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).
There are three ways to name an angle symbol ∠.
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:
Angles are measured in degrees, a unit measuring the amount of rotation from one side to another.
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.
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.
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 angle bisector will divide an angle into two congruent angles.
If \(\overline{BD}\) is the angle bisector of ∠ABC, then:
Angle Bisector Postulate: Every angle has exactly one angle bisector.
A pair of angles are complementary if the sum of their measures is 90°.
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°.
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°
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.
Vertical Angles Theorem: Vertical angles are congruent.