Question

The diagram shows a triangle PQR.
XYZR is a parallelogram where
X is the midpoint of RP,
Y is the midpoint of PQ,
and Z is the midpoint of QR.

Prove that triangle PYX is congruent
to triangle YQZ.

Total marks: 4

Answer 1

Answer:

Triangle ΔPYX is congruent to triangle ΔYQZ by Angle-Angle-Side condition of congruency

Step-by-step explanation:

The given

Statement                ${}$                              Reason

PQR is a triangle               ${}$                    Given

XYZR is a parallelogram           ${}$            Given

X is the midpoint of RP           ${}$              Given

Y  is the midpoint of PQ          ${}$              Given

Z  is the midpoint of QR          ${}$              Given

Therefore;

QY = YP, Definition of midpoint

XY is parallel to QR, and YZ  is parallel to  RP Midpoint theorem

Therefore, ∠PYX ≅ ∠PQR and ∠QZY ≅ ∠YXP Corresponding angles of parallel lines

From which we have;

Triangle ΔPYX is congruent to triangle ΔYQZ by Angle-Angle-Side condition of congruency.