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Geometry Lines and angles

Angle theorems (vertical, complementary, supplementary)

20 practice questions 2 video lessons Theory + worked examples

Angle Relationships

Common Core Geometry • Standard G-CO.9 • Lines & Angles

Angle Relationships is the opening topic of Lines & Angles in the Common Core State Standards. It is aligned to Standard G-CO.9, which requires students to prove theorems about lines and angles, including that vertical angles are congruent.

Angle relationships include vertical angles (equal), complementary angles (summing to \(90^\circ\)), and supplementary angles (summing to \(180^\circ\)).

Common Core Geometry › Lines & Angles › Angle Relationships  —  Standard G-CO.9

Practice 20 questions
Practice questions

Every question with a fully worked solution.

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Watch 2 video(s)
  • Angle Pair Relationships: Adjacent, Vertical, Complementary, Supplementary Watch
  • Complementary, Supplementary & Vertical Angles - Geometry Watch
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Theory

When lines and rays meet, angles form predictable relationships:

  • Vertical angles — opposite angles formed by two crossing lines; they are congruent.
  • Complementary angles — two angles that sum to \(90^\circ\).
  • Supplementary angles — two angles that sum to \(180^\circ\).
  • Linear pair — two adjacent angles on a straight line; they are always supplementary.
These give equations. Set the sum equal to \(90^\circ\) or \(180^\circ\), or set vertical angles equal, then solve.
Vertical angles are congruent When two lines cross, the opposite (vertical) angles are congruent. 1 2 3 4 vertical angles: ∠1≅∠2, ∠3≅∠4
Vertical angles are congruent: \(\angle 1\cong\angle 2\).
Complementary and supplementary angles Complementary angles add to 90 degrees; supplementary angles add to 180 degrees. complementary: sum = 90° supplementary: sum = 180°
Complementary angles sum to \(90^\circ\); supplementary to \(180^\circ\).

The angle relationships:

\[\text{vertical: }\angle 1=\angle 2;\quad \text{complementary: }\angle a+\angle b=90^\circ;\quad \text{supplementary: }\angle a+\angle b=180^\circ\]
vertical angles are equal; complementary sum to 90; supplementary sum to 180
A linear pair is supplementary — a reliable source of a \(180^\circ\) equation.

How to find an unknown angle

  1. Identify the relationship (vertical, complementary, supplementary, linear pair).
  2. Write the equation: equal, or sum \(=90^\circ\) or \(180^\circ\).
  3. Solve for the unknown and substitute back if needed.
Example 1 — Vertical angles
Two lines cross. One angle is \(65^\circ\). Find its vertical angle.
Solution

Vertical angles are congruent (equal).

\(\text{vertical angle}\)\(=\)\(65^\circ\)
the vertical angle is 65 degrees
Example 2 — Complementary
An angle is complementary to \(28^\circ\). Find it.
Solution

Complementary angles sum to \(90^\circ\).

\(90^\circ-28^\circ\)\(=\)\(62^\circ\)
the complement is 62 degrees
Example 3 — Supplementary
An angle is supplementary to \(115^\circ\). Find it.
Solution

Supplementary angles sum to \(180^\circ\).

\(180^\circ-115^\circ\)\(=\)\(65^\circ\)
the supplement is 65 degrees
Example 4 — Solve with a linear pair
Angles \((2x+10)^\circ\) and \((3x)^\circ\) form a linear pair. Find \(x\).
Solution

A linear pair is supplementary, so the two angles sum to \(180^\circ\).

\((2x+10)+3x\)\(=\)\(180\)
\(5x+10\)\(=\)\(180\)
\(5x\)\(=\)\(170\)
\(x\)\(=\)\(34\)
x equals 34

Common pitfalls

Complementary is \(90^\circ\), supplementary is \(180^\circ\). “C” comes before “S” just as \(90<180\).
Vertical angles are equal, not supplementary. Don't confuse them with the linear-pair neighbor.
Only adjacent angles on a line form a linear pair. Two random supplementary angles need not be adjacent.

Frequently asked questions

What are vertical angles?

The opposite angles formed when two lines cross. They are always congruent (equal).

What is the difference between complementary and supplementary angles?

Complementary angles sum to \(90^\circ\); supplementary angles sum to \(180^\circ\).

What is a linear pair?

Two adjacent angles whose non-common sides form a straight line. A linear pair is always supplementary.

How do you find an unknown angle in these relationships?

Write an equation from the relationship — equal for vertical, sum to \(90^\circ\) or \(180^\circ\) otherwise — and solve.

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