Angle

Revision as of 20:21, 21 June 2019 by Rockmanex3 (talk | contribs) (Angle Chasing!)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

Overview

An angle is the union of two rays with a common endpoint. The common endpoint of the rays is called the vertex of the angle, and the rays themselves are called the sides of the angle.

[asy] draw((0,0)--(3,5),EndArrow); draw((0,0)--(5,0),EndArrow); [/asy]

There are many notations for angles. The most common form is $\angle ABC$, read "angle ABC", where $A,C$ are points on the sides of the angle and $B$ is the vertex of the angle. Note that the same angle can be denoted many different ways by choosing different points along the side of the angle. If there is no ambiguity, this notation can be shortened to simply $\angle B$.

Angle Measure

The measure of $\angle ABC$ is denoted $m\angle ABC$, read "measure of angle ABC". There are different units for measuring angles. The three most common are degrees, radians and gradians.

If two angles are congruent, they have the same angle measure.

A ray drawn from the vertex of the angle, such that the angle formed by this ray and one of the sides is congruent to the angle formed by this ray and the other side, is called the angle bisector.

Classifying Angles

  • A straight angle is an angle formed by a pair of opposite rays, or a line. A straight angle has a measure of $180^\circ=\pi\; \mathrm{ rad}$.
  • A right angle is an angle that is supplementary to itself. A right angle has a measure of $90^\circ=\frac{\pi}{2}\;\mathrm{ rad}$.
  • An acute angle has a measure greater than zero but less than that of a right angle, i.e. $\angle ABC$ is acute if and only if $0^\circ<m\angle ABC<90^\circ$.
  • An obtuse angle has a measure greater than that of a right angle but less than that of a straight angle, i.e. $\angle ABC$ is obtuse if and only if $90^\circ<m\angle ABC<180^\circ$.
  • A reflex angle is an angle with measure greater than a straight angle, but less than 360 degrees, or $2\pi$ radians.

Angle Chasing

Angle chasing is a technique where solvers apply angle properties determine the measures of unknown angles. It is commonly used in geometry problems.

Properties Used in Angle Chasing

  • Two angles that are complementary add to $90^\circ$. Two angles that are supplementary add to $180^\circ$. Supplementary angles can be found when two lines intersect each other.
  • Vertical angles are congruent to each other.
  • Parallel lines can create equal or supplementary angles.
  • An angle bisector splits an angle into two congruent angles. For instance, if $\angle ABC$ is bisected by $BD$, then $\angle ABD = \angle CBD$.
  • If angles are founded in a polygon, one can use angle formulas to find the unknown angle.
    • If two polygons are congruent, corresponding angles are congruent.
  • If angles are found in a circle, one can use angle properties and arc measure.
    • Finding cyclic quadrilaterals can be a useful strategy in angle chasing since angles opposite with each other are supplementary.
Invalid username
Login to AoPS