Question

Given: Quadrilateral DEFG is a parallelogram.



Prove: GH ≅ EH
DH ≅ FH

Proof:
Statement Reason
1. Quadrilateral DEFG is a parallelogram. given
2.
definition of a parallelogram
3. Draw and . These line segments are
transversals cutting two pairs of parallel lines:
and and and . drawing line segments
4. Place point H where and intersect. defining a point
5. ∠HGD ≅ ∠HEF
∠HDG ≅ ∠HFE

6. DG ≅ EF Opposite sides of a parallelogram are congruent.
7. ASA criterion for congruence
8. GH ≅ EH
DH ≅ FH Corresponding sides of congruent triangles are congruent.
1
What is the missing statement for step 7 in this proof?
A.
ΔDGH ≅ ΔFEH
B.
ΔGHF ≅ ΔEHD
C.
ΔDGF ≅ ΔFED
D.
ΔDEF ≅ ΔEDG

Answer 1

Answer:

The correct option is A.

Step-by-step explanation:

It is given that  DEFG is a parallelogram.

Draw the diagonals DF and EG. Place point H where DF and EG intersect.

In triangle HGD and HEF

,

∠HGD ≅ ∠HEF                            (Alternate Interior angle)

∠HDG ≅ ∠HFE                      (Alternate Interior angle)

By the definition of a parallelogram, the opposite sides of a parallelogram are congruent.

DG ≅ EF                                      (Opposite sides of parallelogram)

According to ASA postulate, two triangles are congruent if any two angles and their included side are equal in both triangles.

So, by using ASA criterion for congruence we get,

ΔDGH ≅ ΔFEH

Since corresponding sides of congruent triangles are congruent, therefore

GH ≅ EH                      (CPCTC)

DH ≅ FH                     (CPCTC)

Option A is correct.