Chapter 5.1, Problem 13WE

To determine

To prove: that parallelogram $EF\cong HG$ and $FG\cong EH$ arecorresponding parts of congruent triangles, and opposite sides of a parallelogram are congruent.

Answer to Problem 13WE

The corresponding parts of congruent triangles are:

  $EF\cong HG$

  $FG\cong EH$

The opposite angels of parallelogram are congruent.

Explanation of Solution

Given information:

Theorem: Opposite sides of a parallelogram are congruent.

Given: EFGH is a parallelogram.

Calculation:

It is given that $EFGH$ is a parallelogram.

The line $EF\parallel HG$ and $EG$ is a transversal.

Thus, the alternate interior angles are equal

  $\angle 4\cong \angle 3$

  $\angle 2\cong \angle 1$

  $\ln \text{Δ}HEG$ and $\text{Δ}GFE$

  $\angle 4\cong \angle 3$ From equation (1)

  $EG\cong GE$ (Common side of a triangle)

  $\angle 2\cong \angle 1$ From equation (1)

By angle side angle property the triangles are congruent.

  $△HEG\cong \text{Δ}GFE$ (A.S.A Test)

Thus, by corresponding parts of congruent triangles

  $EF\cong HG$

  $FG\cong EH$

Therefore, in a parallelogram opposite angels are congruent.