Question

1. (19) find m∠sru in the diagram.

a. 54
b. 45
c. 24
d. 48
e. none of the above

1. (18) find m∠tru in the diagram.

a. 180
b. 144
c. 132
d. 130
e. none of the above
refer to the diagram below for problems 17 - 19. from each given statement, select the definition, property, postulate, or theorem that leads to the statement you are asked to prove.

1. (18) given: lines ad and xy intersect at point o prove: ∠aox≅∠doy

a. pairs of vertical angles are congruent.
b. if two angles are a linear pair, then they are congruent.
c. betweenness of rays
d. definition of congruent angles
e. if two angles are complementary to the same angle, then they are congruent.

1. (20) given: ad⊥cb; ∠aox and ∠xob are adjacent angles prove: ∠aox and ∠xob are complementary

a. if two angles are complementary to the same angle, then they are congruent.
b. definition of complementary angles
c. if the exterior sides of a pair of adjacent angles are perpendicular, then the angles are complementary.
d. perpendicular lines intersect to form four right angles.
e. if two angles are a linear pair, then they are supplementary.

Explanation:

Step1: Find the value of x for questions 15 and 16

Since $(2x + 24)^{\circ}$ and $4x^{\circ}$ are vertical - angles, they are equal. So, $2x+24 = 4x$.
Subtract $2x$ from both sides: $24=4x - 2x$, which simplifies to $2x = 24$. Then $x = 12$.

Step2: Find $m\angle SRU$ (question 15)

$\angle SRU=4x^{\circ}$. Substitute $x = 12$ into the expression. So, $m\angle SRU=4\times12^{\circ}=48^{\circ}$.

Step3: Find $m\angle TRU$ (question 16)

$\angle TRU$ and $(2x + 24)^{\circ}$ are a linear - pair. A linear - pair of angles is supplementary, so $m\angle TRU=180-(2x + 24)$.
Substitute $x = 12$: $m\angle TRU=180-(2\times12 + 24)=180-(24 + 24)=180 - 48=132^{\circ}$.

Step4: Answer question 17

Since $\angle AOX$ and $\angle DOY$ are vertical angles, by the property that pairs of vertical angles are congruent, the correct answer is A.

Step5: Answer question 18

Given $AD\perp CB$ and $\angle AOX$ and $\angle XOB$ are adjacent angles. If the exterior sides of a pair of adjacent angles are perpendicular, then the angles are complementary. So the correct answer is C.

Answer:

1. D. 48
2. C. 132
3. A. Pairs of vertical angles are congruent.
4. C. If the exterior sides of a pair of adjacent angles are perpendicular, then the angles are complementary.