Question

in the triangle below, with right angle \\( \angle b \\), suppose that \\( m \angle a = (4x + 24)^\circ \\) and \\( m \angle c = (3x + 24)^\circ \\).

find the degree measure of each angle in the triangle.
\\( m \angle a = \square^\circ \\)
\\( m \angle b = \square^\circ \\)
\\( m \angle c = \square^\circ \\)

Explanation:

Step1: Recall triangle angle sum

In a triangle, the sum of angles is \(180^\circ\). Since \(\angle B\) is a right angle, \(m\angle B = 90^\circ\). So, \(m\angle A + m\angle B + m\angle C = 180^\circ\). Substituting the given expressions and \(m\angle B = 90^\circ\), we get \((4x + 24) + 90 + (3x + 24) = 180\).

Step2: Solve for \(x\)

Combine like terms: \(4x + 3x + 24 + 90 + 24 = 180\) → \(7x + 138 = 180\). Subtract 138 from both sides: \(7x = 180 - 138 = 42\). Divide by 7: \(x = \frac{42}{7} = 6\).

Step3: Find \(m\angle A\)

Substitute \(x = 6\) into \(m\angle A = (4x + 24)^\circ\): \(4(6) + 24 = 24 + 24 = 48^\circ\).

Step4: Find \(m\angle C\)

Substitute \(x = 6\) into \(m\angle C = (3x + 24)^\circ\): \(3(6) + 24 = 18 + 24 = 42^\circ\).

Step5: Confirm \(m\angle B\)

Since \(\angle B\) is a right angle, \(m\angle B = 90^\circ\).

Answer:

\(m\angle A = \boldsymbol{48}^\circ\)
\(m\angle B = \boldsymbol{90}^\circ\)
\(m\angle C = \boldsymbol{42}^\circ\)