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Lines, Angles,
& Triangles

Master the angle relationships, parallel line properties, and triangle theorems tested on every SAT Math section.

SAT Math

What Is an Angle?

An angle is formed when two rays share a common endpoint called the vertex. We measure angles in degrees (\(^{\circ}\)), where a full rotation equals \(360^{\circ}\).

Vertex
\(\theta\)
 Ray 1 Ray 2

Types of Angles

Acute
Less than \(90^{\circ}\)
Right
Exactly \(90^{\circ}\)
Obtuse
Between \(90^{\circ}\) and \(180^{\circ}\)
Straight
Exactly \(180^{\circ}\)

Vertical & Linear Angle Pairs

\(a\)
\(a\)
\(b\)
\(b\)
 Vertical angles are equal
1
Vertical angles are the opposite angles formed by two intersecting lines. They are always equal: \(a = a\).
2
Linear pair: Two adjacent angles on a straight line are supplementary: \(a + b = 180^{\circ}\).

Parallel Lines & a Transversal

ℓ1 ℓ2 t
\(\alpha\)
\(\alpha\)
\(\beta\)
\(\beta\)

The Triangle Angle Sum Theorem

\(\alpha\)
\(\beta\)
\(\gamma\)
 A B C

The Exterior Angle Theorem

\(e\)
\(\alpha\)
\(\beta\)
 A B C D

Complementary vs. Supplementary

Complementary Supplementary
Sum \(90^{\circ}\) \(180^{\circ}\)
Example \(30^{\circ} + 60^{\circ}\) \(110^{\circ} + 70^{\circ}\)
Where it appears Right-angle problems Straight lines, linear pairs
Memory trick C for Corner (\(90^{\circ}\)) S for Straight (\(180^{\circ}\))

Triangle Classifications

Equilateral
All sides equal, all angles \(60^{\circ}\)
Isosceles
Two equal sides, two equal base angles
Scalene
No equal sides, no equal angles
Acute
All angles \(< 90^{\circ}\)
Right
One angle = \(90^{\circ}\)
Obtuse
One angle \(> 90^{\circ}\)

The Isosceles Triangle Theorem

\(a\)
\(a\)
\(\theta\)
\(\theta\)
 A B C

How to Solve Angle Problems

1
Label everything. Mark known angles and identify parallel lines, transversals, and triangles in the figure.
2
Choose your tool. Straight line = \(180^{\circ}\). Triangle = \(180^{\circ}\). Vertical angles = equal. Parallel lines = corresponding/alternate angles equal.
3
Write an equation. Express unknown angles in terms of a variable and set up the equation using the appropriate rule.
4
Solve and verify. Solve for the variable, then check: do all angles in the triangle sum to \(180^{\circ}\)? Do supplementary pairs sum correctly?

Example 1: Triangle Angles

\(50^{\circ}\)
\(65^{\circ}\)
\(x^{\circ}\)
 P Q R

Example 2: Parallel Lines

ℓ m
\(3x+10\)
\(5x-30\)

Example 3: Exterior Angle

\(40^{\circ}\)
\(2x^{\circ}\)
\(110^{\circ}\)
 A B C D

Common Mistakes

✕ Wrong sum
Using \(360^{\circ}\) instead of \(180^{\circ}\) for triangle angle sums. Remember: \(360^{\circ}\) is for angles around a point.
✕ Misidentifying angles
Confusing alternate interior angles with co-interior angles. Alternate = equal. Co-interior = supplementary.
✕ Forgetting to check parallel
Applying parallel-line angle rules when lines are not stated to be parallel. Always confirm from the problem.

Key Angle Rules Summary

Rule Relationship Result
Straight line Adjacent angles Sum = \(180^{\circ}\)
Vertical angles Opposite at intersection Equal
Triangle sum Interior angles Sum = \(180^{\circ}\)
Exterior angle One exterior angle = sum of remote interiors
Parallel + transversal Corresponding / alternate Equal
Isosceles triangle Base angles Equal

Question 1 of 5

In a triangle, two angles measure \(45^{\circ}\) and \(78^{\circ}\). What is the measure of the third angle?

Question 2 of 5

Two parallel lines are cut by a transversal. One angle measures \(125^{\circ}\). What is the measure of the alternate interior angle?

Question 3 of 5

An exterior angle of a triangle is \(130^{\circ}\). One of the remote interior angles is \(55^{\circ}\). What is the other remote interior angle?

Question 4 of 5

In an isosceles triangle, the vertex angle (the angle between the two equal sides) measures \(40^{\circ}\). What is the measure of each base angle?

Question 5 of 5

Two angles are supplementary. One angle is \(4x - 10\) and the other is \(2x + 40\). What is the value of \(x\)?