Angles, Parallel Lines, and Triangle Angle Sums — SAT Math

Introduction

Angle relationships built from parallel lines, transversals, and triangles appear across the Geometry and Trigonometry domain, which makes up about 15% of SAT Math questions. Every angle theorem in this note can be applied in one or two steps — so the real skill is recognizing which relationship a question is using and setting up the correct equation immediately.

Core Concept

Angles at a Point and on a Line

Relationship	Definition	Key Equation
Vertical angles	Opposite angles formed when two lines cross	Equal: $\angle 1 = \angle 3$
Supplementary angles	Two angles forming a straight line (linear pair)	Sum to $180°$
Complementary angles	Two angles forming a right angle	Sum to $90°$

Parallel Lines Cut by a Transversal

When a transversal crosses two parallel lines, eight angles are formed (four at each intersection). You only need to remember three rules:

Angle Pair	Location	Relationship
Corresponding	Same side of transversal, same position at each line	Equal
Alternate interior	Between the parallel lines, opposite sides of transversal	Equal
Same-side interior (co-interior)	Between the parallel lines, same side of transversal	Supplementary (sum $= 180°$ )

Alternate exterior angles (outside the parallel lines, opposite sides) are also equal. Corresponding + alternate interior + alternate exterior: equal. Same-side interior: supplementary.

Triangle Angle Theorems

Triangle Angle Sum: The three interior angles of any triangle sum to $180°$ .

\alpha + \beta + \gamma = 180°

Exterior Angle Theorem: An exterior angle of a triangle equals the sum of the two non-adjacent interior angles.

\text{exterior angle} = \alpha + \beta

This is a direct consequence of the angle sum: if the third interior angle is $\gamma$ , then exterior $= 180° - \gamma = \alpha + \beta$ .

Isosceles Triangle: The two base angles are equal. If the vertex angle is $V$ , then each base angle $= \dfrac{180° - V}{2}$ .

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