Question

unit 3 test: creating & verifying proofs

1. put each reason in its appropriate place. reasons may be used more than once.

given:
prove
a. def. of bisector b. def. of midpoint c. sss d. sas e. asa
f. aas g. given h. reflexive property i. cpctc

1. fill each box with the appropriate statement or reason. reasons may be used more

than once.
given:
y is the midpoint of
prove:
a. def. of midpoint b. def. of bisector c. given d. sss e.sas f. asa
g. aas h. hl i. corresponding angles j. alternate interior angles k.
l. m. n. o. p.

1. fill each box with the appropriate statement or reason. reasons may be used more

than once.
given:
prove:
a. def of midpoint b. sss c. sas d.asa e. aas f. hl
g. corresponding angles h. alternate interior angles i. vertical angles
j. k. l
m. n. o.

Explanation:

1.

• Statement 1: \(AC\) bisects \(\angle BAD\) is given information.
• Statement 2: By the definition of a bisector (\(A\)), if a line bisects an angle, it divides the angle into two equal parts. So \(\angle BAC=\angle DAC\).
• Statement 3: \(AC = AC\) is due to the reflexive property (\(H\)) (a segment is congruent to itself).
• Statement 4: \(\angle B\cong\angle D\) is given (\(G\)).
• Statement 5: For \(\triangle ABC\cong\triangle ADC\), we have two angles (\(\angle BAC=\angle DAC\) and \(\angle B\cong\angle D\)) and a non - included side (\(AC = AC\)), so by \(AAS\) (\(F\)).
• Statement 6: \(\overline{BC}\cong\overline{DC}\) is by \(CPCTC\) (\(I\)) (corresponding parts of congruent triangles are congruent).

• Statement 1: \(\overline{ZV}\parallel\overline{YU}\) is given (\(C\)).
• Statement 2: \(\angle Z\cong\angle WYU\) because of the corresponding angles postulate (\(I\)) (if two parallel lines are cut by a transversal, corresponding angles are congruent).
• Statement 3: \(\overline{VY}\parallel\overline{UW}\) is given (\(C\)).
• Statement 4: \(\angle V\cong\angle U\) (corresponding angles from \(\overline{VY}\parallel\overline{UW}\)).
• Statement 5: \(Y\) is the midpoint of \(ZW\) is given (\(C\)).
• Statement 6: \(\overline{ZY}\cong\overline{YW}\) by the definition of a midpoint (\(A\)).
• Statement 7: \(\triangle ZVY\cong\triangle YUW\) by \(ASA\) (\(F\)) (\(\angle Z\cong\angle WYU\), \(\overline{ZY}\cong\overline{YW}\), \(\angle V\cong\angle U\)).

• Statement 1: \(\overline{MQ}\parallel\overline{OP}\) is given (\(C\)).
• Statement 2: \(\angle Q\cong\angle P\) because of the alternate interior angles theorem (\(H\)) (if two parallel lines are cut by a transversal, alternate interior angles are congruent).
• Statement 3: \(\angle MNQ\cong\angle PNO\) because of the vertical angles theorem (\(I\)) (vertical angles are congruent).
• Statement 4: \(\overline{MQ}\cong\overline{OP}\) is given (\(C\)).
• Statement 5: \(\triangle MQN\cong\triangle OPN\) by \(AAS\) (\(E\)) (\(\angle Q\cong\angle P\), \(\angle MNQ\cong\angle PNO\), \(\overline{MQ}\cong\overline{OP}\)).

Answer:

1. \(G\)
2. \(A\)
3. \(H\)
4. \(G\)
5. \(F\)
6. \(I\)

2.