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Lines, Angles &
Triangles

Master the fundamental geometry tested on the GRE: angle relationships, parallel line properties, and triangle rules you need for quantitative reasoning.

GRE Quantitative

Understanding Angles

An angle is formed when two rays share a common endpoint called the vertex. Angles are measured in degrees, where a full rotation equals \(360^\circ\).

Acute
\(0^\circ < \theta < 90^\circ\)
Right
\(\theta = 90^\circ\) exactly
Obtuse
\(90^\circ < \theta < 180^\circ\)
Straight
\(\theta = 180^\circ\) exactly

Key Angle Pairs

a b Supplementary
\(a + b = 180^\circ\)
 c d Complementary
\(c + d = 90^\circ\)
Supplementary
Two angles that sum to \(180^\circ\). They form a straight line.
Complementary
Two angles that sum to \(90^\circ\). They form a right angle.

Vertical Angles & Linear Pairs

a a b b

Parallel Lines & a Transversal

l m t → → 1 2 3 4 5 6 7 8

Eight Angles, Only Two Values

When a transversal crosses two parallel lines, the eight angles created have only two distinct measures. Every angle equals either \(\alpha\) or \(180^\circ - \alpha\).

1
Corresponding angles: Same position at each intersection: equal.
2
Alternate interior angles: Opposite sides, between lines: equal.
3
Alternate exterior angles: Opposite sides, outside lines: equal.
4
Co-interior (same-side interior): Same side, between lines: supplementary (sum to \(180^\circ\)).

Triangle Angle Sum Theorem

a b c A B C

Exterior Angle Theorem

50° 60° 70° 110° A B C

Triangle Classifications

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

Isosceles & Equilateral Properties

α α Isosceles 60° 60° 60° Equilateral

The Triangle Inequality

For any triangle with sides \(a\), \(b\), and \(c\), every pair of sides must satisfy:

Equivalently, the longest side must be less than the sum of the other two. Also, the third side must be greater than the absolute difference of the other two:

The Pythagorean Theorem

a b c (hypotenuse)

Special Right Triangles

45° 45° s s
\(s\sqrt{2}\)
 45-45-90 60° 30° s
\(s\sqrt{3}\)
\(2s\)
 30-60-90
45-45-90
Sides: \(s : s : s\sqrt{2}\). The legs are equal; hypotenuse is leg \(\times \sqrt{2}\).
30-60-90
Sides: \(s : s\sqrt{3} : 2s\). Short leg opposite \(30^\circ\); hypotenuse is \(2 \times\) short leg.

How to Solve Angle & Triangle Problems

1
Identify the configuration: Are lines parallel? Is there a transversal? Is the triangle right, isosceles, or equilateral?
2
Label what you know: Write given angles and sides on the figure. Mark equal angles and sides.
3
Apply relationships: Use angle sum \(= 180^\circ\), supplementary/vertical angles, or parallel line rules to set up equations.
4
Solve for unknowns: Use algebra. For sides, apply Pythagorean theorem or special triangle ratios.
5
Verify: Check that angles sum correctly and the triangle inequality holds.

Example: Parallel Lines

65° x l m l ∥ m

Find \(x\). Lines \(l\) and \(m\) are parallel, cut by a transversal.

Example: Isosceles Triangle

40° x x A B C

Triangle ABC is isosceles with AC = BC. The vertex angle at C is \(40^\circ\). Find \(x\).

Example: Special Right Triangle

30° 60° 6 ? 12

The hypotenuse is \(12\) and the short leg (opposite \(30^\circ\)) is \(6\). Find the missing side.

Common Mistakes to Avoid

Trusting the Figure
GRE figures are not to scale. Never estimate angles or lengths from the drawing alone.
Mixing Up Ratios
Confusing 45-45-90 ratios (\(1:1:\sqrt{2}\)) with 30-60-90 ratios (\(1:\sqrt{3}:2\)).
Forgetting the Inequality
Assuming any three lengths form a triangle. Check: the longest side must be less than the sum of the other two.

Quick Reference Table

Rule Formula / Relationship When to Use
Angle Sum \(a + b + c = 180^\circ\) Any triangle
Exterior Angle ext = sum of remote interiors One angle extended
Vertical Angles Opposite angles are equal Intersecting lines
Parallel Lines Corr. = Alt. Int. = Alt. Ext. Transversal + parallel
Pythagorean \(a^2 + b^2 = c^2\) Right triangles
45-45-90 \(s : s : s\sqrt{2}\) Isosceles right triangle
30-60-90 \(s : s\sqrt{3} : 2s\) Half of equilateral
Triangle Ineq. \(|a-b| < c < a+b\) Valid side lengths

Question 1 of 5

Two angles are supplementary. One angle measures \(3x + 10\) degrees and the other measures \(2x + 20\) degrees. What is the value of \(x\)?

Question 2 of 5

In a triangle, two angles measure \(45^\circ\) and \(65^\circ\). What is the measure of the third angle?

Question 3 of 5

Which of the following cannot be the lengths of the sides of a triangle?

Question 4 of 5

In a 45-45-90 triangle, the hypotenuse is \(10\). What is the length of each leg?

Question 5 of 5

Lines \(l\) and \(m\) are parallel. A transversal creates an angle of \(130^\circ\) with line \(l\). What is the measure of the alternate interior angle at line \(m\)?