Question

practice & problem solving
find the value of each variable and the measure of each labeled angle.

1. (2x + 22)° (3x - 6)°
2. (2x + 32)° (3x - 5)°
3. construct arguments write a proof. given: m∠tuv = 90 prove: x = 12
4. construct arguments write an indirect proof by proving the contrapositive. given: gj = 48 prove: x ≠ 12

Explanation:

Step1: Identify vertical - angle property

Vertical angles are equal. In problem 44, the two given angles $(2x + 22)^{\circ}$ and $(3x-6)^{\circ}$ are vertical angles, so $2x + 22=3x - 6$.

Step2: Solve for $x$

Subtract $2x$ from both sides: $22=x - 6$. Then add 6 to both sides, we get $x = 28$.

Step3: Find the angle measure

Substitute $x = 28$ into $(2x + 22)^{\circ}$, we have $2\times28+22=56 + 22=78^{\circ}$.
In problem 45, the two given angles $(2x + 32)^{\circ}$ and $(3x-5)^{\circ}$ are vertical angles. So $2x+32 = 3x - 5$.

Step4: Solve for $x$ in problem 45

Subtract $2x$ from both sides: $32=x - 5$. Add 5 to both sides, $x=37$.

Step5: Find the angle measure in problem 45

Substitute $x = 37$ into $(2x + 32)^{\circ}$, we get $2\times37+32=74 + 32 = 106^{\circ}$.

Answer:

For problem 44: $x = 28$, angle measure is $78^{\circ}$. For problem 45: $x = 37$, angle measure is $106^{\circ}$.