Question

∠q and ∠r are complementary. the measure of ∠q is 26° less than the measure of ∠r. find the measure of each angle.

m∠q =

m∠r =

Explanation:

Step1: Define variables and complementary angle property

Let \( m\angle R = x \). Then \( m\angle Q = x - 26^\circ \) (since \( \angle Q \) is \( 26^\circ \) less than \( \angle R \)). Complementary angles sum to \( 90^\circ \), so \( m\angle Q + m\angle R = 90^\circ \). Substituting the expressions for the angles: \( (x - 26^\circ) + x = 90^\circ \).

Step2: Solve for \( x \) (measure of \( \angle R \))

Combine like terms: \( 2x - 26^\circ = 90^\circ \). Add \( 26^\circ \) to both sides: \( 2x = 90^\circ + 26^\circ = 116^\circ \). Divide both sides by 2: \( x = \frac{116^\circ}{2} = 58^\circ \). So \( m\angle R = 58^\circ \).

Step3: Find measure of \( \angle Q \)

Substitute \( x = 58^\circ \) into \( m\angle Q = x - 26^\circ \): \( m\angle Q = 58^\circ - 26^\circ = 32^\circ \).

Answer:

\( m\angle Q = 32^\circ \), \( m\angle R = 58^\circ \)