Question

16.) recall that the sum of the measures of the angles of a triangle is 180°. in the triangle below, angle c has the same measure as angle b, and angle a measures 42° less than angle b. find the measure of each angle.

Explanation:

Step1: Define variables for angles

Let the measure of angle \( B \) be \( x^\circ \). Then, since angle \( C \) has the same measure as angle \( B \), angle \( C = x^\circ \). Angle \( A \) measures \( 42^\circ \) less than angle \( B \), so angle \( A=(x - 42)^\circ \).

Step2: Use triangle angle sum formula

The sum of the angles of a triangle is \( 180^\circ \), so we set up the equation:
\( (x - 42)+x + x=180 \)

Step3: Simplify and solve the equation

Combine like terms:
\( 3x- 42=180 \)
Add 42 to both sides:
\( 3x=180 + 42=222 \)
Divide both sides by 3:
\( x=\frac{222}{3}=74 \)

Step4: Find the measure of each angle

• Angle \( B = x = 74^\circ \)
• Angle \( C = x = 74^\circ \)
• Angle \( A=x - 42=74 - 42 = 32^\circ \)

Answer:

Angle \( A = 32^\circ \), Angle \( B = 74^\circ \), Angle \( C = 74^\circ \)