Complementary, Supplementary, and Vertical Angles

In geometry, when lines intersect or angles are grouped together, they often form special relationships. Understanding these relationships allows us to find unknown angle measures easily.

Complementary Angles

Complementary angles are two angles whose measures add up to exactly $90^\circ$. When placed adjacent to each other, they form a right angle.

• Example: If one angle is $40^\circ$, its complement is $90^\circ - 40^\circ = 50^\circ$.

Supplementary Angles

Supplementary angles are two angles whose measures add up to exactly $180^\circ$. When placed adjacent to each other, they form a straight line (also known as a straight angle).

Example Problem: Two supplementary angles are given as $x$ and $2x + 30$. Find $x$.

Since they are supplementary, their sum must be $180^\circ$. We can write and solve an equation:

$$x + (2x + 30) = 180$$

Combine like terms:

$$3x + 30 = 180$$

Subtract $30$ from both sides:

$$3x = 150$$

Divide by $3$:

$$x = 50$$

Vertical Angles

Vertical angles are the opposite angles formed by the intersection of two lines. A key property of vertical angles is that they are always equal to each other.

Example Problem: Two vertical angles are given as $3x$ and $x + 40$. Find $x$.

Since vertical angles are equal, we set their expressions equal to each other:

$$3x = x + 40$$

Subtract $x$ from both sides:

$$2x = 40$$

Divide by $2$:

$$x = 20$$

Summary

• Complementary Angles: Two angles that add to $90^\circ$.
• Supplementary Angles: Two angles that add to $180^\circ$.
• Vertical Angles: Opposite angles at an intersection that are always equal.