Question

in $\triangle abc$, if $m\angle a$ is seven more than four times $m\angle b$ and $m\angle c$ is eleven more than $m\angle b$, find the measure of $\angle a$.

Explanation:

Step1: Use the triangle angle - sum theorem

The sum of the interior angles of a triangle is \(180^{\circ}\). So, \(m\angle A + m\angle B+m\angle C=180^{\circ}\). Given \(m\angle A = 4x + 7\), \(m\angle B=x\), and \(m\angle C=x + 11\). Substitute these into the angle - sum formula: \((4x + 7)+x+(x + 11)=180\).

Step2: Simplify the left - hand side of the equation

Combine like terms: \(4x+x+x+7 + 11=180\), which simplifies to \(6x+18 = 180\).

Step3: Solve for \(x\)

Subtract 18 from both sides: \(6x=180 - 18\), so \(6x=162\). Then divide both sides by 6: \(x=\frac{162}{6}=27\).

Step4: Find \(m\angle A\)

Since \(m\angle A = 4x + 7\), substitute \(x = 27\) into the formula. \(m\angle A=4\times27+7\). First, calculate \(4\times27 = 108\), then \(108+7=115\).

Answer:

\(115^{\circ}\)