Question

the angle measurements in the diagram are represented by the following expressions.

$\angle a = 5x + 34^{\circ}$ $\angle b = 2x + 76^{\circ}$

solve for $x$ and then find the measure of $\angle a$:

$\angle a = \square ^ {\circ}$

Explanation:

Step1: Use the property of same - side interior angles

Since the two lines are parallel and cut by a transversal, \(\angle A+\angle B = 180^{\circ}\).
So, \((5x + 34)+(2x+76)=180\).

Step2: Simplify the equation

Combine like terms: \(5x+2x+34 + 76=180\), which gives \(7x+110 = 180\).
Subtract \(110\) from both sides: \(7x=180 - 110\), so \(7x=70\).
Divide both sides by \(7\): \(x=\frac{70}{7}=10\).

Step3: Find the measure of \(\angle A\)

Substitute \(x = 10\) into the expression for \(\angle A\): \(\angle A=5x+34\).
\(\angle A=5\times10 + 34=50+34=84\).

Answer:

\(84\)