Question

write the letter of the property, definition, or postulate that justifies each statement. 1. if 2kl = kl + mn, then kl = mn 2. if ∠j and ∠k are vertical angles, then ∠j≅∠k 3. m is the midpoint of $overline{ab}$, then am = mb. 4. if q is between p and r, then pq + qr = pr 5. qr = qr 7. if m∠pqr = m∠rst, then ∠pqr≅∠rst 8. jk = lm, then $overline{jk}congoverline{lm}$ 9. if pq = tu, then tu = pq 10. if ∠pqr and ∠stu are complementary angles, then m∠pqr + m∠stu = 90°. 11 if m ∠a = m ∠b and m ∠b = m ∠c, then m ∠a = m ∠c 12 if 2xy = yz, then xy = ½yz 13 if ∠5 and ∠6 form a linear, then ∠5 and ∠6 are supplementary angles. 14 if $overline{ab}congoverline{bc}$ and $overline{bc}congoverline{ce}$, then $overline{ab}congoverline{ce}$ 15 if ∠x and ∠y are supplementary and ∠x and ∠z are supplementary, then ∠y≅∠z 16 if m∠abc = 90°, then ∠abc is a right angle. 17 if ∠1 and ∠2 form a right angle, then ∠1 and ∠2 are complementary angles. 18 if x = 5, then 2x = 10 20 if rs + tu = xy and tu = wv, then rs + wv = xy 21 if ab = cd, then ab + ef = cd + ef 22 if m∠3 + m∠4 = 180°, then ∠3 and ∠4 are supplementary angles. 23. find x. (2x - 6)° (x + 12)° x = 24. find x (2x - 6)° (x + 12)° x = a. addition property of equality b. subtraction property of equality c. multiplication property of equality d. division property of equality e. substitution property f. reflexive property (of = or ≅) g. symmetric property (of = or ≅) h. transitive property (of = or ≅) i. definition of congruence j. definition of midpoint k. segment addition postulate m. definition of angle bisector n. definition of complementary ∠s q. definition of supplementary ∠s r. definition of perpendicular s. definition of a right angle t. angle addition postulate u. vertical angles theorem w. complement theorem x. linear pair (supplement) theorem y. congruent complements theorem z. congruent supplements theorem

Explanation:

Step1: Analyze 1

Subtract $KL$ from both sides of $2KL = KL+MN$. This is the Subtraction Property of Equality. So the answer for 1 is B.

Step2: Analyze 2

Vertical angles are congruent by the Vertical Angles Theorem. So the answer for 2 is U.

Step3: Analyze 3

By the definition of a mid - point, if $M$ is the midpoint of $\overline{AB}$, then $AM = MB$. So the answer for 3 is J.

Step4: Analyze 4

If a point $Q$ is between two points $P$ and $R$, then $PQ + QR=PR$ by the Segment Addition Postulate. So the answer for 4 is K.

Step5: Analyze 5

$QR = QR$ is an example of the Reflexive Property. So the answer for 5 is F.

Step6: Analyze 7

If the measures of two angles are equal, then the angles are congruent by the Definition of Congruence. So the answer for 7 is I.

Step7: Analyze 8

If two segments have equal lengths, then they are congruent by the Definition of Congruence. So the answer for 8 is I.

Step8: Analyze 9

If $PQ = TU$, then $TU = PQ$ shows the Symmetric Property. So the answer for 9 is G.

Step9: Analyze 10

If two angles are complementary, then the sum of their measures is $90^{\circ}$ by the Definition of Complementary Angles. So the answer for 10 is N.

Step10: Analyze 11

If $m\angle A=m\angle B$ and $m\angle B = m\angle C$, then $m\angle A=m\angle C$ by the Transitive Property. So the answer for 11 is H.

Step11: Analyze 12

Divide both sides of $2XY = YZ$ by 2. This is the Division Property of Equality. So the answer for 12 is D.

Step12: Analyze 13

If two angles form a linear pair, then they are supplementary by the Linear Pair (Supplement) Theorem. So the answer for 13 is X.

Step13: Analyze 14

If $\overline{AB}\cong\overline{BC}$ and $\overline{BC}\cong\overline{CE}$, then $\overline{AB}\cong\overline{CE}$ by the Transitive Property of Congruence. So the answer for 14 is H.

Step14: Analyze 15

If $\angle X$ and $\angle Y$ are supplementary and $\angle X$ and $\angle Z$ are supplementary, then $\angle Y\cong\angle Z$ by the Congruent Supplements Theorem. So the answer for 15 is Z.

Step15: Analyze 16

If $m\angle ABC = 90^{\circ}$, then $\angle ABC$ is a right - angle by the Definition of a Right Angle. So the answer for 16 is S.

Step16: Analyze 17

If two angles form a right - angle, then they are complementary by the definition of complementary angles. So the answer for 17 is N.

Step17: Analyze 18

Multiply both sides of $x = 5$ by 2. This is the Multiplication Property of Equality. So the answer for 18 is C.

Step18: Analyze 20

Since $TU = WV$, substitute $WV$ for $TU$ in $RS + TU=XY$. This is the Substitution Property. So the answer for 20 is E.

Step19: Analyze 21

Add $EF$ to both sides of $AB = CD$. This is the Addition Property of Equality. So the answer for 21 is A.

Step20: Analyze 22

If the sum of the measures of two angles is $180^{\circ}$, then they are supplementary by the Definition of Supplementary Angles. So the answer for 22 is Q.

Step21: Analyze 23

Since vertical angles are congruent, we set $2x−6=x + 12$.
Subtract $x$ from both sides: $2x−x−6=x−x + 12$, which gives $x−6=12$.
Add 6 to both sides: $x=12 + 6=18$.

Step22: Analyze 24

Since vertical angles are congruent, we set $2x−6=x + 12$.
Subtract $x$ from both sides: $2x−x−6=x−x + 12$, which gives $x−6=12$.
Add 6 to both sides: $x=18$.

Answer:

1. B
2. U
3. J
4. K
5. F
6. I
7. I
8. G
9. N
10. H
11. D
12. X
13. H
14. Z
15. S
16. N
17. C
18. E
19. A
20. Q
21. 18
22. 18