Question

decide whether enough information is given to prove that the triangles are congruent. if there is enough information, state the congruence postulate you would use.

1. $\triangle xyw, \triangle zwy$ 11. $\triangle mae, \triangle tae$ 12. $\triangle khj, \triangle klj$
2. $\triangle dka, \triangle tks$ 14. $\triangle env, \triangle lov$ 15. $\triangle tra, \triangle gra$

complete the proof by supplying the reasons.

1. given: $o$ is the midpoint of $\overline{mq}$. $o$ is the midpoint of $\overline{np}$. prove: $\triangle mon \cong \triangle qop$

Explanation:

Step1: Reason for Statement 1

Given

Step2: Reason for Statement 2

Definition of midpoint (If \(O\) is the mid - point of \(\overline{MQ}\), then \(MO = QO\) or \(\overline{MO}\cong\overline{QO}\))

Step3: Reason for Statement 3

Given

Step4: Reason for Statement 4

Definition of midpoint (If \(O\) is the mid - point of \(\overline{NP}\), then \(NO = PO\) or \(\overline{NO}\cong\overline{PO}\))

Step5: Reason for Statement 5

Vertical angles theorem (\(\angle MON\) and \(\angle QOP\) are vertical angles, and vertical angles are congruent)

Step6: Reason for Statement 6

\(SAS\) (Side - Angle - Side) congruence postulate (\(\overline{MO}\cong\overline{QO}\), \(\angle MON\cong\angle QOP\), \(\overline{NO}\cong\overline{PO}\))

Answer:

1. Given
2. Definition of midpoint
3. Given
4. Definition of midpoint
5. Vertical angles theorem
6. \(SAS\) congruence postulate