Complementary and Supplementary Angles: Definitions, Properties, and Problem Solving

Complementary and Supplementary Angles

Introduction to Complementary & Supplementary Angles

Complementary and supplementary angles are fundamental concepts in geometry and trigonometry, often encountered in precalculus. Understanding these relationships is essential for solving angle-based problems and for further study in trigonometry.

• Complementary angles are two angles whose measures add up to 90°.

• Supplementary angles are two angles whose measures add up to 180°.

Complementary Angles	Supplementary Angles
Sum of angles: $x + y = 90^{\circ}$ Example: $60^{\circ} + 30^{\circ} = 90^{\circ}$ Form right angles (90°)	Sum of angles: $x + y = 180^{\circ}$ Example: $150^{\circ} + 30^{\circ} = 180^{\circ}$ Form straight lines (180°)

Note: Complementary and supplementary angles are always assumed to be positive.

Finding Complementary and Supplementary Angles

To find the complement or supplement of a given angle, subtract the angle from 90° or 180°, respectively.

• Complement of $x$: $90^{\circ} - x$

• Supplement of $x$: $180^{\circ} - x$

Example 1: Find the complement and supplement of $28^{\circ}$.

• Complement: $90^{\circ} - 28^{\circ} = 62^{\circ}$

• Supplement: $180^{\circ} - 28^{\circ} = 152^{\circ}$

Example 2: Find the complement and supplement of $100^{\circ}$.

• Complement: $90^{\circ} - 100^{\circ} = -10^{\circ}$ (not possible, as complements must be positive)

• Supplement: $180^{\circ} - 100^{\circ} = 80^{\circ}$

Practice: Find the complement and supplement of a $45^{\circ}$ angle.

• Complement: $90^{\circ} - 45^{\circ} = 45^{\circ}$

• Supplement: $180^{\circ} - 45^{\circ} = 135^{\circ}$

Solving Problems with Complementary & Supplementary Angles

Problems may involve finding unknown angles using the properties of complementary or supplementary angles. In right triangles, the two non-right angles are always complementary.

• When angles are expressed in terms of variables, set up equations based on their sum (90° for complementary, 180° for supplementary).

Example 3: Find each angle in a right triangle if one angle is $x$ and the other is $90^{\circ} - x$.

• Since the sum must be $90^{\circ}$, if $x = 50^{\circ}$, then the other angle is $90^{\circ} - 50^{\circ} = 40^{\circ}$.

Example 4: Solve for $x$ if $50 - 2x$ and $17x - 20$ are supplementary angles.

• Set up the equation: $(50 - 2x) + (17x - 20) = 180$

• Solve: $50 - 2x + 17x - 20 = 180$

• $15x + 30 = 180$

• $15x = 150$

• $x = 10$

Substitute $x$ back to find each angle:

• $50 - 2(10) = 30^{\circ}$

• $17(10) - 20 = 150^{\circ}$

Check: $30^{\circ} + 150^{\circ} = 180^{\circ}$

Summary Table: Complementary vs. Supplementary Angles

Type	Sum	Example	Formula
Complementary	$90^{\circ}$	$60^{\circ} + 30^{\circ}$	$x + y = 90^{\circ}$
Supplementary	$180^{\circ}$	$120^{\circ} + 60^{\circ}$	$x + y = 180^{\circ}$