Question

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Explanation:

Step1: Recall triangle angle sum

The sum of angles in a triangle is \(180^\circ\). So, \(m\angle R + m\angle S + m\angle T = 180^\circ\).
Given \(m\angle R = 2x + 5\), \(m\angle S = x + 1\), \(m\angle T = 7x - 16\). Substitute into the equation:
\((2x + 5) + (x + 1) + (7x - 16) = 180\)

Step2: Simplify the equation

Combine like terms:
\(2x + x + 7x + 5 + 1 - 16 = 180\)
\(10x - 10 = 180\)

Step3: Solve for x

Add 10 to both sides:
\(10x = 180 + 10\)
\(10x = 190\)
Divide by 10:
\(x = \frac{190}{10} = 19\)

Step4: Find each angle

• \(m\angle R = 2x + 5 = 2(19) + 5 = 38 + 5 = 43^\circ\)
• \(m\angle S = x + 1 = 19 + 1 = 20^\circ\)
• \(m\angle T = 7x - 16 = 7(19) - 16 = 133 - 16 = 117^\circ\)

Answer:

\(x = 19\)
\(m\angle R = 43^\circ\)
\(m\angle S = 20^\circ\)
\(m\angle T = 117^\circ\)