Big Picture

Angles are formed when two rays share an endpoint. Angles are classified by their measure. Acute angles measure less than 90°, right angles measure exactly 90°, obtuse angles measure between 90 to 180°, and straight angles are 180°. Angles also have relationships with each other. For example, two angles sharing a side are adjacent angles, and two adjacent angles that add up to 180° are linear pairs

Key Terms

Angles

An angle is formed when two rays (\(\overrightarrow{AB} \text{ and }\overrightarrow{AT}\)) share a vertex (point A).

• The two rays are the sides of the angle.
• The interior of an angle is the set of all points between the sides of the angle.
• The exterior is the set of all points outside the angle.

Naming Angles

There are three ways to name an angle symbol ∠.

• By number: ∠1
• By the single letter at the vertex:  ∠A.
• We can use the vertex to name the angle if there is only one angle at point A.
• By three points that form the angle:  ∠BAT or  ∠TAB
  
• If named with three letters, the middle letter will be the vertex
• If two or more angles share same vertex, we MUST use three points to name the angle!

In the diagram below, we cannot name the angle  ∠U because it is not clear which angle we are referring to. The figure has three angles:

• ∠XUZ or ∠ZUX
• ∠XUY or ∠YUX
• ∠YUZ or ∠ZUY

Measuring Angles

Angles are measured in degrees, a unit measuring the amount of rotation from one side to another.

• The symbol for degree is °.
• A full circle has 360°, so a degree is 1 ⁄ 360 of a circle.
• The measure of an angle is denoted by m∠

An angle’s degrees can be measured with a protractor. A protractor is a half-circle measuring device with angle measures marked for each degree. To use a protractor, line up the vertex of the angle with the center of the protractor.

• To use a protractor, line up the vertex of the angle with the center of the protractor.

Protractor Postulate

Protractor Postulate: For every angle, there is a number between 0 and 180 that is the measure of the angle in degrees. The angle’s measure on a protractor is the absolute value of the difference of the numbers shown on the protractor.

• You do not need to measure the angle from 0°:

Angle Addition Postulate

Angle Addition Postulate: The measure of any angle can be found by adding the measures of the smaller angles that comprise it.

       If R is in the interior of  ∠BAT, then m∠BAR + m∠RAT = m∠BAT

Classifying Angles

Angles can be classified by their measure:

Congruent Angles

Congruent angles are angles with the same measure. Arc marks are used to show that the angles are congruent (use symbol ≅   to indicate congruence).

not equal different angle marks

equal angles same arc marks

Right Angle Theorem: If two angles are right angles, then the angles are congruent.

Angle Bisectors

An angle bisector will divide an angle into two congruent angles.

If \(\overline{BD}\) is the angle bisector of  ∠ABC, then:

• ∠ABD ≅ ∠DBC
• m ∠ABD = \(\frac{1}{2}\)m∠ABC

Angle Bisector Postulate: Every angle has exactly one angle bisector.

Angle Pairs

Complimentary Angles

A pair of angles are complementary if the sum of their measures is 90°.

• The angles do not need to be congruent or touching.

Same Angle Complements Theorem: If two angles are complementary to the same angle (or to congruent angles), then the angles are congruent.

Supplementary Angles

A pair of angles are supplementary if their measures sum to 180°.

• The angles do not need to be congruent or touching.

Same Angle Supplements Theorem: If two angles are supplementary to the same angle (or to congruent angles), then the angles are congruent.

Tip: c comes before s, just like 90° comes before 180°, so complementary angles add up to 90° and suplementary angles add up to 180°

Notes