Chapter 9, Problem 11CUR

Solution

To prove: $\overline{AB}\cong \overline{DC}$

Given:

The diagram:

  $\angle 1\cong \angle 2$

  $\angle 2\cong \angle 3$

Concept Used:

Postulate 11: corresponding angles between parallel lines must be equal.

Theorem 3-5: Alternate interior angles between parallel lines must be equal.

Theorem 5-1: Opposite sides of a parallelogram are congruent.

Explanation:

Consider the figure shown below:

It is known that a parallelogram is a quadrilateral whose both pairs of opposite sides are parallel.

From the above figure, it can be observed that the angles $\angle 1$ and $\angle 2$ are the corresponding angles which are formed when the lines $\overline{AB}$ and $\overline{DC}$ are cut by the transversal $\overline{AD}$ .

It is given that,

  $\angle 1\cong \angle 2$

Thus, by the postulate 11, it can be said that the lines $\overline{AB}$ and $\overline{DC}$ are parallel.

Similarly, the angles $\angle 2$ and $\angle 3$ are the corresponding angles which are formed when the lines $\overline{AD}$ and $\overline{BC}$ are cut by the transversal $\overline{DC}$ .

It is given that,

  $\angle 2\cong \angle 3$

Thus, by the theorem 3-5, it can be said that the lines $\overline{AD}$ and $\overline{BC}$ are parallel.

That is, for the quadrilateral

  $ABCD$ whose both pairs of opposite sides are parallel.

So, it can be said that $ABCD$ is a parallelogram.

Now by theorem 5-1, it is known that opposite sides of a parallelogram are congruent. So,

  $\overline{AB}\cong \overline{DC}$