Question

1. in the accompanying diagram, $overline{ab}$ and $overline{cd}$ intersect at e, $mangle aec=(5x + 12)^{circ}$, and $mangle bed=(8x - 3)^{circ}$. find the number of degrees in the measure of $angle aec$.

Explanation:

Step1: Identify vertical - angle relationship

Vertical angles are equal. $\angle AEC$ and $\angle BED$ are vertical angles, so $m\angle AEC=m\angle BED$.
$5x + 12=8x - 3$

Step2: Solve the equation for $x$

Subtract $5x$ from both sides:
$12 = 8x-5x - 3$
$12=3x - 3$
Add 3 to both sides:
$12 + 3=3x$
$15 = 3x$
Divide both sides by 3:
$x=\frac{15}{3}=5$

Step3: Find the measure of $\angle AEC$

Substitute $x = 5$ into the expression for $m\angle AEC$:
$m\angle AEC=5x + 12$
$m\angle AEC=5\times5+12$
$m\angle AEC=25 + 12$
$m\angle AEC=37^{\circ}$

Answer:

$37^{\circ}$