Question

in the triangle below, with right angle $\angle c$, suppose that $m\angle b=(3x + 28)^{circ}$ and $m\angle d=(4x - 8)^{circ}$. find the degree measure of each angle in the triangle.

Explanation:

Step1: Use the triangle angle - sum property

In a right - triangle \( \triangle BCD\) with \( \angle C = 90^{\circ}\), we know that \(m\angle B+m\angle D+m\angle C=180^{\circ}\). Substituting the given expressions for \(m\angle B=(3x + 28)^{\circ}\), \(m\angle D=(4x - 8)^{\circ}\), and \(m\angle C = 90^{\circ}\) into the angle - sum formula:

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Step2: Solve for \(x\)

Subtract \(110\) from both sides of the equation \(7x+110 = 180\):

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Divide both sides by \(7\): \(x=\frac{70}{7}=10\)

Step3: Find \(m\angle B\)

Substitute \(x = 10\) into \(m\angle B=(3x + 28)^{\circ}\). Then \(m\angle B=(3\times10 + 28)^{\circ}=(30+28)^{\circ}=58^{\circ}\)

Step4: Find \(m\angle D\)

Substitute \(x = 10\) into \(m\angle D=(4x - 8)^{\circ}\). Then \(m\angle D=(4\times10-8)^{\circ}=(40 - 8)^{\circ}=32^{\circ}\)

Answer:

\(m\angle B = 58^{\circ}\), \(m\angle C=90^{\circ}\), \(m\angle D = 32^{\circ}\)