Angle questions on the SAT test whether you know a few fundamental rules. Once you have these down, you can chain them together to solve multi-step problems.
Angles at a Point and on a Line
- Angles around a point sum to
- Angles on a straight line sum to (supplementary)
- Vertical angles are equal (formed by two intersecting lines)
Example 1: Two lines cross. One angle is . The vertical angle is also . The adjacent angles are .
Parallel Lines Cut by a Transversal
When a line (transversal) crosses two parallel lines, it creates 8 angles. You only need to know two angle measures — all the rest follow.
Key relationships:
- Corresponding angles are equal (same position at each intersection)
- Alternate interior angles are equal (opposite sides of the transversal, between the parallels)
- Alternate exterior angles are equal
- Co-interior (same-side interior) angles are supplementary (sum to )
Example 2: Lines and are parallel. A transversal creates an angle of at line . What is the alternate interior angle at line ?
They're equal.
Example 3: In the same setup, what is the co-interior angle?
Triangle Angle Sum
The three interior angles of a triangle always add up to .
Example 4: A triangle has angles of and . Find the third angle.
Exterior Angle Theorem
An exterior angle of a triangle equals the sum of the two non-adjacent interior angles.
Example 5: A triangle has interior angles of and . The exterior angle at the third vertex is:
Isosceles and Equilateral Triangles
- Isosceles: two equal sides → two equal base angles
- Equilateral: all sides equal → all angles are
Example 6: An isosceles triangle has a vertex angle of . Find the base angles.
Polygon Angle Formulas
For a polygon with sides:
- Sum of interior angles:
- Each interior angle (regular polygon):
- Sum of exterior angles: always
Example 7: What is the measure of each interior angle of a regular hexagon ()?
Example 8: What is each exterior angle of a regular octagon?
Multi-Step SAT Problem
Example 9: In the figure, lines and are parallel. If angle and angle , find angle at the intersection inside the triangle.
Using the parallel lines: the angle alternate to angle 1 is .
In the triangle:
Practice Problems
Problem 1: Two angles of a triangle are and . What is the exterior angle at the third vertex?
Solution
Third interior angle:
Exterior angle: (or directly: )
Problem 2: What is the sum of interior angles of a pentagon?
Solution
Problem 3: Lines and are parallel. A transversal creates a angle with line . What is the corresponding angle at line ?
Solution
(corresponding angles are equal when lines are parallel).
Key Takeaways
- Vertical angles are equal; supplementary angles sum to
- Parallel lines + transversal: corresponding and alternate interior angles are equal
- Triangle angles sum to ; exterior angle = sum of the two remote interior angles
- Polygon interior angle sum:
- Regular polygon exterior angles always sum to
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