Answer:
1.<3,<5 are supplementary and <4, <6 are supplementary -Exterior sides in opposite rays
2. <5=<6 and <1=<2 - Given
3. <3=<4 - Supplements to = angles
4. MN=MN - Reflexive
5. Triangle MNQ congruent to Triangle MNP - ASA
6. MQ=MP - CPCTE
Step-by-step explanation:
Its given that <1=<2, <5=<6
At point N, pairs <3,<5 and <4,<6 are supplementary angles as adjacent angles whose exterior sides are always supplementary,
hence ∠3+∠5=180° and ∠4+∠6=180°
⇒∠3=180°-∠5 and ∠4=180°-∠6
but its given that ∠5=∠6,
⇒∠3=∠4=180°-∠5
hence ∠3=∠4 (they supplement to equal angles)
Also from the figure we can see that MN=MN (reflexive relation)
Therefore, ΔMNQ≅ΔMNP (by ASA criteria) as
- ∠1=∠2 (given)
- MN=MN (reflexive relation)
- ∠3=∠4 (proved above)
therefore, MQ=MP (congruent parts of congruent triangles are equal,i.e. CPCTE)
