Question

Triangle A C B is a right triangle. Angle C A B is 90 degrees, angle A B C is (4 x) degrees, and angle A C B is (7 x minus 20) degrees.
What is the m∠ACB?

10°
50°
90°
180°

Answer 1

In a right triangle, the sum of the angles is always 180 degrees. We know that angle \( CAB \) is 90 degrees. So we can write the equation:

\[
m∠CAB + m∠ABC + m∠ACB = 180°
\]

Substituting the known angle:

\[
90° + (4x) + (7x - 20°) = 180°
\]

Combine the terms:

\[
90° + 4x + 7x - 20° = 180°
\]
\[
70° + 11x = 180°
\]

Now, isolate \( 11x \):

\[
11x = 180° - 70°
\]
\[
11x = 110°
\]

Now, solve for \( x \):

\[
x = \frac{110°}{11} = 10°
\]

Now, we can find the measures of angles \( ABC \) and \( ACB \):

1. Calculate \( m∠ABC \):
\[
m∠ABC = 4x = 4(10°) = 40°
\]

2. Calculate \( m∠ACB \):
\[
m∠ACB = 7x - 20° = 7(10°) - 20° = 70° - 20° = 50°
\]

Therefore, the measure of angle \( ACB \) (or \( m∠ACB \)) is \( \boxed{50°} \).