Question

1. ∠g and ∠h are complementary angles. if ( mangle g = (6x - 15)^circ ) and ( mangle h = (3x + 6)^circ ), find ( mangle h ). 35. ∠1 and ∠2 are vertical angles. if ( mangle 1 = (5x + 12)^circ ) and ( mangle 2 = (6x - 11)^circ ), find ( mangle 1 ).

Explanation:

Problem 34

Step1: Recall complementary angles sum to 90°

Since \( \angle G \) and \( \angle H \) are complementary, \( m\angle G + m\angle H = 90^\circ \). Substitute the given expressions: \( (6x - 15) + (3x + 6) = 90 \).

Step2: Solve for x

Combine like terms: \( 9x - 9 = 90 \). Add 9 to both sides: \( 9x = 99 \). Divide by 9: \( x = 11 \).

Step3: Find \( m\angle H \)

Substitute \( x = 11 \) into \( m\angle H = (3x + 6)^\circ \): \( 3(11) + 6 = 33 + 6 = 39 \).

Step1: Recall vertical angles are equal

Since \( \angle 1 \) and \( \angle 2 \) are vertical angles, \( m\angle 1 = m\angle 2 \). Substitute the given expressions: \( 5x + 12 = 6x - 11 \).

Step2: Solve for x

Subtract \( 5x \) from both sides: \( 12 = x - 11 \). Add 11 to both sides: \( x = 23 \).

Step3: Find \( m\angle 1 \)

Substitute \( x = 23 \) into \( m\angle 1 = (5x + 12)^\circ \): \( 5(23) + 12 = 115 + 12 = 127 \).

Answer:

\( 39^\circ \)

Problem 35