Chapter 5.1, Problem 14WE

To determine

To prove: that parallelograms $\angle A\cong \angle C\text{ and }\angle B\cong \angle D$ are corresponding parts of congruent triangles, and opposite sides of a parallelogram are congruent.

Answer to Problem 14WE

The corresponding parts of congruent triangles: $\angle A\cong \angle C$

The corresponding parts of congruent triangles: $\angle B\cong \angle D$

The opposite angels of parallelogram are congruent.

Explanation of Solution

Given information:

Theorem: Opposite sides of a parallelogram are congruent.

Given: $ABCD$ is a parallelogram.

Calculation:

In triangle $\text{Δ}ADB$ and $\text{Δ}BCD$ .

If two parallel sides are cut by a transversal then the alternate interior angles are equal.

  $ABD\cong BDC$ (Alternate interior angle) $ADB\cong DBC$

  $BD\cong DB$ (Common side of a triangle)

By the angle side angle property the triangles are congruent

  $\text{Δ}ADB\cong \text{Δ}BCD$ (A.S.A Test)

Therefore, by the corresponding parts of congruent triangles

  $\angle A\cong \angle C$

Similarly,

Show that $\text{Δ}ADC\cong \text{Δ}ABC$

By the corresponding parts of congruent triangles

  $\angle B\cong \angle D$

Therefore, in a parallelogram opposite angels are congruent.