Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "Angle"

 
(→‎Angle Chasing)
 
(17 intermediate revisions by 9 users not shown)
Line 1: Line 1:
−
==Definition==
+
==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.
+
An '''angle''' is the [[union]] of two [[ray]]s with a common [[endpoint]].  The common endpoint of the rays is called the [[vertex]] of the angle, and the rays themselves are called the [[side]]s of the angle.
  
−
{{image}}
+
<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 <math>\angle ABC</math>, read "angle ABC", where <math>A,C</math> are points on the sides of the angle and <math>B</math> is the vertex of the angle.
+
There are many [[notation]]s for angles. The most common form is <math>\angle ABC</math>, read "angle ABC", where <math>A,C</math> are points on the sides of the angle and <math>B</math> 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 <math>\angle B</math>.
  
 
==Angle Measure==
 
==Angle Measure==
−
The measure of <math>\angle ABC</math> is denoted <math>m\angle ABC</math>, read "measure of angle ABC". There are three units for measuring angles: [[degrees]], [[radian|radians]], and [[grad]].
+
The measure of <math>\angle ABC</math> is denoted <math>m\angle ABC</math>, read "measure of angle ABC". There are different units for measuring angles.  The three most common are [[degree (geometry) | degrees]], [[radian]]s and [[gradian]]s.
  
−
==Special Angles==
+
If two angles are [[congruent (geometry)|congruent]], they have the same angle measure.
−
===Straight Angle===
 
−
A '''straight angle''' is an angle formed by a pair of opposite rays, or a line. A straight angle has a measure of <math>180^\circ=\pi\; rad</math>.
 
−
===Right Angle===
 
−
A '''right angle''' is an angle that is [[supplementary]] to itself. A right angle has a measure of <math>90^\circ=\frac{\pi}{2}\; rad</math>.
 
  
−
An '''acute angle''' has a measure greater than zero but less than that of a right angle, i.e.
+
A ray drawn from the vertex of the angle, such that the angle formed by this ray and one of the sides is [[congruent (geometry)|congruent]] to the angle formed by this ray and the other side, is called the [[angle bisector]].
−
<math>\angle ABC</math> is acute<math>\Leftrightarrow 0^\circ<m\angle ABC<90^\circ</math>.
 
  
−
An '''obtuse angle''' has a measure greater than that of a right angle but less than that of a straight angle, i.e. <math>\angle ABC</math> is obtuse<math>\Leftrightarrow 90^\circ<m\angle ABC<180^\circ</math>.
+
==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 <math>180^\circ=\pi\; \mathrm{ rad}</math>.
 +
* A [[right angle]] is an angle that is [[supplementary]] to itself. A right angle has a measure of <math>90^\circ=\frac{\pi}{2}\;\mathrm{ rad}</math>.
 +
* An [[acute angle]] has a measure greater than zero but less than that of a right angle, i.e. <math>\angle ABC</math> is acute if and only if <math>0^\circ<m\angle ABC<90^\circ</math>.
 +
* An [[obtuse angle]] has a measure greater than that of a right angle 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>.
 +
* A [[reflex angle]] is an angle with measure greater than a straight angle, but less than 360 [[degree]]s, or <math>2\pi</math> [[radian]]s.
 +
 
 +
==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. Usually this can use a variety of ways including [[circle]]s, new [[line]]s, or transforming the figure somehow. Lots of angle chasing problems require you to think intuitively.
 +
 
 +
===Properties Used in Angle Chasing===
 +
 
 +
* Two angles that are complementary add to <math>90^\circ</math>.  Two angles that are supplementary add to <math>180^\circ</math>.  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 <math>\angle ABC</math> is bisected by <math>BD</math>, then <math>\angle ABD = \angle CBD</math>.
 +
** If side lengths are known, the [[angle bisector theorem]] can be used to determine that a line bisects an angle.
 +
* 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.
 +
 
 +
 
 +
 
 +
 
 +
[[Category:Geometry]]
 +
[[Category:Angles]]

Latest revision as of 17:39, 9 May 2024

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. Usually this can use a variety of ways including circles, new lines, or transforming the figure somehow. Lots of angle chasing problems require you to think intuitively.

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.
Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=Angle&oldid=219160"