ACT Math: Angles
Angles is a foundational Plane Geometry topic on the ACT, and every question tends to boil down to a handful of core relationships once you recognize the pattern. This free, complete lesson covers angle relationships (complementary, supplementary, vertical, and linear pairs), parallel lines cut by a transversal, the triangle angle sum and exterior angle theorem, and polygon interior and exterior angle sums. Each idea is followed by a fully worked example, and then a set of original practice problems with instant feedback, in the same four-answer-choice, calculator-allowed format as the current ACT. Everything here is free, and you can keep practicing afterward with the full ACT Math Question Bank linked below.
1. Angle Relationships
Complementary angles sum to 90°. Supplementary angles sum to 180°. When two lines cross, they form two pairs of vertical angles, which are always equal, and adjacent pairs called linear pairs, which are always supplementary.
Worked Example: Two angles form a linear pair. One measures 112°. Find the other.
180° − 112° = 68°
1. Two angles are complementary. One measures 27°. What is the other?
Explanation: 90° − 27° = 63°.
2. Two angles are supplementary. One measures 98°. What is the other?
Explanation: 180° − 98° = 82°.
3. Two lines intersect, forming vertical angles. One measures 47°. What is its vertical angle?
Explanation: Vertical angles are always equal, so the vertical angle also measures 47°.
4. Two angles form a linear pair. One measures 65°. What is the other?
Explanation: A linear pair is always supplementary: 180° − 65° = 115°.
5. Four angles meet at a single point (forming a full circle around it). Three of them measure 80°, 95°, and 110°. What is the measure of the fourth angle?
Explanation: Angles all the way around a single point sum to 360°: 360° − 80° − 95° − 110° = 75°.
2. Parallel Lines and Transversals
When a transversal crosses two parallel lines, it creates eight angles, but only two distinct angle measures appear in the whole figure.
| Angle pair | Relationship |
|---|---|
| Corresponding angles | Equal |
| Alternate interior angles | Equal |
| Alternate exterior angles | Equal |
| Co-interior (same-side interior) angles | Supplementary |
Parallel lines cut by a transversal
Worked Example: Two parallel lines are cut by a transversal. One angle measures 72°. Find its co-interior (same-side interior) angle.
180° − 72° = 108°
1. Two parallel lines are cut by a transversal. One angle measures 58°. What is its corresponding angle?
Explanation: Corresponding angles are equal: 58°.
2. Two parallel lines are cut by a transversal. One angle measures 58°. What is its alternate interior angle?
Explanation: Alternate interior angles are equal: 58°.
3. Two parallel lines are cut by a transversal. One angle measures 58°. What is its co-interior (same-side interior) angle?
Explanation: Co-interior angles are supplementary: 180° − 58° = 122°.
4. Two parallel lines are cut by a transversal. Two co-interior angles measure (3x + 10)° and (2x + 30)°. What is the value of x?
Explanation: Co-interior angles sum to 180°: (3x + 10) + (2x + 30) = 180. This gives 5x + 40 = 180, so 5x = 140 and x = 28.
5. Two parallel lines are cut by a transversal. One angle measures (4x − 10)° and its corresponding angle measures (2x + 30)°. What is the value of x?
Explanation: Corresponding angles are equal: 4x − 10 = 2x + 30. This gives 2x = 40, so x = 20.
3. Triangle Angle Sum and the Exterior Angle Theorem
The interior angles of any triangle always sum to 180°. An exterior angle, formed by extending one side, always equals the sum of the two remote interior angles (the two angles not adjacent to it).
Exterior angle = sum of the two remote interior angles
Worked Example: A triangle has angles measuring 2x, 3x, and 4x. Find the measure of each angle.
2x + 3x + 4x = 180, so 9x = 180 and x = 20
The angles measure 40°, 60°, and 80°.
1. A triangle has angles of 55° and 65°. What is the measure of the third angle?
Explanation: 180° − 55° − 65° = 60°.
2. An isosceles triangle has a vertex angle of 50°. What is the measure of each base angle?
Explanation: The two base angles of an isosceles triangle are equal. Together they take up 180° − 50° = 130°, so each is 130° ÷ 2 = 65°.
3. A triangle has angles x, 2x, and 3x. What is the measure of the largest angle?
Explanation: x + 2x + 3x = 180, so 6x = 180 and x = 30. The largest angle is 3(30) = 90°.
4. An exterior angle of a triangle measures 115°. What is the sum of the two remote interior angles?
Explanation: By the exterior angle theorem, an exterior angle always equals the sum of the two remote interior angles: 115°.
5. An exterior angle of a triangle measures 130°, and one remote interior angle measures 70°. What is the other remote interior angle?
Explanation: 130° − 70° = 60°.
4. Polygon Interior and Exterior Angle Sums
The sum of the interior angles of any convex polygon with n sides is 180°(n − 2). The sum of the exterior angles of any convex polygon, one at each vertex, is always 360°, regardless of the number of sides.
Worked Example: Find the sum of the interior angles of a hexagon (6 sides).
180°(6 − 2) = 180°(4) = 720°
1. Find the sum of the interior angles of a pentagon (5 sides).
Explanation: 180°(5 − 2) = 180°(3) = 540°.
2. Each interior angle of a regular polygon measures 144°. How many sides does the polygon have?
Explanation: Each exterior angle is 180° − 144° = 36°. Since the exterior angles of a regular polygon are all equal and sum to 360°, the number of sides is 360° ÷ 36° = 10.
3. What is the sum of the exterior angles of any convex polygon, one at each vertex?
Explanation: No matter how many sides a convex polygon has, its exterior angles (one per vertex) always sum to exactly 360°, a full turn around the shape.
4. Find the measure of each interior angle of a regular octagon (8 sides).
Explanation: Sum of interior angles: 180°(8 − 2) = 1,080°. Since a regular octagon has 8 equal angles: 1,080° ÷ 8 = 135°.
Common Mistakes to Avoid
- Mixing up complementary and supplementary. Complementary sums to 90°, supplementary sums to 180°. Confusing the two flips your entire answer.
- Treating co-interior angles as equal instead of supplementary. Corresponding, alternate interior, and alternate exterior angles are equal; co-interior (same-side interior) angles are the one pair that's supplementary instead.
- Forgetting that the exterior angle theorem only applies to triangles. "Exterior angle equals the sum of the two remote interior angles" is specific to triangles; for other polygons, use the general 180°(n − 2) interior angle sum instead.
- Using the interior angle sum formula for exterior angles, or vice versa. Interior angles sum to 180°(n − 2), which depends on the number of sides. Exterior angles always sum to 360°, no matter how many sides the polygon has.
- Assuming a polygon is regular without being told so. The "divide the total by the number of sides" shortcut for finding one interior angle only works when every angle in the polygon is equal.
Frequently Asked Questions
Is there a fast way to remember all the parallel-line angle relationships?
Sort every pair into one of two buckets: angles in matching positions relative to the two intersections (corresponding, alternate interior, alternate exterior) are equal, while angles on the same side, between the parallel lines (co-interior), are supplementary. That single distinction covers the whole topic.
Why does the exterior angle of a triangle equal the sum of the remote interior angles?
The exterior angle and its adjacent interior angle form a linear pair, so together they sum to 180°. Since all three interior angles of the triangle also sum to 180°, the exterior angle must equal whatever's left after removing its adjacent interior angle, which is exactly the sum of the other two.
How can I verify the interior angle sum formula for myself?
Any convex polygon with n sides can be split into (n − 2) triangles by drawing diagonals from one vertex. Since each triangle contributes 180°, the total interior angle sum is 180°(n − 2).
Where can I practice more problems like these?
The Angles quizzes in the ACT Math Question Bank include additional original problems on this topic, along with quizzes covering every other topic tested on the ACT Math section.