Question

Given that NPQR, MN = MR, and NPQR, fill in the reasons below to prove that AMPQ is isosceles.

a) NPQ = AMP, PQ = PQ
b) NPQ = AMP, MP = MP
c) NPQ = AMP, NP = MP
d) NPQ = AMP, NP = PQ

Answer 1

To prove that triangle AMPQ is isosceles, we used the congruency of triangles NPQ and AMP. With this congruency, by CPCTC, NP = MP, which indicates that triangle AMPQ has at least two congruent sides, hence it is isosceles.

Step-by-step explanation:

We are tasked with proving that triangle AMPQ is isosceles. To do so, we must show that two sides of the triangle are congruent. Given that MN = MR, and triangle NPQR is presumably congruent to triangle AMPR (though the full conditions for congruency are not provided, this is implied by the choices given), we will deduce the congruency of sides in triangle AMPQ.

• If ∆NPQ ≈ ∆AMP, then by CPCTC (Corresponding Parts of Congruent Triangles are Congruent), we know NP = MP.
• With NP = MP, we can state that triangle AMPQ has at least two congruent sides, thus making it isosceles.

The correct sequence of reasons leading to triangle AMPQ being isosceles is (c) NPQ = AMP, NP = MP. This shows that two sides of triangle AMPQ are congruent, meeting the definition of an isosceles triangle.