Question

in the triangle below, with right angle ∠b, suppose that m∠c = (4x - 8)° and m∠d = (3x + 21)°. find the degree measure of each angle in the triangle. m∠b = \\(\square\\)° m∠c = \\(\square\\)° m∠d = \\(\square\\)°

Explanation:

Step1: Identify triangle type

It's a right triangle at \( \angle B \), so \( m\angle B = 90^\circ \). The sum of angles in a triangle is \( 180^\circ \), so \( m\angle C + m\angle D + m\angle B = 180^\circ \). Substituting \( m\angle B = 90^\circ \), we get \( (4x - 8) + (3x + 21) + 90 = 180 \).

Step2: Solve for \( x \)

Simplify the equation: \( 4x - 8 + 3x + 21 + 90 = 180 \)
Combine like terms: \( 7x + 103 = 180 \)
Subtract 103: \( 7x = 77 \)
Divide by 7: \( x = 11 \)

Step3: Find \( m\angle C \)

Substitute \( x = 11 \) into \( m\angle C = (4x - 8)^\circ \):
\( 4(11) - 8 = 44 - 8 = 36^\circ \)

Step4: Find \( m\angle D \)

Substitute \( x = 11 \) into \( m\angle D = (3x + 21)^\circ \):
\( 3(11) + 21 = 33 + 21 = 54^\circ \)

Answer:

\( m\angle B = \boldsymbol{90}^\circ \)
\( m\angle C = \boldsymbol{36}^\circ \)
\( m\angle D = \boldsymbol{54}^\circ \)