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Solve Problems With
Angle Relationships

Learn to identify supplementary, complementary, vertical, and adjacent angles, then write and solve equations for unknown angles.

7th Grade Math

What Are Angle Relationships?

When two or more angles share a special connection, we call it an angle relationship. These relationships give us rules we can use to find missing angle measures without a protractor.

Complementary Angles

\(35^\circ\)
\(55^\circ\)
 A B C O

Supplementary Angles

\(130^\circ\)
\(50^\circ\)
 A B C O

Vertical Angles

\(63^\circ\)
\(63^\circ\)
\(117^\circ\)
\(117^\circ\)

Adjacent Angles

\(40^\circ\)
\(75^\circ\)
 A B C O

The Four Core Rules

Complementary
\(a + b = 90^\circ\)
Supplementary
\(a + b = 180^\circ\)
Vertical
\(a = b\) (opposite angles)
Adjacent
Share a vertex and side, no overlap

Writing Equations from Angles

\(x\)
\(110^\circ\)
 A B C

Complementary vs. Supplementary

Complementary Supplementary
Sum \(90^\circ\) \(180^\circ\)
Forms a... Right angle Straight line
Both angles are... Acute (each < \(90^\circ\)) One may be obtuse
Memory trick "C" for "Corner" (\(90^\circ\)) "S" for "Straight" (\(180^\circ\))

Step-by-Step Strategy

1
Identify the relationship. Are the angles complementary, supplementary, vertical, or adjacent on a line?
2
Write the equation. Use the matching rule: sum = \(90^\circ\), sum = \(180^\circ\), or angles are equal.
3
Solve for the unknown. Isolate the variable using basic algebra.
4
Check your answer. Substitute back in: does the sum match the rule?

Example 1: Complementary Angles

\(27^\circ\)
\(x\)

Example 2: Vertical Angles

\((3x + 10)^\circ\)
\(85^\circ\)

Example 3: Multi-Step Problem

\((2x+5)^\circ\)
\(65^\circ\)
\(x^\circ\)

Example 4: Combined Relationships

\((4x)^\circ\)
\((5x+18)^\circ\)

Common Mistakes

✕ Mixing Up \(90^\circ\) and \(180^\circ\)
Using \(180^\circ\) for complementary or \(90^\circ\) for supplementary. Remember: C = Corner = \(90^\circ\), S = Straight = \(180^\circ\).
✕ Calling Adjacent Angles Equal
Adjacent angles share a side, but that does not mean they are equal. Only vertical angles are guaranteed equal.
✕ Forgetting to Distribute
In an expression like \(2(x + 15)\), forgetting to distribute gives the wrong equation. Always expand before solving.

Angle Relationships Summary

Relationship Rule Look For
Complementary \(a + b = 90^\circ\) Right angle / L-shape
Supplementary \(a + b = 180^\circ\) Straight line
Vertical \(a = b\) Crossing lines, opposite angles
Adjacent Share vertex + side Side-by-side, no overlap

Question 1 of 5

Two angles are complementary. One angle measures \(38^\circ\). What is the measure of the other angle?

Question 2 of 5

Two angles form a straight line. One measures \((3x + 12)^\circ\) and the other measures \(78^\circ\). What is the value of \(x\)?

Question 3 of 5

Two lines intersect. One of the angles formed is \(47^\circ\). What is the measure of the angle vertical to it?

Question 4 of 5

Angles \(A\) and \(B\) are supplementary. If \(\angle A = (5x - 10)^\circ\) and \(\angle B = (3x + 30)^\circ\), what is the measure of \(\angle A\)?

Question 5 of 5

Three angles meet at a point on a straight line. They measure \(x^\circ\), \(2x^\circ\), and \(60^\circ\). What is the value of \(x\)?