Question

1. in △abc, if (overline{ac}congoverline{cb}), (mangle a=(3x + 18)^{circ}), (mangle b=(7x - 58)^{circ}), and (mangle c=(2x - 6)^{circ}), find the value of (x) and the measure of each angle.(x=)(mangle a=)(mangle b=)(mangle c=)13. in △qrs, if (overline{qr}congoverline{rs}), (mangle q=(8x - 17)^{circ}), (mangle r=(19x + 4)^{circ}), and (mangle s=(5x + 1)^{circ}), find the value of (x) and the measure of each angle.(x=)(mangle q=)(mangle r=)(mangle s=)

Explanation:

Step1: Use the property of isosceles triangle for problem 12

Since \(AC = CB\) in \(\triangle ABC\), \(\angle A=\angle B\). So \(3x + 18=7x-58\).

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Then \(\angle A=3x + 18=3\times19+18=57 + 18=75^{\circ}\), \(\angle B = 7x-58=7\times19-58=133 - 58=75^{\circ}\), \(\angle C=2x-6=2\times19-6=38 - 6=32^{\circ}\)

Step2: Use the property of isosceles triangle for problem 13

Since \(QR = RS\) in \(\triangle QRS\), \(\angle Q=\angle S\). So \(8x-17=5x + 1\)

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Then \(\angle Q=8x-17=8\times6-17=48-17 = 31^{\circ}\), \(\angle S=5x + 1=5\times6+1=30 + 1=31^{\circ}\), \(\angle R=19x + 4=19\times6+4=114+4 = 118^{\circ}\)

Answer:

For problem 12: \(x = 19\), \(m\angle A=75^{\circ}\), \(m\angle B=75^{\circ}\), \(m\angle C=32^{\circ}\)

For problem 13: \(x = 6\), \(m\angle Q=31^{\circ}\), \(m\angle R=118^{\circ}\), \(m\angle S=31^{\circ}\)