Question

1. in \\( \triangle a b c \\), if \\( m \angle a \\) is thirteen less than \\( m \angle c \\) and \\( m \angle b \\) is eleven less than four times \\( m \angle c \\), find the measure of each angle. \\( m \angle a= \\) \\( m \angle b= \\) \\( m \angle c= \\)

Explanation:

Step1: Define variables

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

Step2: Use the triangle - angle sum theorem

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:

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