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

**Angle: **Figure formed when two rays share a common endpoint.

**Vertex: **Common endpoint of an angle.

**Degree: **The measure of an angle; one degree is equivalent to 1⁄360 of a circle.

**Acute Angle: **An angle that measures between 0 to 90°.

**Right Angle: **An angle that measures exactly 90°.

**Obtuse Angle: **An angle that measures between 90 to 180°.

**Straight Angle: **An angle that measures exactly 180°.

**Congruent Angles: **Angles that have the same measure.

**Angle Bisector: **A ray that divides an angle into two congruent angles, each with a measure equal to exactly half of the original angle.

**Linear Pair: **A pair of adjacent angles whose non-common sides are opposite rays.

**Adjacent Angles: **Two angles that have the same vertex and share one side but do not overlap (no common interior points).

**Vertical Angles: **Two non-adjacent angles formed by intersecting lines.

**Theorem: **A statement that can be proven using postulates, definitions, logic, etc.

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.

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

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:* 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: *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

Angles can be classified by their measure:

Measures between 0 to 90°

Measures between 90 to 180°

Measures 90°

Marked with a small square

When two lines intersect at a right angle, they are perpendicular

Measures 180°

Formed by two opposite rays, looks just like a straight line

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.

An a**ngle 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.

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.

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.

A **linear pair **is a pair of** adjacent angles** whose non-common sides form a line.

\(\overrightarrow{PN}\) is the common side. ∠MNP and ∠PNO are adjacent angles.

*Linear Pair Postulate: *If two angles are a linear pair, then they are supplementary.

Vertical angles are two non-adjacent angles formed by intersecting lines.

- ∠1 and ∠3 are vertical angles

- ∠2 and ∠4 are vertical angle

*Vertical Angles Theorem: *Vertical angles are congruent*.*

- ∠1 ≅ ∠3 and ∠2 ≅ ∠4