Questions: In triangle ABC, 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=

In triangle ABC, 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=

Transcript text: In $\triangle ABC$, 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. \[ \begin{array}{l} m \angle A= \\ m \angle B= \\ m \angle C= \end{array} \]

Solution

Solution Steps

Step 1: Define the variables

Let $ m \angle C = x $. Then:

$ m \angle A = x - 13 $ (since $ m \angle A $ is thirteen less than $ m \angle C $).
$ m \angle B = 4x - 11 $ (since $ m \angle B $ is eleven less than four times $ m \angle C $).

Step 2: Use the triangle angle sum property

In any triangle, the sum of the interior angles is $ 180^\circ $. Therefore: \[ m \angle A + m \angle B + m \angle C = 180^\circ \] Substitute the expressions for $ m \angle A $, $ m \angle B $, and $ m \angle C $: \[ (x - 13) + (4x - 11) + x = 180 \]

Step 3: Simplify and solve for $ x $

Combine like terms: \[ x - 13 + 4x - 11 + x = 180 \] \[ 6x - 24 = 180 \] Add $ 24 $ to both sides: \[ 6x = 204 \] Divide both sides by $ 6 $: \[ x = 34 \]

Step 4: Find the measures of the angles