🎯 Learning Objectives
- Angle relationships at a single point and along a straight line.
- Parallel lines cut by a transversal: every angle pair you need.
- How to spot vertical, supplementary, and complementary angles.
- Angle relationships in polygons.
- 10 SAT-style practice questions.
1. Angle Basics
Angle Type | Measure | Visual Cue |
Acute | less than 90° | Sharp wedge |
Right | exactly 90° | Square corner |
Obtuse | between 90° and 180° | Wide opening |
Straight | exactly 180° | Flat line |
Reflex | between 180° and 360° | Wraps around |
2. Angle Pairs You Must Know
Pair Type | Sum / Relationship | When You See |
Complementary | Sum to 90° | Two angles in a right triangle |
Supplementary | Sum to 180° | Two angles on a straight line |
Vertical | Equal | Two lines crossing: opposite angles |
Adjacent | Share a side | Side-by-side angles |
💡 Pro Tip: Vertical Angles Are Equal
Whenever two lines cross, the two opposite angles are ALWAYS equal. No calculation needed: just spot the X-shape.
3. Parallel Lines Cut by a Transversal
When a transversal crosses two parallel lines, eight angles are formed: but they fall into just two groups: four angles are equal, and the other four are equal, and the two groups are supplementary (sum to 180°).
Three Equal-Angle Pairs
Pair Name | Position | Relationship |
Corresponding | Same position at each intersection | Equal |
Alternate interior | Between the two lines, opposite sides | Equal |
Alternate exterior | Outside the two lines, opposite sides | Equal |
Co-interior (same-side interior) | Between the two lines, same side | Sum = 180° |
⚡ Strategy: The Z, F, and C
- Z-shape: Alternate interior angles → equal.
- F-shape: Corresponding angles → equal.
- C-shape: Co-interior angles → supplementary (180°).
Trace the transversal and identify the angle relationship before writing an equation.
4. Angles in a Polygon
n = number of sides
Sum of interior angles = (n − 2) × 180°
Common Polygons
Shape | Sides (n) | Sum of Interior Angles |
Triangle | 3 | 180° |
Quadrilateral | 4 | 360° |
Pentagon | 5 | 540° |
Hexagon | 6 | 720° |
Octagon | 8 | 1080° |
Each interior angle of a regular polygon = (n − 2) × 180° / n
(true for any polygon)
Sum of EXTERIOR angles of any polygon = 360°
5. The 'All Triangles Sum to 180°' Rule
In any triangle, the three interior angles add to 180°. This is the most-used geometry fact on the SAT: every angle-chase question eventually leans on it.
📝 Worked Example
In a triangle, two angles measure 35° and 70°. What is the third angle?
180 − 35 − 70 = 75°.
6. Strategies
⚡ Spot 'X' = vertical angles
Crossed lines = two pairs of equal angles.
⚡ Use the Z, F, C tracing
Parallel lines + transversal? Trace the letter with your pencil to identify the rule.
⚡ For polygons, use (n − 2) × 180°
One formula handles every interior-angle question.
⚡ Exterior Angles of a Convex Polygon Sum to 360°
Useful for regular polygons: each exterior angle = 360/n.
⚡ Mark every angle on the figure as you go
Saves rework when later questions reuse the same diagram.
7. Summary
📌 Key Takeaways
- Complementary: sum to 90°.
- Supplementary: sum to 180°.
- Vertical angles: equal.
- Parallel + transversal: Z, F equal; C supplementary.
- Triangle: angles sum to 180°.
- Polygon interior: (n − 2) × 180°.
- Polygon exterior: always 360° total.
8. Practice: 10 Questions
Q1. Two lines cross. One angle is 35°. What is the angle directly opposite?
A) 35° B) 55° C) 90° D) 145°
Hint: Vertical angles are equal.
Q2. Two angles are supplementary. One measures 110°. The other?
A) 70° B) 80° C) 90° D) 110°
Hint: 180 − 110 = 70.
Q3. Two angles are complementary. One is 32°. The other?
A) 32° B) 58° C) 90° D) 148°
Hint: 90 − 32 = 58.
Q4. A triangle has angles 50° and 60°. What is the third?
A) 60° B) 70° C) 80° D) 110°
Hint: 180 − 50 − 60 = 70.
Q5. What is the sum of interior angles of a hexagon?
A) 360° B) 540° C) 720° D) 900°
Hint: (6 − 2) × 180 = 720.
Q6. Each interior angle of a regular octagon?
A) 108° B) 120° C) 135° D) 144°
Hint: 1080 / 8 = 135.
Q7. Two parallel lines are cut by a transversal. Two corresponding angles are…
A) Equal B) Supplementary C) Complementary D) Different
Hint: Corresponding angles → equal.
Q8. A pentagon has interior angles 110°, 100°, 105°, 115°, and x. Find x.
A) 100° B) 105° C) 110° D) 115°
Hint: Sum = 540; 540 − 430 = 110.
Q9. Sum of exterior angles of a 20-sided polygon?
A) 360° B) 540° C) 1800° D) 3240°
Hint: For a convex polygon, the exterior angles total 360° when one is taken at each vertex.
Q10. Angle A and angle B are alternate interior angles. A = 75°. B = ?
A) 15° B) 75° C) 105° D) 90°
Hint: Alternate interior angles equal.
Answer Key & Worked Solutions
# | Answer | Reasoning |
Q1. | A) 35° | Vertical angles equal. |
Q2. | A) 70° | 180 − 110. |
Q3. | B) 58° | 90 − 32. |
Q4. | B) 70° | 180 − 110. |
Q5. | C) 720° | (6−2) × 180. |
Q6. | C) 135° | 1080 / 8. |
Q7. | A) Equal | Corresponding angles. |
Q8. | C) 110° | 540 − 430. |
Q9. | A) 360° | For a convex polygon, the exterior angles total 360° when one is taken at each vertex. |
Q10. | B) 75° | Alternate interior. |