Question

∠a=(4x + 4)°
∠g=(10x + 8)°
establish an equation that represents the relationship in the diagram.
solve the equation for the variable.
what are the measures of ∠a and ∠g?
in the diagram, line g is parallel to line k, and line t intersects lines g and k.
what angle pair is represented? how do you describe the angle measures?
(-3x + 165)°
g
(6x + 30)°
k
t
write an equation to solve for the value of x.
solve the equation for the variable.
what is the measure of the angles in the diagram?

Explanation:

Step1: Identify angle - relationship

Since the lines are parallel, the corresponding angles are equal. For the first part with $\angle A=(4x + 4)^{\circ}$ and $\angle G=(10x+8)^{\circ}$, we set up the equation $4x + 4=10x + 8$.

Step2: Solve the equation for $x$

Subtract $4x$ from both sides:
$4=10x - 4x+8$
$4 = 6x+8$
Subtract 8 from both sides:
$4-8=6x$
$-4 = 6x$
Divide both sides by 6:
$x=-\frac{4}{6}=-\frac{2}{3}$

Step3: Find the measures of $\angle A$ and $\angle G$

For $\angle A$:
$\angle A=4x + 4=4\times(-\frac{2}{3})+4=\frac{-8 + 12}{3}=\frac{4}{3}=1\frac{1}{3}^{\circ}$
For $\angle G$:
$\angle G=10x + 8=10\times(-\frac{2}{3})+8=\frac{-20 + 24}{3}=\frac{4}{3}=1\frac{1}{3}^{\circ}$

For the second part:

Step1: Identify angle - relationship

The angles $(-3x + 165)^{\circ}$ and $(6x + 30)^{\circ}$ are corresponding angles (because line $g$ is parallel to line $k$ and line $t$ is a transversal), so we set up the equation $-3x+165=6x + 30$.

Step2: Solve the equation for $x$

Add $3x$ to both sides:
$165=6x+3x + 30$
$165=9x + 30$
Subtract 30 from both sides:
$165 - 30=9x$
$135=9x$
Divide both sides by 9:
$x = 15$

Step3: Find the measure of the angles

For the first angle:
$-3x+165=-3\times15 + 165=-45+165 = 120^{\circ}$
For the second angle:
$6x + 30=6\times15+30=90 + 30=120^{\circ}$

Answer:

For the first part: $x=-\frac{2}{3}$, $\angle A = 1\frac{1}{3}^{\circ}$, $\angle G=1\frac{1}{3}^{\circ}$
For the second part: $x = 15$, the measure of both angles is $120^{\circ}$