Question

1. in △abc, if m∠a is thirteen less than m∠c and m∠b is eleven less than four times m∠c, find the measure of each angle.

m∠a =
m∠b =
m∠c =

Explanation:

Step1: Set up variables

Let \(m\angle C = x\). Then \(m\angle A=x - 13\) and \(m\angle B = 4x-11\).

Step2: Use the angle - sum property of a triangle

Since the sum of the interior angles of a triangle is \(180^{\circ}\), we have the equation \((x-13)+(4x - 11)+x=180\).
Simplify the left - hand side: \(x-13 + 4x-11+x=(1 + 4+1)x-(13 + 11)=6x-24\).
So, \(6x-24 = 180\).

Step3: Solve for \(x\)

Add \(24\) to both sides of the equation: \(6x-24+24=180 + 24\), which gives \(6x=204\).
Divide both sides by \(6\): \(x=\frac{204}{6}=34\).

Step4: Find \(m\angle A\)

Substitute \(x = 34\) into \(m\angle A=x - 13\). Then \(m\angle A=34-13 = 21^{\circ}\).

Step5: Find \(m\angle B\)

Substitute \(x = 34\) into \(m\angle B = 4x-11\). Then \(m\angle B=4\times34-11=136 - 11=125^{\circ}\).

Answer:

\(m\angle A = 21^{\circ}\), \(m\angle B=125^{\circ}\), \(m\angle C = 34^{\circ}\)