Question

in the triangle below, with right angle $\angle r$, suppose that $m\angle p=(2x + 24)\degree$ and $m\angle q=(5x - 11)\degree$.
find the degree measure of each angle in the triangle.
$m\angle p=\square\degree$
$m\angle q=\square\degree$
$m\angle r = 90\degree$

Explanation:

Step1: Use triangle angle sum property

In a triangle, the sum of interior angles is \(180^\circ\). Since \(\angle R = 90^\circ\), we have \(m\angle P + m\angle Q + m\angle R = 180^\circ\). Substituting the given angles: \((2x + 24) + (5x - 11) + 90 = 180\).

Step2: Simplify the equation

Combine like terms: \(2x + 5x + 24 - 11 + 90 = 180\) → \(7x + 103 = 180\).

Step3: Solve for x

Subtract 103 from both sides: \(7x = 180 - 103 = 77\). Then divide by 7: \(x = \frac{77}{7} = 11\).

Step4: Find \(m\angle P\)

Substitute \(x = 11\) into \(m\angle P = (2x + 24)^\circ\): \(2(11) + 24 = 22 + 24 = 46^\circ\).

Step5: Find \(m\angle Q\)

Substitute \(x = 11\) into \(m\angle Q = (5x - 11)^\circ\): \(5(11) - 11 = 55 - 11 = 44^\circ\).

Answer:

\(m\angle P = 46^\circ\)
\(m\angle Q = 44^\circ\)
\(m\angle R = 90^\circ\)