Angle Relationships

By the end of this lesson, you will be able to identify and calculate complementary, supplementary, and vertical angles, and use parallel line angle relationships.

Types of Angles

Acute: $0° < \theta < 90°$

Right: $\theta = 90°$ (the square corner symbol marks this)

Obtuse: $90° < \theta < 180°$

Straight: $\theta = 180°$ (a straight line)

Complementary and Supplementary

Complementary angles sum to 90°.

If one angle is 35°, its complement is $90° - 35° = 55°$.

Supplementary angles sum to 180°.

If one angle is 110°, its supplement is $180° - 110° = 70°$.

On the AP/test: complementary → think "C" for corner (90°); supplementary → think "S" for straight line (180°).

Adjacent Angles and Linear Pairs

Adjacent angles share a vertex and a side.

Linear pair: two adjacent angles that form a straight line (sum = 180°).

Vertical Angles

A
     |
C----|----D
     |
     B

When two lines intersect, four angles are formed. The vertical angles (opposite each other) are congruent (equal).

$\angle AOC = \angle BOD$ (vertical angles) $\angle AOD = \angle BOC$ (vertical angles) $\angle AOC + \angle AOD = 180°$ (linear pair)

Parallel Lines Cut by a Transversal

When a transversal crosses two parallel lines, several pairs of equal angles are formed:

Corresponding angles (same position, different parallel lines): congruent

Alternate interior angles (between the parallel lines, opposite sides of transversal): congruent

Alternate exterior angles (outside the parallel lines, opposite sides): congruent

Co-interior (same-side interior) angles: supplementary (sum = 180°)

Solving for Unknown Angles

Example: Two lines intersect. One angle is $3x + 15°$ and the vertical angle is $4x - 10°$. Find $x$.

Vertical angles are equal: $3x + 15 = 4x - 10$

$25 = x$ → angles = $3(25)+15 = 90°$ ✓

Example: Two parallel lines are cut by a transversal. One angle is $5x + 20°$ and the co-interior angle is $3x + 40°$. Find $x$.