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

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.

according to the triangle sum theorem, all the angles in a triangle must add up to 180 degrees. if we know one angle is already 90, we do 180-90=90. the other 2 angles must add up to 90, exactly.

step-by-step explanation: