Examples - Angle Relationships Find the measures 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 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 \( x \). Then, the measure of the larger angle can be expressed as: \[ 4x + 5 \] 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 \( 5 \): \[ x = 17 \] Now that we have the measure of the smaller angle, we can find the larger angle: \[ 4x + 5 = 4(17) + 5 = 68 + 5 = 73 \] 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 \( y \). Then, the measure of the larger angle can be expressed as: \[ y + 18 \] 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: \[ 2y = 162 \] Now, divide by \( 2 \): \[ y = 81 \] Now that we have the measure of the smaller angle, we can find the larger angle: \[ y + 18 = 81 + 18 = 99 \] 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 \( 99^\circ \)