Step-by-step explanation:
The proof can be had by making use of the AAS congruence postulate (twice) and CPCTC.
We start by showing ΔPQY≅ΔPRX, then by showing ΔXQN≅ΔYRN. The proof is then a result of CPCTC.
Proof
1. PQ≅PR, ∠Q≅∠R . . . . given
2. ∠P≅∠P . . . . reflexive property of congruence
3. ΔPQY≅ΔPRX . . . . AAS congruence postulate
4. PX≅PY . . . . CPCTC
5. PX+XQ=PQ, PY+YR=PR . . . . segment sum theorem
6. PX+XQ = PY +YR . . . . substitution property
7. PX +XQ = PX +YR . . . . substitution property
8. XQ = YR . . . . subtraction property of equality
9. ∠XNQ≅∠YNR . . . . vertical angles are congruent
10. ΔXNQ≅ΔYNR . . . . AAS congruence postulate
11. XN ≅ YN . . . . CPCTC
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Additional comment
You probably did steps 1-3 in part (a) of the problem.