Question

in the figure, m∠2 = 92 and m∠12 = 74. find the measure of each angle. tell which postulate(s) or theorem(s) you used.

1. ∠10
2. ∠8
3. ∠9
4. ∠5
5. ∠11
6. ∠13
7. find the value of the variable(s) in each figure. explain your reasoning.

(9x + 12)°
3x°
(4y - 10)°

1. find the value of the variable(s) in each figure. explain your reasoning.

(5y - 4)°
3y°
(2x + 13)°

Explanation:

Step1: Identify vertical - angle relationship

Vertical angles are congruent. $\angle2$ and $\angle8$ are vertical angles. Since $m\angle2 = 92$, then $m\angle8=92$ (Vertical - Angles Theorem).

Step2: Identify corresponding - angle relationship

$\angle12$ and $\angle10$ are corresponding angles. Since $m\angle12 = 74$, then $m\angle10 = 74$ (Corresponding - Angles Postulate).

Step3: Find $\angle9$

$\angle9$ and $\angle10$ are supplementary (linear - pair). Since $m\angle10 = 74$, then $m\angle9=180 - 74=106$ (Linear - Pair Postulate).

Step4: Find $\angle5$

$\angle5$ and $\angle12$ are alternate - exterior angles. Since $m\angle12 = 74$, then $m\angle5 = 74$ (Alternate - Exterior Angles Theorem).

Step5: Find $\angle11$

$\angle11$ and $\angle12$ are supplementary (linear - pair). Since $m\angle12 = 74$, then $m\angle11=180 - 74 = 106$ (Linear - Pair Postulate).

Step6: Find $\angle13$

$\angle13$ and $\angle12$ are vertical angles. Since $m\angle12 = 74$, then $m\angle13 = 74$ (Vertical - Angles Theorem).

Step7: Solve for $x$ in the first variable - figure

$(9x + 12)$ and $3x$ are vertical angles. So, $9x+12 = 3x$.
Subtract $3x$ from both sides: $9x-3x + 12=3x-3x$, which gives $6x+12 = 0$.
Subtract 12 from both sides: $6x+12 - 12=0 - 12$, so $6x=-12$.
Divide both sides by 6: $x=-2$.
$(4y - 10)$ and $3x$ are corresponding angles. Substitute $x = - 2$, so $3x=-6$.
Then $4y-10=-6$.
Add 10 to both sides: $4y-10 + 10=-6 + 10$, so $4y = 4$.
Divide both sides by 4: $y = 1$.

Step8: Solve for $y$ in the second variable - figure

$(5y - 4)$ and $3y$ are supplementary. So, $(5y - 4)+3y=180$.
Combine like - terms: $5y+3y-4 = 180$, which gives $8y-4 = 180$.
Add 4 to both sides: $8y-4 + 4=180 + 4$, so $8y=184$.
Divide both sides by 8: $y = 23$.
$(2x + 13)$ and $3y$ are corresponding angles. Substitute $y = 23$, so $3y = 69$.
Then $2x+13 = 69$.
Subtract 13 from both sides: $2x+13 - 13=69 - 13$, so $2x=56$.
Divide both sides by 2: $x = 28$.

Answer:

1. $m\angle10 = 74$ (Corresponding - Angles Postulate)
2. $m\angle8 = 92$ (Vertical - Angles Theorem)
3. $m\angle9 = 106$ (Linear - Pair Postulate)
4. $m\angle5 = 74$ (Alternate - Exterior Angles Theorem)
5. $m\angle11 = 106$ (Linear - Pair Postulate)
6. $m\angle13 = 74$ (Vertical - Angles Theorem)
7. $x=-2,y = 1$
8. $x = 28,y = 23$