angle CAB</math>?
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• 2010 AIME II Problems
  Find the number of second-degree polynomials <math>f(x)</math> with integer coefficients and integer zeros f ...th> on <math>\overline{AB}</math> is chosen such that <math>\angle APC = 2\angle ACP</math> and <math>CP = 1</math>. The ratio <math>\frac{AP}{BP}</math> ca
  8 KB (1,246 words) - 21:58, 10 August 2020
• 2010 AMC 10B Problems/Problem 6
  ...h <math>\angle COB = 50^\circ</math>. What is the degree measure of <math>\angle CAB</math>? ...\cong \angle OAC</math>. They also sum to <math>50^{\circ}</math>, so each angle is <math>\boxed{\textbf{(B)}\ 25}</math>.
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• 2010 AMC 10A Problems/Problem 7
  ...45 degrees. She then switches directions to southeast, meaning a 90 degree angle change. The distance now from traveling north for one mile, and her current
  3 KB (370 words) - 00:16, 29 June 2022
• Python reference
  Let's figure out the sine of a 30-degree angle in Python! ...tions f(g(x))). Take 30, convert it to radians, then take the sine of that angle.
  28 KB (4,762 words) - 21:20, 12 June 2023
• 1975 IMO Problems
  ...Q = 30^\circ, \angle ABR = \angle BAR = 15^\circ</math>. Prove that <math>\angle QRP = 90^\circ</math> and <math>QR = RP</math>. (that is, <math>P</math> is homogeneous of degree <math>n</math>),
  2 KB (416 words) - 16:06, 29 January 2021
• 2011 AMC 10B Problems/Problem 18
  ...<math>\angle AMD = \angle CMD</math>. What is the degree measure of <math>\angle AMD</math>? ...BMC = 30^{\circ}</math>. If we let <math>x</math> be the measure of <math>\angle AMD,</math> then
  5 KB (782 words) - 14:29, 1 April 2024
• 2011 AMC 12B Problems
  ...ths in the ratio <math>2 : 3</math>. What is the degree measure of <math>\angle{BAC}</math>? ...t <math>\angle AMD=\angle CMD</math>. What is the degree measure of <math>\angle AMD</math>?
  13 KB (1,978 words) - 16:28, 12 July 2020
• 2011 AMC 12B Problems/Problem 6
  ...ths in the ratio <math>2 : 3</math>. What is the degree measure of <math>\angle{BAC}</math>? ...er to solve this problem, use of the tangent-tangent intersection theorem (Angle of intersection between two tangents dividing a circle into arc length A an
  2 KB (370 words) - 13:35, 26 January 2021
• 2011 AIME II Problems/Problem 3
  ...sequence]] with integer values. Find the degree measure of the smallest [[angle]]. The average angle in an 18-gon is <math>160^\circ</math>. In an arithmetic sequence the avera
  3 KB (547 words) - 01:13, 31 January 2024
• 2011 AIME II Problems
  ...etic sequence with integer values. Find the degree measure of the smallest angle. ...C</math>, <math>AB=20</math> and <math>AC=11</math>. The angle bisector of angle <math>A</math> intersects <math>BC</math> at point <math>D</math>, and poin
  8 KB (1,301 words) - 08:43, 11 October 2020
• Tridecagon
  It has an internal angle degree of ~152.308 degrees.
  174 bytes (22 words) - 03:15, 24 September 2018
• 1999 AMC 8 Problems
  What is the degree measure of the smaller angle formed by the hands of a clock at 10 o clock? The degree measure of angle <math>A</math> is
  17 KB (2,394 words) - 19:51, 8 May 2023
• Mock Geometry AIME 2011 Problems
  ...uare, and let <math>AB_1C_1D_1</math> be its image after a <math>30</math> degree rotation about point <math>A.</math> The area of the region consisting of a ...lie on a plane, and their pairwise intersections meet at angles of integer degree. How many possible values are there for the smallest of these angles?
  8 KB (1,349 words) - 19:10, 14 June 2022
• 2011 AMC 10B Problems
  ...c}</math> larger than the other. What is the degree measure of the largest angle in the triangle? ...</math> are in the ratio <math>4 : 5</math>. What is the degree measure of angle <math>BCD</math>?
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• 2011 AMC 10B Problems/Problem 7
  ...c}</math> larger than the other. What is the degree measure of the largest angle in the triangle? The sum of two angles in a triangle is <math>\frac{6}{5}</math> of a right angle <math>\longrightarrow \frac{6}{5} \times 90 = 108</math>
  1 KB (176 words) - 13:11, 4 July 2013
• 2011 AMC 10B Problems/Problem 9
  ...to twice the ratio of sides (in similar figures) because area is a second-degree property of similar figures. So like solution 3, the ratio of sides is <mat ...AA</math> similarity (<math>\angle B=\angle B</math> and <math>\angle EDB=\angle ACB</math>). Thus, let the side length of <math>ED</math> equal <math>3x</m
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• 2011 AMC 10B Problems/Problem 17
  ...</math> are in the ratio <math>4 : 5</math>. What is the degree measure of angle <math>BCD</math>? ...gle sum theorem, <math>4x+5x+90=180 \longrightarrow x=10</math> and <math>\angle ABE = 50</math>.
  5 KB (699 words) - 04:53, 21 January 2023
• 2007 AMC 10B Problems
  ...nd <math>\angle AOB=140^\circ</math>. What is the degree measure of <math>\angle ABC?</math> ...\angle A= \angle B = 90^\circ.</math> What is the degree measure of <math>\angle E?</math>
  15 KB (2,297 words) - 12:57, 19 February 2020
• 2007 AMC 10B Problems/Problem 4
  ...angle AOB=140^\circ,</math> as shown. What is the degree measure of <math>\angle ABC?</math> Because all the [[central angle]]s of a circle add up to <math>360^\circ,</math>
  928 bytes (140 words) - 12:19, 4 July 2013
• 2007 AMC 10B Problems/Problem 7
  ...\angle A= \angle B = 90^\circ.</math> What is the degree measure of <math>\angle E?</math> ...iangle, <math>\angle AEC = 90</math> and <math>\angle CED = 60.</math> Use angle addition:
  1 KB (181 words) - 12:54, 4 June 2021
• 2007 AMC 10B Problems/Problem 15
  ...angle B=3 \angle C=4 \angle D.</math> What is the degree measure of <math>\angle A,</math> rounded to the nearest whole number? 360 &= \angle A + \angle B + \angle C + \angle D\\
  997 bytes (137 words) - 12:22, 4 July 2013
• 2011 AMC 12B Problems/Problem 10
  ...t <math>\angle AMD=\angle CMD</math>. What is the degree measure of <math>\angle AMD</math>? ...</math>. Therefore, <math>\angle BMC = 30^\circ</math>. Therefore <math>\angle AMD=\boxed{\mathrm{(E)}\ 75^\circ}</math>
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• 1999 AMC 8 Problems/Problem 2
  What is the degree measure of the smaller angle formed by the hands of a clock at 10 o'clock?
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• 1999 AMC 8 Problems/Problem 21
  The degree measure of angle <math>A</math> is Angle-chasing using the small triangles:
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• 2005 CEMC Gauss (Grade 7) Problems
  //put a label at the end of the 75 degree unit vector (and position it out 75 degrees) //put a label at the end of the 75 degree unit vector (rotate label 75 degrees to create a tick)
  16 KB (2,317 words) - 03:54, 24 October 2014
• 2010 PUMaC Problems
  ...h>AC = 5</math>, and <math>BC = 6</math>. An angle bisector is drawn from angle <math>A</math>, and meets <math>BC</math> at <math>M</math>. What is the n ...twice that of rectangle <math>AGPE</math>. Given that the value of <math>\angle FAH</math> in degrees is <math>x</math>, find the nearest integer to <math>
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• 2009 PUMaC Problems
  Given that <math>P(x)</math> is the least degree polynomial with rational coefficients such that ...h>A</math> falls on line <math>EF</math>. How large, in degrees, is <math>\angle ABM</math>?
  25 KB (4,154 words) - 16:27, 2 September 2011
• 1985 AHSME Problems
  ...missing piece (the mouth) has central angle <math>\usepackage{gensymb} 60\degree</math>. What is the perimeter of the monster in cm? ...kage{gensymb} A = 20 \degree</math> and <math>\usepackage{gensymb} B = 25 \degree</math>, then the value of <math>\left(1+\tan A\right)\left(1+\tan B\right)<
  17 KB (2,488 words) - 03:26, 20 March 2024
• 1985 AHSME Problems/Problem 2
  ...missing piece (the mouth) has central angle <math>\usepackage{gensymb} 60\degree</math>. What is the perimeter of the monster in cm?
  1 KB (181 words) - 18:35, 19 March 2024
• 1985 AHSME Problems/Problem 16
  ...kage{gensymb} A = 20 \degree</math> and <math>\usepackage{gensymb} B = 25 \degree</math>, then the value of <math>\left(1+\tan A\right)\left(1+\tan B\right)< ...package{gensymb} 25 \degree + 20 \degree = 45 \degree</math>, we apply the angle sum formula <cmath>\tan(A+B) = \frac{\tan A+\tan B}{1-\tan A\tan B},</cmath
  5 KB (904 words) - 22:25, 19 March 2024
• Mock Geometry AIME 2011 Problems/Problem 1
  ...uare, and let <math>AB_1C_1D_1</math> be its image after a <math>30</math> degree rotation about point <math>A.</math> The area of the region consisting of a ...<math> \triangle AED_1\cong\triangle AEB </math>, since <math> \angle ABE=\angle AD_1E=90^\circ </math>, <math> AD_1=AB=1 </math>, and <math> AE=AE </math>,
  2 KB (376 words) - 22:06, 23 December 2022
• Mock Geometry AIME 2011 Problems/Problem 2
  ...lie on a plane, and their pairwise intersections meet at angles of integer degree. How many possible values are there for the smallest of these angles? ...es. One line can be translated across the plane, without rotation, and the angle between the two lines will remain the same.
  1 KB (174 words) - 22:06, 23 December 2022
• 2012 AMC 10A Problems/Problem 4
  ...20^\circ </math>. What is the smallest possible degree measure for <math>\angle CBD</math>? .../math> share ray <math>AB</math>. In order to minimize the value of <math>\angle CBD</math>, <math>D</math> should be located between <math>A</math> and <ma
  882 bytes (131 words) - 13:56, 1 July 2023
• 2012 AMC 10A Problems/Problem 10
  ...metic sequence. What is the degree measure of the smallest possible sector angle? ...>a_1+11d</math>, where <math>d</math> is the common difference. Since each angle measure must be an integer, <math>d</math> must also be an integer. We can
  4 KB (592 words) - 13:54, 1 July 2023
• 2012 AMC 10A Problems
  ...e ABD = 20^\circ </math>. What is the smallest possible degree measure for angle <math>CBD</math>? ...metic sequence. What is the degree measure of the smallest possible sector angle?
  13 KB (1,994 words) - 01:31, 22 December 2023
• 2012 AMC 12A Problems
  ...etic sequence. What is the degree measure of the smallest possible sector angle? ...that side of length <math>9</math>. Let <math>\theta</math> be the acute angle formed by that side and the median. What is <math>\sin{\theta}</math>?
  14 KB (2,197 words) - 13:34, 12 August 2020
• 1989 AHSME Problems/Problem 19
  ...a</math>, and <math>5\theta</math> for some <math>\theta</math>. By Circle Angle Sum, we obtain <math>3\theta+4\theta+5\theta=360</math>. Solving yields <ma ...cutting off length <math>4</math> is <math>60</math> degrees, and the last angle of the triangle is <math>75</math> degrees.
  4 KB (649 words) - 10:04, 20 May 2021
• 2012 AMC 12B Problems/Problem 15
  ...lar disk of radius 12, along 2 radii to form 2 sectors, one with a central angle of 120. He makes two circular cones using each sector to form the lateral s ...defined by the central angles, the smaller arc, a <math>120</math> degree angle, is half the size of the larger sector. so the smaller arc is <math>8\pi</m
  2 KB (259 words) - 13:42, 2 September 2020
• 2015 AIME I Problems
  ...gle AHG</math> by <math>12^\circ</math>. Find the degree measure of <math>\angle BAG</math>. Let <math>f(x)</math> be a third-degree polynomial with real coefficients satisfying
  10 KB (1,615 words) - 21:48, 13 January 2024
• 2012 AIME II Problems/Problem 13
  ...eta</math> be the measure of angle <math>BAD_1</math> (also the measure of angle <math>BAD_2</math>); by the Law of Cosines, Angle <math>E_1AC = E_1AD_1 + D_1AB + BAC = 60 + \theta + 60 = 120 + \theta</math
  13 KB (2,052 words) - 18:02, 5 February 2024
• 2006 IMO Problems
  ...or of the triangle satisfies <math>\angle PBA + \angle PCA = \angle PBC + \angle PCB</math>. Let <math>P(x)</math> be a polynomial of degree <math>n>1</math> with integer coefficients, and let <math>k</math> be a pos
  2 KB (402 words) - 23:18, 12 April 2021
• 2017 AMC 8 Problems
  ...in the ratio <math>3:3:4</math>, what is the degree measure of the largest angle of the triangle? ...ilateral <math>ABCD</math> shown below, <math>\angle BCD</math> is a right angle, <math>AB=12</math>, <math>BC=4</math>, <math>CD=3</math>, and <math>AD=13<
  12 KB (1,771 words) - 21:13, 20 January 2024
• 1988 USAMO Problems/Problem 4
  ...at <math>\angle CO_aB=\beta+\gamma</math>, implying that <math>\angle BAC+\angle BO_aC=\alpha+\beta+\gamma=180^{\circ}</math>. Therefore quadrilateral <math ...math> (to check, prove <math>\angle IAM = \angle AIM = \frac{\angle BAC + \angle ABC}{2}</math>), so that <math>M</math> is the circumcenter of <math>AIC</m
  3 KB (504 words) - 19:25, 14 October 2021
• 1953 AHSME Problems
  ...h> and <math>\angle{C}</math> respectively. <math>AD</math> bisects <math>\angle{A}</math> and meets <math>BC</math> at <math>D</math>. Each angle of a rectangle is trisected. The intersections of the pairs of trisectors a
  21 KB (3,123 words) - 14:24, 20 February 2020
• 1980 AHSME Problems
  The degree of <math>(x^2+1)^4 (x^3+1)^3</math> as a polynomial in <math>x</math> is ...equilateral triangle and ABCD and DEFG are squares. The measure of <math>\angle GDA</math> is
  15 KB (2,302 words) - 10:47, 30 April 2021
• 1984 USAMO Problems
  ...cute angle. Determine the greatest and least values of <math>\angle APC + \angle BPD</math>. <math>P(x)</math> is a polynomial of degree <math>3n</math> such that
  2 KB (336 words) - 12:33, 18 July 2016
• Thales' theorem
  ...circle, <math>A, B, C</math> with <math>AC</math> being a diameter, <math>\angle ABC=90^{\circ}</math>. ...and has measure <math>180^{\circ}</math>. Thus, the intercepted angle has degree measure <math>\frac{180}{2} = 90</math>.
  2 KB (334 words) - 00:42, 5 June 2021
• 2013 AMC 10B Problems/Problem 7
  ...so, because the two adjacent angles are one sixth of the circle apart, the angle opposite them is thirty degrees. This is a <math>30-60-90</math> triangle. ...ths. The radius is <math>1</math>, so the side opposite the thirty degree angle in the triangle is also <math>1</math>. From the properties of <math>30-60
  3 KB (417 words) - 20:21, 12 January 2024
• Geometry Solutions
  ...t's take a look at the 3/4 circle. We know it is 3/4 because there is a 90 degree triangle cut out of it. Hence, each of its angles measure 60 degrees. Namely, <math>\angle ABC</math> measures 60 degrees.
  23 KB (3,182 words) - 12:30, 5 April 2014
• ASIA TEAM Problems
  ...h> lies between <math>P</math> and <math>M</math>, and <math>\angle ABC = \angle APC</math>. Let <math>AP</math> intersect the circle again at <math>D</math Kelvin the frog generates a polynomial <math>P(x)</math> of degree <math>2012</math> such that <math>P(k)=2^{2012-k}3^k</math> for <math>k=1,2
  10 KB (1,710 words) - 23:23, 10 January 2020
• 1962 AHSME Problems/Problem 20
  Now we know that the sum of the degree measures of the angles of a pentagon is <math>180(5-2)=540</math>. ...ns that <math>a+2n</math> is one of the angles we defined earlier, so that angle must have a measure of <math>\boxed{108\textbf{ (A)}}</math>.
  1 KB (247 words) - 20:00, 29 October 2020
• 1968 AHSME Problems
  ...ath>CDE</math> and <math>DCE</math>, and let <math>S'</math> represent the degree-sum of angles <math>BAD</math> and <math>ABC.</math> If <math>r=S/S'</math> ...math> is on segment <math>BC</math>. Line <math>AE</math> extended bisects angle <math>FEC</math>. If <math>DE</math> has length <math>5</math> inches, then
  16 KB (2,571 words) - 14:13, 20 February 2020
• 2014 AMC 12A Problems/Problem 12
  ...ler circle. Therefore, <math>m\angle AO_1B = 30^{\circ}</math> and <math>m\angle AO_2B = 60^{\circ}</math>. Note that <math>\Delta AO_2B</math> is equilater ...in{30} = \dfrac{1}{2}</math>, and we can use the angle sum formula or half angle formula to compute <math>\sin{15} = \dfrac{\sqrt{6} - \sqrt{2}}{4}</math>.
  7 KB (1,191 words) - 23:37, 23 June 2022
• 1993 AHSME Problems/Problem 17
  ...ace positions, each section has a <math>\tfrac{360}{12} = 30</math> degree angle from the center. So each section t is a <math>30-60-90</math> triangle wit
  2 KB (301 words) - 03:42, 12 December 2018
• 1969 AHSME Problems
  ...<math>x+75^{\circ} , 2x+25^{\circ},3x-22^{\circ}</math>. Then one interior angle of the triangle is: ...dividing <math>x^{100}</math> by <math>x^2-3x+2</math> is a polynomial of degree less than <math>2</math>. Then <math>R</math> may be written as:
  16 KB (2,662 words) - 14:12, 20 February 2020
• 1987 AHSME Problems
  How many polynomial functions <math>f</math> of degree <math>\ge 1</math> satisfy ...math>\triangle ABC</math> has <math>\angle A =45^{\circ}</math> and <math>\angle B =30^{\circ}</math>. A line <math>DE</math>, with <math>D</math> on <math>
  16 KB (2,291 words) - 13:45, 19 February 2020
• 1983 AHSME Problems
  ...>ED</math>. Each side has length <math>1</math>. Also, <math>\angle FAB = \angle BCD = 60^\circ</math>. Triangle <math>ABC</math> has a right angle at <math>C</math>. If <math>\sin A = \frac{2}{3}</math>, then <math>\tan B<
  15 KB (2,309 words) - 23:43, 2 December 2021
• 2014 AMC 8 Problems
  ...math> measures <math>70^\circ</math>. What is the degree measure of <math>\angle ADB</math>?
  13 KB (1,957 words) - 12:08, 13 January 2024
• 2014 AMC 8 Problems/Problem 9
  ...math> measures <math>70^\circ</math>. What is the degree measure of <math>\angle ADB</math>? ...th>\angle ADB</math> and <math>\angle BDC</math> are supplementary, <math>\angle ADB = 180 - 40 = \boxed{\textbf{(D)}~140}</math>.
  1 KB (194 words) - 11:14, 2 July 2023
• 2015 AMC 10A Problems/Problem 17
  ...<math>\frac{\sqrt{3}}{3}</math> (can be find out easily by using 60 degree angle of the triangle) so the third must be <math>-\frac{\sqrt{3}}{3}</math>. Sin
  3 KB (562 words) - 00:45, 24 October 2023
• 2015 AMC 10A Problems/Problem 14
  ...It would already have completed three 180 degree turns. Therefore, two 180 degree turns would be completed at 4:00. ...raveled on a <math>60\pi</math> arrow path. This is a ratio of 1/3, so the angle it carves is 120 degrees, which leads us to the correct answer of 4 o' clo
  9 KB (1,380 words) - 09:00, 1 December 2023
• 2015 AMC 12B Problems/Problem 19
  In <math>\triangle ABC</math>, <math>\angle C = 90^\circ</math> and <math>AB = 12</math>. Squares <math>ABXY</math> and ...a point <math>P</math> on <math>\overline{YZ}</math>. Now, we must do some angle chasing to prove that <math>\triangle WBX</math> is similar to <math>\trian
  11 KB (1,898 words) - 22:42, 16 June 2023
• 2015 AIME I Problems/Problem 15
  ...ath>. Then <math>ON=4</math>, <math>MN=3</math> (because of the 120 degree angle), and so <math>OM=5</math>.
  9 KB (1,407 words) - 19:37, 17 February 2024
• 2015 AIME I Problems/Problem 6
  ...gle AHG</math> by <math>12^\circ</math>. Find the degree measure of <math>\angle BAG</math>. Let <math>x</math> be the degree measurement of <math>\overarc{ED}=\overarc{DC}=\overarc{CB}=\overarc{BA}</m
  5 KB (782 words) - 16:04, 21 July 2023
• 2015 Final tour - Azerbaijan in lower age category
  ...and (measure of angle DBA)+(measure of angle DCB)=180. Find the measure of angle DBA. (10 points)
  914 bytes (121 words) - 20:12, 23 January 2017
• 2017 AIME I Problems
  A circle circumscribes an isosceles triangle whose two congruent angles have degree measure <math>x</math>. Two points are chosen independently and uniformly a ...h> and <math>b</math> respectively, and <math>\angle OQP</math> and <math>\angle ORP</math> are both right angles. The probability that <math>QR \leq 100</m
  7 KB (1,163 words) - 16:43, 2 June 2022
• 2017 UNM-PNM Statewide High School Mathematics Contest II Problems
  Find a second-degree polynomial with integer coefficients, <math>p(x) = ax^2 + bx + c</math>, su Let <math>ABC</math> be a right triangle with right angle at <math>C</math>. Suppose <math>AC = 12</math> and <math>BC = 5</math> and
  5 KB (846 words) - 03:36, 19 January 2019
• 2016 AMC 12B Problems/Problem 4
  ...s is twice as large as the complement of the other. What is the sum of the degree measures of the two angles? ...le. The larger angle will be represented as <math>x</math> and the smaller angle will be represented as <math>y</math>, in degrees. This implies that
  2 KB (254 words) - 20:49, 6 November 2021
• 2016 AMC 12B Problems
  ...s is twice as large as the complement of the other. What is the sum of the degree measures of the two angles? ...ame time Bob looks due west from his house and sees the same airplane. The angle of elevation of the airplane is <math>30^\circ</math> from Alice's position
  14 KB (2,037 words) - 19:09, 29 July 2023
• 2016 AIME I Problems/Problem 6
  ...ath> be the center of the inscribed circle, and let the bisector of <math>\angle ACB</math> intersect <math>AB</math> at <math>L</math>. The line through <m ...}</math>. Also, <math>AI</math> bisects <math>\angle LAC</math>, so by the Angle Bisector Theorem <math>\frac{CI}{IL} =\frac{AC}{AL}= \frac{5}{3}</math>. Th
  14 KB (2,397 words) - 20:04, 27 August 2023
• 1982 AHSME Problems/Problem 18
  ...irc</math> and <math>\angle FHB=60^\circ</math>. Find the cosine of <math>\angle BHD</math>. Looking at square GHDC, we see that <math>\angle DHC=45</math>, which implies that <math>DC=CH=1</math> and <math>DH=\sqrt{2
  2 KB (294 words) - 08:42, 15 April 2016
• 2016 AMC 8 Problems
  ...ints, one of which is <math>E</math>. What is the degree measure of <math>\angle CED</math>?
  12 KB (1,665 words) - 12:37, 1 April 2024
• 2016 AMC 8 Problems/Problem 23
  ...ints, one of which is <math>E</math>. What is the degree measure of <math>\angle CED</math>? ...angle{BED}=m\angle{BDE}=30^{\circ}</math>. Similarly, <math>m\angle{AEC}=m\angle{ACE}=30^{\circ}</math>.
  2 KB (337 words) - 19:37, 28 December 2023
• 2017 AMC 12A Problems/Problem 11
  ...e forgot to include one angle. What is the degree measure of the forgotten angle? ...math>\equiv 2160-2017=143\mod 180</math>. Since the polygon is convex, the angle is <math>\leq 180</math>, so the answer is <math>\boxed{(D)\ =\ 143}</math>
  1 KB (212 words) - 14:56, 10 June 2023
• 2017 AMC 12A Problems
  ...e forgot to include one angle. What is the degree measure of the forgotten angle? ...d <math>5</math> so that one vertex of the square coincides with the right-angle vertex of the triangle. A square with side length <math>y</math> is inscrib
  15 KB (2,418 words) - 16:58, 7 November 2022
• 2017 AIME I Problems/Problem 6
  ...circumscribed around an isosceles triangle whose two congruent angles have degree measure <math>x</math>. Two points are chosen independently and uniformly a ...he central angle from two points on a circle is always twice the inscribed angle from those two points), and the probability that a point is chosen on the a
  4 KB (726 words) - 16:18, 5 January 2024
• 2017 AIME I Problems/Problem 8
  ...h> and <math>b</math> respectively, and <math>\angle OQP</math> and <math>\angle ORP</math> are both right angles. The probability that <math>QR \leq 100</m Noting that <math>\angle OQP</math> and <math>\angle ORP</math> are right angles, we realize that we can draw a semicircle with
  9 KB (1,539 words) - 15:47, 17 February 2024
• 2017 AIME I Problems/Problem 10
  <math>\angle Z_3Z_1Z_2</math> looks similar to <math>\angle Z_3ZZ_2</math>, so let’s try to prove that they are congruent. We can sho ...equal to the angle between the two vectors since we need to rotate by that angle to lay <math>V_2</math> flat on <math>V_1</math>.
  13 KB (2,252 words) - 15:46, 6 January 2024
• 2017 AMC 8 Problems/Problem 6
  ...in the ratio <math>3:3:4</math>, what is the degree measure of the largest angle of the triangle? ...ngle, <math>3x+3x+4x=180^{\circ}\implies x=18^{\circ}</math>. The greatest angle is <math>4x</math> and after substitution we get <math>\boxed{\textbf{(D) }
  2 KB (326 words) - 17:37, 31 March 2023
• 2018 AMC 12A Problems/Problem 23
  ...overline{UG},</math> respectively. What is the degree measure of the acute angle formed by lines <math>MN</math> and <math>PA?</math> We will refer to the <b>Diagram</b> section. In this solution, all angle measures are in degrees.
  12 KB (1,878 words) - 22:11, 23 October 2021
• 2018 AMC 12A Problems
  ...planted square <math>S</math> so that from the air it looks like the right angle symbol. The rest of the field is planted. The shortest distance from <math> ...e the midpoint of <math>\overline{AC}</math>. The angle bisector of <math>\angle BAC</math> intersects <math>\overline{DE}</math> and <math>\overline{BC}</m
  15 KB (2,380 words) - 18:52, 7 April 2022
• 2018 AMC 12B Problems
  ...of <math>\triangle ABC</math> has length <math>10</math>. The bisector of angle <math>A</math> meets <math>\overline{BC}</math> at <math>D</math>, and <mat Consider polynomials <math>P(x)</math> of degree at most <math>3</math>, each of whose coefficients is an element of <math>\
  14 KB (2,118 words) - 15:36, 28 October 2021
• 2018 AMC 12B Problems/Problem 23
  ...ct sphere with center <math>C.</math> What is the degree measure of <math>\angle ACB?</math> ...itude is the same as <math>245^\circ \text{ E}</math> longitude, so <math>\angle ACD=135^\circ.</math>
  6 KB (984 words) - 22:39, 3 December 2021
• Bibhorr Formula
  ...angle]] which is equated to linear [[variable|variables]] is the Bibhorr [[angle]]. ...ं and shortest side छ, the [[angle]] opposite the medium side (Bibhorr angle) बि is given as:
  2 KB (245 words) - 06:45, 9 September 2018
• 2013 Indonesia MO Problems
  ...iangle and <math>\omega</math> be its circumcircle. The bisector of <math>\angle BAC</math> intersects <math>\omega</math> at [another point] <math>M</math> Let <math>P</math> be a quadratic (polynomial of degree two) with a positive leading coefficient and negative discriminant. Prove t
  4 KB (648 words) - 00:29, 13 August 2018
• 2019 AMC 10A Problems
  ...the quadrilateral <math>BCDE</math>. What is the degree measure of <math>\angle BFC ?</math>
  13 KB (2,024 words) - 16:07, 22 April 2024
• 2019 AIME II Problems/Problem 10
  ...math>n</math> is a multiple of <math>3</math>, and negative otherwise. The degree measure of <math>\theta</math> is <math>\tfrac{p}{q}</math>, where <math>p< (This is because if it isn't the same value, the terminal angle will gradually shift from the first quadrant into different quadrants, maki
  7 KB (1,109 words) - 00:40, 28 January 2024
• Circle Theorems
  ...is <math>2</math> times the measure of the interior degree measure of the angle created by the chords. This theorem is rather visual, so refer to the pictu ...: If <math>2</math> radii form a straight line or a <math>180^\circ</math> angle, this line is the diameter.
  3 KB (428 words) - 17:58, 29 April 2019
• 2019 AMC 10A Problems/Problem 13
  ...the quadrilateral <math>BCDE</math>. What is the degree measure of <math>\angle BFC ?</math> ...B=20^{\circ}</math> and <math>\angle DBC=50^{\circ}</math> by the triangle angle sum on <math>\triangle ECB</math> and <math>\triangle DBC</math>.
  4 KB (654 words) - 11:15, 4 November 2022
• 2019 AIME I Problems
  ...where <math>g(x)</math> is a polynomial with complex coefficients and with degree at most <math>2016</math>. The value of ...th> are the vertices of a right triangle in the complex plane with a right angle at <math>f(z)</math>. There are positive integers <math>m</math> and <math>
  8 KB (1,331 words) - 06:57, 4 January 2021
• 2019 AIME II Problems
  ...math>n</math> is a multiple of <math>3</math>, and negative otherwise. The degree measure of <math>\theta</math> is <math>\tfrac{p}{q}</math>, where <math>p<
  7 KB (1,254 words) - 14:45, 21 August 2023
• What is the definition of Pure Mathematics?
  ''A quadratic equation is an equation of the second degree, meaning it contains at least one term that is squared. The standard form i ...he spherical angle formed by two great circles on a sphere is the dihedral angle between the planes determined by the great circles. Most people mesure angl
  35 KB (5,882 words) - 18:08, 28 June 2021
• 2021 JMPSC Sprint Problems/Problem 3
  ...all angles marked with one black curve are equal, find the measure of the angle with a question mark. ...circle. We are given two of the black curves, and a <math>90</math> degree angle, in which all three of them add up to <math>210</math> degrees.
  1 KB (178 words) - 10:41, 12 July 2021
• 2019 AMC 10C Problems/Problem 24
  Subcase 4: the four points are on a slanted rectangle that forms a 45 degree angle with the "ground": <math>\dbinom{3}{2} \times \dbinom{3}{2} \times 8 = 72</ ...se 7: the four points are the vertices of a parallelogram with a 45 degree angle to all axes: 16 ways
  2 KB (378 words) - 13:21, 7 January 2020
• 2020 AMC 12A Problems/Problem 12
  ...}</math> and one angle is <math>90^{\circ}</math>. This means the last leg angle must also be <math>45^{\circ}</math>. Since the slope of the line is <math>\frac{3}{5}</math>, and the angle we are rotating around is x, then <math>\tan x = \frac{3}{5}</math>
  7 KB (1,145 words) - 20:27, 5 November 2023
• 2020 AMC 12B Problems/Problem 7
  ...ath>xy</math>-coordinate plane intersect to form a <math>45^{\circ}</math> angle. One line has slope equal to <math>6</math> times the slope of the other li ...green triangle with the red angle (45 deg) and blue angle, where sine blue angle is <math>1/(\sqrt{1+36m^2})</math> from right triangle w vertices <math>(0,
  5 KB (895 words) - 15:03, 8 June 2023
• 2020 AIME I Problems/Problem 1
  ...B}</math> such that <math>AE=ED=DB=BC.</math> The degree measure of <math>\angle ABC</math> is <math>\tfrac{m}{n},</math> where <math>m</math> and <math>n</ If we set <math>\angle{BAC}</math> to <math>x</math>, we can find all other angles through these t
  6 KB (968 words) - 15:01, 24 January 2024
• 2020 AIME I Problems
  ...B}</math> such that <math>AE=ED=DB=BC.</math> The degree measure of <math>\angle ABC</math> is <math>\tfrac{m}{n},</math> where <math>m</math> and <math>n</ ...{AD}</math> intersects the bisectors of <math>\angle ABC</math> and <math>\angle ACB</math> in points <math>E</math> and <math>F,</math> respectively. Given
  7 KB (1,257 words) - 17:51, 5 February 2022
• 2021 HMC 10
  ...side length <math>AB=7, BC=24, AC=25</math>. The angle bisector of <math>\angle{ABC}</math> intersects <math>AC</math> at <math>D</math>. Find <math>AD</ma Let <math>f(x)</math> be a polynomial with degree of <math>5</math> such that <math>f(-2)=f(-1)=f(1)=f(2)=f(3)=13</math> and
  11 KB (1,745 words) - 16:44, 6 October 2021
• 2020 AIME II Problems/Problem 4
  ...ine{AB}</math> to the horizontal segment <math>\overline{A'B'}</math>, the angle of rotation is <math>90^\circ</math> degrees clockwise. For any point <math ...he angle of rotation must be <math>90^{\circ}</math>, and therefore <math>\angle P = 90^{\circ}</math>. Therefore, <math>CPC'</math> is a 45-45-90 right tri
  10 KB (1,542 words) - 13:29, 19 January 2024
• 2021 AIME I Problems/Problem 15
  ...^2+dy^2=g(x,y)</math> for polynomials <math>f</math> and <math>g</math> of degree at most <math>1</math>, whenever <math>(a,b),(c,d)</math> are linearly inde ...ate axis <math>2\theta,</math> its length <math>\rho_1(\theta),</math> the angle between the vector connecting the focus of the second parabola and its poin
  10 KB (1,742 words) - 02:31, 13 November 2023
• 1955 AHSME Problems/Problem 34
  ...the bigger circle, we can simply subtract twice the <math>60</math> degree angle from the circle to get <math>360 - 120 = 240</math> degrees. When plugged i
  2 KB (360 words) - 22:07, 16 August 2020
• 2022 AIME I Problems/Problem 8
  ...row{O\omega_B}</math> is <math>6</math> and is at a <math>30</math> degree angle from the horizontal. Therefore, the coordinates of this point are <math>(3\ ...is easier to find <math>OX</math> as they lie on the same vertical, <math>\angle XOY</math> is <math>120</math> degrees so we can make use of <math>30-60-90
  12 KB (1,955 words) - 21:11, 31 January 2024
• 2022 AMC 12B Problems
  ...> and the <math>y</math>-axis at point <math>B</math>. What is <math>\tan(\angle ABC)</math>? ...he remainder is <math>2x + 1</math>. There is a unique polynomial of least degree with these two properties. What is the sum of the squares of the coefficien
  15 KB (2,233 words) - 13:02, 10 November 2023
• 2021 Fall AMC 10A Problems
  ...>, and <math>ABCD</math> is a square. What is the degree measure of <math>\angle AFE</math>?
  14 KB (2,162 words) - 21:33, 2 November 2023
• 2021 AIME II Problems
  ...ath> in that order. Denote by <math>\theta</math> the measure of the acute angle formed by the diagonals of the quadrilateral. Then <math>\tan \theta</math> ...> are in the ratio <math>13 : 2 : 17, </math> the degree measure of <math>\angle BAC</math> can be written as <math>\frac{m}{n},</math> where <math>m</math>
  8 KB (1,429 words) - 14:31, 26 February 2024
• 2021 AIME II Problems/Problem 14
  ...> are in the ratio <math>13 : 2 : 17, </math> the degree measure of <math>\angle BAC</math> can be written as <math>\frac{m}{n},</math> where <math>m</math> In this solution, all angle measures are in degrees.
  10 KB (1,526 words) - 16:50, 25 December 2022
• 2022 AMC 10B Problems
  ...icular to <math>\overline{BE}</math>. What is the degree measure of <math>\angle{BFC}</math>? ...he remainder is <math>2x + 1</math>. There is a unique polynomial of least degree with these two properties. What is the sum of the squares of the coefficien
  15 KB (2,224 words) - 13:10, 20 February 2024
• Jadhav Sin Function Theorem
  ...ame [https://en.wikipedia.org/wiki/Sine sine function ratio] with know one angle. This Formula based theorem is a superior alternative to complex sine funct ...ngle <math>x </math> is in [https://en.wikipedia.org/wiki/Degree_(angle) degree measure].
  2 KB (374 words) - 03:28, 29 July 2021
• 2021 Fall AMC 12B Problems/Problem 8
  ...angle's height to the base. What is the measure, in degrees, of the vertex angle of this triangle? Set the vertex angle to be <math>a</math>, we derive the equation:
  2 KB (293 words) - 22:07, 10 July 2023
• 2021 Fall AMC 10B Problems/Problem 17
  ...problem in terms of Euclidean geometry, using the observation that <math>\angle POP'' = 90^{\circ}</math>, and that both <math>\ell</math> and <math>m</mat ...>. Since the sum of both sides combined must be <math>90^{\circ}</math> by angle addition,
  8 KB (1,331 words) - 22:44, 16 December 2023
• 2021 Fall AMC 10B Problems/Problem 18
  ...>45^\circ{}</math>, and this point rotates <math>30^\circ{}</math>, so the angle that <math>OC</math> makes with the horizontal is <math>75^\circ{}</math>. ...th>OD</math> forms a right angle with <math>BC</math>. Furthermore, <math>\angle DOC = 90^\circ{}-75^\circ{} = 15^\circ{}</math>.
  18 KB (3,011 words) - 22:05, 26 September 2023
• 2021 Fall AMC 12A Problems
  ...>, and <math>ABCD</math> is a square. What is the degree measure of <math>\angle AFE</math>? The angle bisector of the acute angle formed at the origin by the graphs of the lines <math>y = x</math> and <mat
  15 KB (2,452 words) - 19:37, 7 June 2023
• 2021 Fall AMC 12A Problems/Problem 6
  ...>, and <math>ABCD</math> is a square. What is the degree measure of <math>\angle AFE</math>? ...e ADE}{2}=10^\circ.</math> Finally, we get <math>\angle AFE = 180^\circ - \angle DFE = \boxed{\textbf{(D) }170}</math> degrees.
  3 KB (453 words) - 21:21, 12 July 2023
• Right triangles
  ...a triangle that has one 90 degree angle. There are special cases of right angle triangles and they are very important. for instance the [[30-60-90 triangle
  208 bytes (35 words) - 14:24, 7 January 2022
• 30-60-90 triangle
  It is a triangle with a 30 degree angle, a 60 degree angle,a 90 degree angle. it is special because it side lengths are always in the same ratio. the le
  241 bytes (47 words) - 14:27, 7 January 2022
• Chebyshev polynomials of the first kind
  From the [[Trigonometric identities#Angle addition identities|cosine sum and difference identities]] we have <cmath>\ ...at is, for <math>x \in [-1,1]</math>. However, <math>T_{mn}(x)</math> is a degree-<math>mn</math> polynomial, and so is <math>T_m(T_n(x))</math>, so the fact
  10 KB (1,919 words) - 15:24, 26 June 2023
• AoPSWiki:Sandbox
  == Angle Measures == If you want to draw an angle measure on an angle use
  28 KB (4,808 words) - 19:43, 3 June 2022
• 2022 AMC 10B Problems/Problem 20
  Let <math>ABCD</math> be a rhombus with <math>\angle ADC = 46^\circ</math>. Let <math>E</math> be the midpoint of <math>\overlin ...icular to <math>\overline{BE}</math>. What is the degree measure of <math>\angle BFC</math>?
  11 KB (1,742 words) - 20:06, 8 September 2023
• 2023 AMC 10A Problems
  ...at is standing in the same field as far from Abdul as possible so that the angle formed by his lines of sight to Abdul and Chiang measures <math>60^\circ</m Let <math>P(x)</math> be the unique polynomial of minimal degree with the following properties:
  16 KB (2,411 words) - 00:18, 7 May 2024
• 2022 SSMO Relay Round 3 Problems
  Let <math>T=</math> TNYWR. In cyclic quadrilateral <math>ABCD,</math> <math>\angle{BAD}=60^{\circ},</math> and <math>BC=CD=T.</math> If <math>AB</math> is a p Let <math>T=</math> TNYWR. Let <math>f(x)</math> be a polynomial of degree 10, such that <math>f(i)=i</math> for all <math>i=1,2,\dots,10</math> and <
  1 KB (182 words) - 21:57, 31 May 2023
• 2019 Mock AMC 10B Problems/Problem 12
  ...t means that <math>\angle COI</math> is <math>60</math> degrees and <math>\angle IOB</math> is <math>120</math> degrees. ...the remaining region, which is a triangle. We know that the area of the 60-degree sector is <math>\frac{1}{6} \cdot (5\sqrt{3})^2 \pi = \frac{25}{2} \pi</mat
  2 KB (276 words) - 20:22, 1 July 2023
• 2023 AMC 12A Problems
  What is the degree measure of the acute angle formed by lines with slopes <math>2</math> and <math>\tfrac{1}{3}</math>? ...(APQRS)</math>. What angle <math>\theta</math><math>\angle PAB=\angle QPC=\angle RQB=\cdots</math> will produce in a length that is <math>120</math> meters?
  15 KB (2,168 words) - 05:11, 4 February 2024
• Know the Facts: Gmaas
  - 2023 Gmaas won the Woman's World Cup by calculating the exact force and angle needed to score (With Gmaasology of course). Spain was so thankful, they ga 385. The degree used to be called the Gmaas Arc, but the name was too long for most people,
  88 KB (14,928 words) - 13:54, 29 April 2024
• 2023 AMC 10B Problems
  ...re <math>EFGH</math>, as shown below. What is the degree measure of <math>\angle{EAB}</math>?
  13 KB (1,959 words) - 10:29, 4 April 2024
• 2023 AMC 12A Problems/Problem 11
  What is the degree measure of the acute angle formed by lines with slopes <math>2</math> and <math>\frac{1}{3}</math>? ...th>k_1=2=\tan \alpha</math>, <math>k_2=\dfrac{1}{3}=\tan \beta</math>. The angle formed by the two lines is <math>\alpha-\beta</math>. <math>\tan(\alpha-\be
  6 KB (907 words) - 23:48, 18 February 2024
• 2023 AMC 10B Problems/Problem 7
  What is the degree measure of <math>\angle EAB</math>? ...>AI</math> bisects angle <math>\angle{DAB}</math>, so <math>\angle{BAI} = \angle{DAI} = 45</math>. Subtracting <math>45</math> from <math>80</math>, we get
  4 KB (577 words) - 02:09, 27 November 2023
• 2024 AMC 8 Problems/Problem 18
  ...aded and unshaded regions are equal in area. What is the measure of <math>\angle{BOC}</math> in degrees? real angleStart = -54; // starting angle of the sector
  5 KB (869 words) - 19:58, 8 May 2024