Question

Rectangle MNOP is shown below. Aiden argues that MNQ is congruent to POQ. What triangle congruence criteria proves that MNQ is congruent to POQ? Select all that apply

1. angle-angle-side
2.angle-side-angle
3.side-side-angle
4.hypotenuse-angle
5.hypotenuse-leg​

Answer 1

Δ MNQ ≅ Δ POQ ( BY Angle Angle Side)

Explanation:

Given:

m∠NQM = 40°

m∠NQO = 100°

MNOP is a Rectangle

So we can say that:

Property of rectangle:

All angles of rectangle are 90°

Opposite sides of Rectangle are equal and parallel to each other

MN ≅ PO (opposite side of rectangle)

∠NMQ ≅ ∠OPQ (Both angles are 90°)

Now By Straight Angle property

m∠NQM + m∠NQO + m∠OQP = 180° (angles of straight line)

Substituting the values we get;

40° + 100° + m∠OQP = 180°

140° + m∠OQP = 180°

m∠OQP = 180° - 140° = 40°

So m∠OQP = m∠NQM = 40°

m∠OQP ≅ m∠NQM (Equals angles are congruent to each other)

Hence In Δ MNQ and Δ POQ

m∠OQP ≅ m∠NQM

∠NMQ ≅ ∠OPQ (Both angles are 90°)

MN ≅ PO (opposite side of rectangle)

Δ MNQ ≅ Δ POQ ( BY Angle Angle Side)

Hence Proved...