Question

in the diagram, g || h, m∠1=(4x + 36)°, and m∠2=(3x - 3)°. what is the measure of ∠3? 21° 60° 120° 159°

Explanation:

Step1: Use vertical - angle property

Since $\angle1$ and $\angle2$ are vertical angles, $m\angle1 = m\angle2$. So, $4x + 36=3x - 3$.

Step2: Solve for $x$

Subtract $3x$ from both sides: $4x-3x + 36=3x-3x - 3$, which gives $x+36=-3$. Then subtract 36 from both sides: $x=-3 - 36=-39$.

Step3: Find $m\angle2$

Substitute $x = - 39$ into the expression for $m\angle2$: $m\angle2=3x - 3=3\times(-39)-3=-117 - 3=-120$. But angle measures are non - negative, we made a mistake above. $\angle1$ and $\angle2$ are supplementary (because $g\parallel h$), so $m\angle1+m\angle2 = 180^{\circ}$.

Step4: Set up the correct equation

$(4x + 36)+(3x - 3)=180$. Combine like terms: $4x+3x+36 - 3=180$, $7x+33 = 180$.

Step5: Solve for $x$

Subtract 33 from both sides: $7x=180 - 33 = 147$. Then divide by 7: $x = 21$.

Step6: Find $m\angle2$

$m\angle2=3x - 3=3\times21-3=63 - 3=60^{\circ}$.

Step7: Find $m\angle3$

$\angle2$ and $\angle3$ are supplementary (linear pair), so $m\angle3=180 - m\angle2$. Substitute $m\angle2 = 60^{\circ}$, then $m\angle3=120^{\circ}$.

Answer:

$120^{\circ}$