Question

1. (1) if $overline{xy}$ bisects $angle wxz$, $mangle wxy = 4x - 1$, and $mangle yxz=2x + 15$, then find $mangle wxz$. (hint: use the definition of a segment or angle bisector to solve this problem.) a. 62 b. 78 c. 54 d. 45 e. none of the above 6. (1) if $l$ bisects $overline{de}$ at point $r$, $dr = 2y + 9$, and $re = 3y - 1$, then find $de$. (hint: use the definition of a segment or angle bisector to solve this problem.) a. 64 b. 76 c. 58 d. 49 e. none of the above 7. (1) complementary angles __ equal. a. are always b. are sometimes c. are never d. cannot be e. none of the above 8. (1) pairs of angles are always congruent. a. complementary b. supplementary c. acute d. vertical e. obtuse 9. (1) the between two points is the length of the line segment joining the points. a. linear pair b. angle c. distance d. midpoint e. bisector 10. (1) a perpendicular bisector is to a line segment and intersects the line segment at its __. a. perpendicular; endpoint b. adjacent; midpoint c. supplementary; angle d. perpendicular; midpoint e. exterior; linear pair

Explanation:

Step1: Solve for x in question 5

Since $\overline{XY}$ bisects $\angle WXZ$, then $m\angle WXY = m\angle YXZ$. So, $4x - 1=2x + 15$.
Subtract $2x$ from both sides: $4x-2x - 1=2x-2x + 15$, which simplifies to $2x-1 = 15$.
Add 1 to both sides: $2x-1 + 1=15 + 1$, so $2x=16$.
Divide both sides by 2: $x = 8$.
Then $m\angle WXZ=m\angle WXY + m\angle YXZ=(4x - 1)+(2x + 15)$. Substitute $x = 8$: $(4\times8-1)+(2\times8 + 15)=(32-1)+(16 + 15)=31+31 = 62$.

Step2: Solve for y in question 6

Since $l$ bisects $\overline{DE}$ at point $R$, then $DR=RE$. So, $2y + 9=3y - 1$.
Subtract $2y$ from both sides: $2y-2y + 9=3y-2y - 1$, which gives $9=y - 1$.
Add 1 to both sides: $y=10$.
$DE=DR + RE=(2y + 9)+(3y - 1)$. Substitute $y = 10$: $(2\times10 + 9)+(3\times10 - 1)=(20+9)+(30 - 1)=29+29 = 58$.

Step3: Answer question 7

Complementary angles are two angles whose sum is $90^{\circ}$. They are not always equal. For example, one could be $30^{\circ}$ and the other $60^{\circ}$. So the answer is C. are never (equal in general).

Step4: Answer question 8

Vertical angles are always congruent. Complementary, supplementary, acute and obtuse angles are not always congruent. So the answer is D. vertical.

Step5: Answer question 9

The distance between two points is the length of the line - segment joining the points. So the answer is C. distance.

Step6: Answer question 10

A perpendicular bisector is perpendicular to a line - segment and intersects the line - segment at its mid - point. So the answer is D. perpendicular; midpoint.

Answer:

1. A. 62
2. C. 58
3. C. are never
4. D. vertical
5. C. distance
6. D. perpendicular; midpoint