Math

Angles and Lines

6 min readEasy5-question drill

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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→Right Triangles →Trigonometry →Coordinate Geometry

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.

An angle measures 51°. What is the measure of its supplement?

129

309

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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Try it yourself

5 practice questions on Angles and Lines, drawn from the question bank. The tutor is one click away if you get stuck.

Where this lesson fits

Angles and Lines

You are here

→Right Triangles →Trigonometry →Coordinate Geometry