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Angles and Lines

4 min readEasy10-question checkpoint

Angle questions on the SAT come down to a few rules — straight lines sum to 180°, vertical angles match, and parallel lines cut by a transversal create predictable equal-or-supplementary angle pairs.

Where this lesson fits

Angles and Lines

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The basics:

A straight line = $180°$ . Two angles on the same line sum to 180° (they're supplementary).
Vertical angles (across from each other where two lines cross) are EQUAL.
Right angle = 90°. Acute < 90°. Obtuse > 90° (but < 180°).
A full revolution = 360°. Sum of angles around a point = 360°.

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Angle pair relationships — what's equal, what's supplementary

Pair	When	Relationship
Vertical angles	Two lines cross	EQUAL
Linear pair (supplementary)	Two angles on a straight line	Sum = 180°
Complementary angles	Two angles together = right angle	Sum = 90°
Corresponding angles	Parallel + transversal, same position	EQUAL
Alternate interior angles	Parallel + transversal, between, opposite	EQUAL
Same-side interior	Parallel + transversal, between, same side	Sum = 180°

Triangle angle sum: the three angles of any triangle sum to 180°.

Polygon angle sum. For an $n$ -sided polygon, the interior angles sum to:

(n - 2) \cdot 180°

Triangle (3 sides): 180°.
Quadrilateral (4 sides): 360°.
Pentagon (5 sides): 540°.
Hexagon (6 sides): 720°.

For a regular polygon (all angles equal), each interior angle is $\frac{(n-2) \cdot 180°}{n}$ .

Parallel lines cut by a transversal. The most-tested setup. A line crossing two parallel lines creates 8 angles. They come in pairs:

Corresponding angles (same position relative to each intersection): EQUAL.
Alternate interior angles (between the parallels, opposite sides of the transversal): EQUAL.
Alternate exterior angles (outside the parallels, opposite sides): EQUAL.
Same-side interior (co-interior) angles: SUPPLEMENTARY (sum to 180°).

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Polygon interior angle sums

Polygon	Sides	Angle sum	Each angle (regular)
Triangle	3	180°	60°
Quadrilateral	4	360°	90° (square)
Pentagon	5	540°	108°
Hexagon	6	720°	120°
Octagon	8	1080°	135°

Quick rule for parallel-line problems: there are usually only TWO distinct angle measures in the whole figure. Every angle is either equal to one or supplementary to it (sums to 180°).

Triangle facts you should know:

Exterior angle of a triangle = sum of the two non-adjacent interior angles. Example: in a triangle with interior angles 50° and 60°, the exterior angle at the third vertex is $50 + 60 = 110°$ .
Isosceles triangle: two equal sides AND two equal angles (opposite those sides).
Equilateral triangle: all sides equal, all angles 60°.

SAT-typical setup: a figure with several lines crossing, asking for an unknown angle marked $x$ . Strategy:

Find the simplest fact that gives you any angle.
Use straight-line sum (180°), vertical angles, parallel-line rules to chain.
Triangle sum (180°) is often the final step.

Watch for isosceles triangles disguised — if a problem says two segments are equal, it's signaling angle equality too.

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Quick check

Identify which rule applies (straight line, vertical angles, triangle sum, parallel-line rules, polygon sum). Chain rules together if needed to find the unknown angle.

The exterior angle of a triangle measures 130°. One of the non-adjacent interior angles is 70°. What is the other non-adjacent interior angle?

50°

70°

110°

60°

Worked examples

Example 1

In the figure (a triangle with one side extended), the interior angles of the triangle are $40°$ , $75°$ , and $x°$ . The exterior angle at the vertex with $x°$ is $y°$ . What is the value of $y$ ?

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Example 2

Two parallel lines are cut by a transversal. One of the angles formed is $52°$ . Which of the following could NOT be the measure of another angle in the figure?

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Common pitfalls

Confusing complementary with supplementary

Complementary = sum to 90°. Supplementary = sum to 180°. SAT angle problems use both — read carefully.

Assuming lines are parallel without being told

Just because lines look parallel in a diagram doesn't mean they are. The problem must STATE parallel (or use ∥ symbols). Otherwise don't apply parallel-line angle rules.

Using triangle sum (180°) on a non-triangle

Quadrilaterals sum to 360°, pentagons to 540°, hexagons to 720°. Use $(n-2) \cdot 180°$ for any $n$ -sided polygon. Don't apply 180° outside of a triangle.

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Forgetting isosceles → equal base angles

If two sides of a triangle are equal, the angles OPPOSITE those sides are also equal. SAT problems often hide this — they tell you a triangle is isosceles and expect you to deduce equal angles.

Key takeaways

Straight line = 180°. Triangle = 180°. Quadrilateral = 360°.
Polygon angle sum: $(n-2) \cdot 180°$ .
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Vertical angles are equal. Linear pair (on a straight line) sums to 180°.
Parallel + transversal: only TWO distinct angle measures appear.
Exterior angle of a triangle = sum of two non-adjacent interior angles.

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Checkpoint

10 questions on Lines, Angles, and Triangles, from our question bank. Score 70% or better across 10 answers to reach Silver and pass this step.

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Where this lesson fits

Angles and Lines

You are here

→Right Triangles →Trigonometry →Coordinate Geometry