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Angle
Relationships

Master the rules that connect angles: complementary, supplementary, vertical, and more to solve for missing measures.

SSAT Middle Level: Quantitative

What Is an Angle?

V Ray 1 Ray 2 angle

An angle is formed when two rays share a common endpoint called the vertex. We measure angles in degrees (\(^{\circ}\)), from \(0^{\circ}\) to \(360^{\circ}\).

Types of Angles

Acute
Less than \(90^{\circ}\)
Right
Exactly \(90^{\circ}\)
Obtuse
Between \(90^{\circ}\) and \(180^{\circ}\)
Straight
Exactly \(180^{\circ}\)
\(45^\circ\) \(90^\circ\) \(120^\circ\) \(180^\circ\)

Why Do Angle Relationships Matter?

Many SSAT problems give you some angle measures and ask you to find the missing ones. Angle relationships are the rules that connect known angles to unknown angles.

1
Complementary angles add to \(90^{\circ}\).
2
Supplementary angles add to \(180^{\circ}\).
3
Vertical angles are always equal.
4
Angles in a triangle add to \(180^{\circ}\).

Complementary Angles

a b dividing ray

If one angle is \(35^{\circ}\), the complement is \(90^{\circ} - 35^{\circ} = 55^{\circ}\).

Supplementary Angles

a b straight line = \(180^\circ\)

If \(\angle a = 130^{\circ}\), then \(\angle b = 180^{\circ} - 130^{\circ} = 50^{\circ}\).

Vertical Angles

a a b b

Also, each pair of adjacent angles at the intersection is supplementary: \(\angle a\) + \(\angle b\) = \(180^{\circ}\).

Angles in a Triangle

A B C a b c

Complementary vs. Supplementary

Complementary Supplementary
Sum \(90^{\circ}\) \(180^{\circ}\)
Visual Form a right angle Form a straight line
Memory Tip "C" comes before "S"
90 comes before 180		 "S" for "Straight"
A straight line = \(180^\circ\)
Example \(30^{\circ}\) and \(60^{\circ}\) \(110^{\circ}\) and \(70^{\circ}\)

How to Find a Missing Angle

1
Identify the relationship. Are the angles complementary, supplementary, vertical, or in a triangle?
2
Write the equation. Use the matching rule: sum = \(90^{\circ}\), sum = \(180^{\circ}\), or angles are equal.
3
Plug in known values. Substitute the angle measures you already know.
4
Solve for the unknown. Subtract or simplify to find the missing angle.
5
Check your answer. Do the angles add up correctly?

Example 1: Supplementary Angles

\(125^\circ\) \(x^\circ\)

Example 2: Vertical Angles

\(72^\circ\) \(x^\circ\) \(y^\circ\) \(y^\circ\)

Example 3: Triangle Angle Sum

A B C \(x^\circ\) \(45^\circ\) \(65^\circ\)

Example 4: Multi-Step Problem

\(40^\circ\) \(y^\circ\) \(85^\circ\)

Common Mistakes

✕ Mixing Up the Sums
Using \(180^{\circ}\) for complementary or \(90^{\circ}\) for supplementary. Remember: C = Corner (\(90^{\circ}\)), S = Straight (\(180^{\circ}\)).
✕ Forgetting Vertical = Equal
Setting vertical angles as supplementary instead of equal. Opposite angles at an intersection are always the same.
✕ Wrong Triangle Sum
Using \(360^{\circ}\) for a triangle. A triangle's interior angles always sum to \(180^{\circ}\), not \(360^{\circ}\).

Summary of Angle Rules

Relationship Rule Key Clue
Complementary \(a + b = 90^{\circ}\) Form a right angle
Supplementary \(a + b = 180^{\circ}\) Form a straight line
Vertical \(a = b\) Opposite at intersection
Triangle Sum \(a + b + c = 180^{\circ}\) Three interior angles

Question 1 of 5

Two angles are complementary. One angle measures \(37^{\circ}\). What is the measure of the other angle?

Question 2 of 5

Two lines cross. One of the angles formed is \(115^{\circ}\). What is the measure of the angle directly opposite it?

Question 3 of 5

A triangle has angles measuring \(50^{\circ}\) and \(80^{\circ}\). What is the measure of the third angle?

Question 4 of 5

Angles \(x\) and \(y\) are supplementary. If \(x = 3y\), what is the value of \(y\)?

Question 5 of 5

Two lines cross. One angle is \(2x + 10\) and the vertical angle across from it is \(3x - 20\). What is \(x\)?