Examples - Angle Relationships Find the messures of two complementary angles if the measure of the larger angle is five more than four times the measure of the smaller angle. The difference between the measures of two supplementary angles is \( 18^{\circ} \). The measure of the smaller angle is \( \quad \) and the measure of the larger angle is

Let's solve each problem step by step. ### Problem 1: Complementary Angles Let the measure of the smaller angle be \( x \). According to the problem, the larger angle is five more than four times the smaller angle, which can be expressed as: \[ \text{Larger angle} = 4x + 5 \] Since the angles are complementary, their measures add up to \( 90^\circ \): \[ x + (4x + 5) = 90 \] Now, combine like terms: \[ 5x + 5 = 90 \] Next, subtract 5 from both sides: \[ 5x = 85 \] Now, divide by 5: \[ x = 17 \] So, the measure of the smaller angle is \( 17^\circ \). Now, we can find the larger angle: \[ \text{Larger angle} = 4(17) + 5 = 68 + 5 = 73 \] Thus, the measures of the two complementary angles are: - Smaller angle: \( 17^\circ \) - Larger angle: \( 73^\circ \) ### Problem 2: Supplementary Angles Let the measure of the smaller angle be \( y \). The larger angle can then be expressed as: \[ \text{Larger angle} = y + 18 \] Since the angles are supplementary, their measures add up to \( 180^\circ \): \[ y + (y + 18) = 180 \] Combine like terms: \[ 2y + 18 = 180 \] Now, subtract 18 from both sides: \[ 2y = 162 \] Next, divide by 2: \[ y = 81 \] So, the measure of the smaller angle is \( 81^\circ \). Now, we can find the larger angle: \[ \text{Larger angle} = 81 + 18 = 99 \] Thus, the measures of the two supplementary angles are: - Smaller angle: \( 81^\circ \) - Larger angle: \( 99^\circ \) ### Summary of Results 1. Complementary angles: \( 17^\circ \) and \( 73^\circ \) 2. Supplementary angles: \( 81^\circ \) and \( 99^\circ \)