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• Obtuse triangle
  ...in which one [[angle]] is an [[obtuse angle]]. Any triangle which is not obtuse is either a [[right triangle]] or an [[acute triangle]]. ...e because the angles of a triangle must sum to <math>180^\circ</math>. Two obtuse angles would sum to more than that.
  1 KB (193 words) - 14:20, 12 June 2022
• Obtuse angle
  An '''obtuse angle''' is an [[angle]] with a [[measure]] greater than 90 [[degree (geome Any triangle with an obtuse angle is considered an [[Obtuse Triangle]]
  508 bytes (62 words) - 17:21, 2 March 2014
• Obtuse Triangle
  #REDIRECT [[Obtuse triangle]]
  29 bytes (3 words) - 17:22, 2 March 2014
• Obtuse triangles
  ...triangle that has one angle that is greater than 90, which is called and [[Obtuse Angle]].
  100 bytes (18 words) - 14:30, 7 January 2022
• Obtuse Angle
  An Obtuse angle is a angle greater than 90.
  43 bytes (8 words) - 14:31, 7 January 2022

Page text matches

• Geometric inequality
  ...a^2 + b^2 = c^2</math>. The Inequality extends this to [[obtuse triangle| obtuse]] and [[acute triangle]]s. The inequality says: ...s of length <math>a \leq b \leq c</math>, <math>a^2+b^2>c^2</math>. For an obtuse triangle with sides <math>a \leq b \leq c</math>, <math>a^2+b^2<c^2</math>.
  7 KB (1,296 words) - 14:22, 22 October 2023
• Orthic triangle
  ...rthic_triangle.png|thumb|right|300px|Case 2: <math>\triangle ABC</math> is obtuse.]] ...right. The two cases of when <math>\triangle ABC</math> is either acute or obtuse each carry different characteristics and must be handled separately.
  8 KB (1,408 words) - 11:54, 8 December 2021
• Triangle
  ===Obtuse=== ...n equilateral obtuse triangle. It is also impossible to have more than one obtuse angle in a triangle.
  4 KB (628 words) - 17:17, 17 May 2018
• Law of Cosines
  ===Obtuse Triangle=== The argument for an obtuse triangle is the same as the proof for an acute triangle.
  6 KB (1,003 words) - 09:11, 7 June 2023
• Trigonometry
  ...ing right triangles, trigonometry is just as useful when used on acute and obtuse triangles. The [[Law of Sines]] and [[Law of Cosines]] mentioned below gene
  8 KB (1,217 words) - 20:15, 7 September 2023
• Right triangle
  *[[Obtuse triangle]]
  3 KB (499 words) - 23:41, 11 June 2022
• 2021 JMC 10
  ...termine two angles of a triangle. What is the probability this triangle is obtuse?
  12 KB (1,784 words) - 16:49, 1 April 2021
• 2001 AMC 12 Problems
  ...th>E=(0,4)</math>. What is the probability that <math>\angle APB</math> is obtuse?
  13 KB (1,957 words) - 12:53, 24 January 2024
• 2006 AMC 12B Problems/Problem 23
  ...le BPA</math>, <math>\angle APC</math> and <math>\angle BPC</math> must be obtuse, since they sum to <math>360^{\circ}</math>. Then, note the well known fact
  7 KB (1,169 words) - 14:04, 10 June 2022
• Right angle
  ** [[Obtuse angle]]
  673 bytes (91 words) - 23:59, 11 June 2022
• Acute triangle
  ...]. Any triangle which is not acute is either a [[right triangle]] or an [[obtuse triangle]]. ...so can be scalene. There are Acute [[triangle]],[[Right triangles]], and [[Obtuse triangles]].
  1 KB (194 words) - 13:09, 20 February 2024
• Obtuse triangle
  ...in which one [[angle]] is an [[obtuse angle]]. Any triangle which is not obtuse is either a [[right triangle]] or an [[acute triangle]]. ...e because the angles of a triangle must sum to <math>180^\circ</math>. Two obtuse angles would sum to more than that.
  1 KB (193 words) - 14:20, 12 June 2022
• Orthocenter
  ...on the vertex opposite the [[hypotenuse]]; and if it is [[obtuse triangle|obtuse]], then the orthocenter is outside the triangle.
  5 KB (829 words) - 13:11, 20 February 2024
• 1970 Canadian MO Problems
  Given a triangle <math>ABC</math> with angle <math>A</math> obtuse and with altitudes of length <math>h</math> and <math>k</math> as shown in
  4 KB (604 words) - 04:32, 8 October 2014
• Angle
  ...le but less than that of a straight angle, i.e. <math>\angle ABC</math> is obtuse if and only if <math>90^\circ<m\angle ABC<180^\circ</math>.
  3 KB (568 words) - 20:21, 21 June 2019
• Straight angle
  ** [[Obtuse angle]]
  568 bytes (78 words) - 13:50, 12 June 2022
• Obtuse angle
  An '''obtuse angle''' is an [[angle]] with a [[measure]] greater than 90 [[degree (geome Any triangle with an obtuse angle is considered an [[Obtuse Triangle]]
  508 bytes (62 words) - 17:21, 2 March 2014
• Euler line
  <i><b>Obtuse triangle</b></i> [[File:Euler line obtuse triangle.png|400px|right]]
  59 KB (10,203 words) - 04:47, 30 August 2023
• Mock AIME 4 2006-2007 Problems/Problem 15
  ...ake the positive sign because [[angle]] <math>B</math> is [[obtuse angle | obtuse]] so <math>\overline{MD}</math> is the longer of the two [[line segment | s
  3 KB (532 words) - 20:29, 31 August 2020
• 2002 USAMO Problems/Problem 2
  Sidenote: A 13, 40, 45 obtuse triangle has an integer area of 252 and an inradius of <math>\frac{36}{7}</
  2 KB (298 words) - 22:32, 6 April 2016
• 2002 USA TST Problems/Problem 1
  ...positely sorted. But if one angle, say <math> \displaystyle A </math>, is obtuse, then we must have <math> \sin B, \sin C \le \sin (B+C) = \sin A </math>, a
  2 KB (380 words) - 20:16, 3 April 2007
• Reflex angle
  ** [[Obtuse angle]]
  1 KB (153 words) - 19:32, 31 December 2019
• De Longchamps point
  De Longchamps circle <math>\Omega</math> of the obtuse triangle <math>ABC</math> is circle centered at de Longchamps point <math>L Let <math>ABC</math> be obtuse triangle <math>(\angle A > 90^\circ)</math> with circumcircle <math>\Omega,
  10 KB (1,780 words) - 09:23, 17 November 2022
• 1995 AHSME Problems
  ...math> is an integer. For how many values of <math>k</math> is the triangle obtuse?
  17 KB (2,387 words) - 22:44, 26 May 2021
• 1995 AHSME Problems/Problem 23
  ...math> is an integer. For how many values of <math>k</math> is the triangle obtuse? By the [[Law of Cosines]], a triangle is obtuse if the sum of the squares of two of the sides of the triangles is less than
  1 KB (183 words) - 14:05, 5 July 2013
• Euclid's Elements
  *An obtuse angle is an angle greater than a right angle. ...that which has a right angle, an obtuse-angled triangle that which has an obtuse angle, and an acute-angled triangle that which has its three angles acute.
  12 KB (2,094 words) - 15:42, 1 December 2015
• 1991 USAMO Problems/Problem 1
  ...> is twice angle <math>B</math>, angle <math>C</math> is [[obtuse triangle|obtuse]], and the three side lengths <math>a, b, c</math> are integers. Determine We find that to satisfy the conditions for an obtuse triangle, <math>\beta \in (0^\circ, 30^\circ)</math> and therefore <math>\g
  5 KB (755 words) - 08:58, 6 May 2023
• 1991 USAMO Problems
  ...,A\,</math> is twice angle <math>\,B,\,</math> angle <math>\,C\,</math> is obtuse, and the three side lengths <math>\,a,b,c\,</math> are integers. Determine
  3 KB (512 words) - 19:17, 18 July 2016
• 2001 USAMO Problems/Problem 4
  ...ath> are the sides of an obtuse triangle. Assume that in this triangle the obtuse angle opposes the side congruent to <math>PA</math>. Prove that <math>\angl ...<math>PA</math>, <math>PB</math>, and <math>PC</math> are the sides of an obtuse triangle and <math>PA > \max\{PB, PC\}</math>.
  8 KB (1,470 words) - 22:24, 18 June 2022
• 2001 USAMO Problems
  ...ath> are the sides of an obtuse triangle. Assume that in this triangle the obtuse angle opposes the side congruent to <math>PA</math>. Prove that <math>\angl
  3 KB (446 words) - 09:56, 20 July 2016
• 1972 USAMO Problems/Problem 2
  ...the sake of contradiction, that each face is non-acute; that is, right or obtuse. Consider triangles <math>\triangle ABC</math> and <math>\triangle ABD</mat ...btuse and all three angles add up to 180 degrees), so the tetrahedron with obtuse triangle faces is impossible.
  8 KB (1,458 words) - 23:42, 27 February 2022
• 2001 AMC 12 Problems/Problem 17
  ...th>E=(0,4)</math>. What is the probability that <math>\angle APB</math> is obtuse? The angle <math>APB</math> is obtuse if and only if <math>P</math> lies inside the circle with diameter <math>AB
  5 KB (792 words) - 15:23, 30 November 2021
• 2009 AMC 12B Problems/Problem 13
  ...hus <math>BC = BD + BC = 5 + 9 = 14</math> or assume that the triangle is obtuse at angle <math>B</math> then <math>BC = DC - BD = 9 -5 = 4</math>. The sum
  883 bytes (137 words) - 18:42, 23 February 2017
• 2021 AIME II Problems/Problem 5
  ...eal numbers <math>s</math>, let <math>\tau(s)</math> denote the set of all obtuse triangles that have area <math>s</math> and two sides with lengths <math>4< ...6</math> and <math>14</math>, exclusive. We also have to consider the word OBTUSE triangles. That means that the two small sides squared is less than the <ma
  14 KB (2,269 words) - 00:43, 2 January 2023
• 2011 AIME I Problems
  ...andom from among the vertices of a regular <math>n</math>-gon determine an obtuse triangle is <math>\frac{93}{125}</math>. Find the sum of all possible valu
  10 KB (1,634 words) - 22:21, 28 December 2023
• 2011 AIME I Problems/Problem 10
  ...s chosen at random from among the vertices of a regular n-gon determine an obtuse triangle is <math>\frac{93}{125}</math> . Find the sum of all possible valu ...meter of the circle. (This is because if an inscribed angle on a circle is obtuse, the arc it spans must be 180 degrees or greater).
  6 KB (980 words) - 03:14, 5 May 2020
• 1999 AHSME Problems/Problem 7
  Five acute angles and one obtuse angle can only sum to less than <math>90\cdot 5 + 180 = 630</math> degrees, Four acute angles and two obtuse angles can only sum to less than <math>90\cdot 4 + 180\cdot 2 = 720</math>
  1 KB (228 words) - 21:18, 5 November 2019
• 1984 AHSME Problems
  In the [[obtuse triangle]] <math> ABC </math>, <math> AM=MB </math>, <math> MD\perp BC </ma
  13 KB (1,879 words) - 14:00, 19 February 2020
• 1984 AHSME Problems/Problem 26
  In the [[obtuse triangle]] <math> ABC </math> with <math> \angle C>90^\circ </math>, <math>
  2 KB (303 words) - 20:28, 2 October 2023
• 1985 AHSME Problems
  Exactly three of the interior angles of a convex polygon are obtuse. What is the maximum number of sides of such a polygon?
  17 KB (2,488 words) - 03:26, 20 March 2024
• 1985 AHSME Problems/Problem 14
  Exactly three of the interior angles of a convex polygon are obtuse. What is the maximum number of sides of such a polygon? Suppose that such a polygon has <math>n</math> sides. Let the three obtuse angle measures, in degrees, be <math>o_1</math>, <math>o_2</math>, and <mat
  2 KB (256 words) - 21:23, 19 March 2024
• 2012 AMC 12A Problems/Problem 22
  ...ace other than the one described in case 1, i.e. <math>t_2</math> makes an obtuse angle with <math>t_1</math>. It is possible to prove that <math>p \cap Q</m
  5 KB (940 words) - 17:13, 4 April 2020
• 1952 AHSME Problems
  <math> \textbf{(A) \ }\text{the triangle is obtuse} </math>
  23 KB (3,556 words) - 15:35, 30 December 2023
• 1954 AHSME Problems
  ...e angle and two unequal acute angles}\\ \textbf{(C)}\ \text{are always one obtuse angle and two equal acute angles}\\ \textbf{(D)}\ \text{are always acute an
  23 KB (3,535 words) - 16:29, 24 April 2020
• 1991 AHSME Problems/Problem 4
  ...le (C) An obtuse right triangle (D) A scalene right triangle (E) A scalene obtuse triangle A right triangle has an angle of 90 degrees, so any obtuse angle would make the sum of those two angles over 180. This contradicts the
  603 bytes (90 words) - 05:47, 24 February 2018
• 1991 AHSME Problems
  <math>\textbf{(C) }</math> An obtuse right triangle <math>\textbf{(E) }</math> A scalene obtuse triangle
  16 KB (2,451 words) - 04:27, 6 September 2021
• Mock AIME 6 2006-2007 Problems
  ...A,B,C)</math> in <math>R</math> exist such that <math>\angle ABC</math> is obtuse?
  7 KB (1,173 words) - 21:04, 7 December 2018
• 1962 AHSME Problems/Problem 23
  ..., we'll show that we can determine <math>DB</math> if <math>A</math> is an obtuse angle. ...ince we've determined the length of <math>DB</math> even though we have an obtuse angle, <math>DB</math> is not <math>\bf{only}</math> determined by what typ
  1 KB (256 words) - 22:39, 10 April 2023
• 1952 AHSME Problems/Problem 15
  <math> \textbf{(A) \ }\text{the triangle is obtuse} </math> ...te if <math> a^2+b^2>c^2 </math>, right if <math> a^2+b^2=c^2 </math>, and obtuse if <math> a^2+b^2<c^2 </math>. If <math> c </math> were equal to <math> 10
  1 KB (169 words) - 23:17, 2 January 2014
• 2014 AMC 10A Problems/Problem 13
  The area of the obtuse triangle is <math>\frac{1}{2}\sin{120} = \frac{\sqrt{3}}{4}</math> ...th> are right, and <math>m\angle CAB=60^{\circ}</math>. So the area of the obtuse triangle is <math>\frac{1}{2}\cdot1\cdot1\cdot\sin\left(120^{\circ}\right)=
  7 KB (1,040 words) - 02:07, 8 October 2022

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