Question

in the triangle below, with right angle ∠r, suppose that m∠q = (3x + 13)° and m∠s = (2x + 2)°. find the degree measure of each angle in the triangle. m∠q = \\(\square\\)° m∠r = \\(\square\\)° m∠s = \\(\square\\)°

Explanation:

Step1: Sum of angles in triangle

In a triangle, the sum of interior angles is \(180^\circ\). Since \(\angle R\) is a right angle, \(m\angle R = 90^\circ\). So, \(m\angle Q + m\angle R + m\angle S = 180^\circ\). Substituting the given expressions: \((3x + 13) + 90 + (2x + 2) = 180\).

Step2: Solve for \(x\)

Combine like terms: \(3x + 2x + 13 + 2 + 90 = 180\) → \(5x + 105 = 180\). Subtract 105 from both sides: \(5x = 180 - 105 = 75\). Divide by 5: \(x = \frac{75}{5} = 15\).

Step3: Find \(m\angle Q\)

Substitute \(x = 15\) into \(m\angle Q = (3x + 13)^\circ\): \(3(15) + 13 = 45 + 13 = 58^\circ\).

Step4: Find \(m\angle S\)

Substitute \(x = 15\) into \(m\angle S = (2x + 2)^\circ\): \(2(15) + 2 = 30 + 2 = 32^\circ\).

Step5: Confirm \(m\angle R\)

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

Answer:

\(m\angle Q = 58^\circ\), \(m\angle R = 90^\circ\), \(m\angle S = 32^\circ\)