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In the given figure, if MR ≅ PN; NR ≅ MP, prove that ∆NOR
≅ ∆MOP.
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| Photo by Math Principles in Everyday Life |
Consider the given figure above
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| Photo by Math Principles in Everyday Life |
Proof:
1. Statement: MR ≅ PN and NR ≅ MP.
Reason: Given items.
2. Statement: PR ≅ PR
Reason: Reflexive property of congruence.
3. Statement: ∆MRP ≅ ∆NPR
Reason: Side Side Side (SSS) Postulate.
4. Statement: ∠MPR ≅ ∠NRP
∠PMR ≅ ∠PNR
∠PRM ≅ ∠RPN
Reason: Since ∆MRP ≅ ∆NPR, then all interior angles of a triangle are congruent to all interior angles of other triangle.
5. Statement: ∠PRM ≅ ∠4 and ∠RPN ≅ ∠3.
Reason: Reflexive property of congruence.
6. Statement: ∠PRM ≅ ∠RPN ≅ ∠3 ≅ ∠4
Reason: Transitive property of congruence.
7. Statement: ∆POR is an isosceles triangle.
Reason: The two angles of an isosceles triangle are congruent. Hence, ∠3 ≅ ∠4 at the base.
8. Statement: OP ≅ OR
Reason: Since ∆POR is an isosceles triangle, then the two sides of an isosceles triangle are congruent.
9. Statement: ∠MPR = ∠1 + ∠3 and ∠NRP = ∠2 + ∠4
Reason: Addition property of angles.
10. Statement: ∠1 ≅ ∠2
Reason: If ∠MPR ≅ ∠NRP and ∠3 ≅ ∠4, then it follows that ∠1 ≅ ∠2.
11. Statement: ∆MOP ≅ ∆NOR
Reason: Side Angle Side (SAS) Postulate.