Question

in triangle abc, the angle c is six times as large as angle a. the measure of angle b is 52° greater than that of angle a. find the measure of the angles.
a. 16°, 68°, and 96°
b. 16°, 68°, and 112°
c. 16°, 84°, and 80°
d. 16°, 84°, and 96°

Explanation:

Step1: Set up variables

Let the measure of angle \(A\) be \(x\). Then angle \(C = 6x\), and angle \(B=x + 52\)

Step2: Use the triangle - angle sum formula

Since the sum of angles in a triangle is \(180^{\circ}\), we have \(x+(x + 52)+6x=180\)

Step3: Simplify the equation

Combine like terms: \(8x+52 = 180\)
Subtract \(52\) from both sides: \(8x=180 - 52=128\)
Divide both sides by \(8\): \(x = 16\)

Step4: Find the measures of other angles

Angle \(B=x + 52=16+52 = 68^{\circ}\)
Angle \(C=6x=6\times16 = 96^{\circ}\)

Answer:

A. \(16^{\circ},68^{\circ},\) and \(96^{\circ}\)