Draw the diagonals and think about what you can prove about the triangles.
See solution.
Practice makes perfect
We are asked to prove that if two sides of a quadrilateral are parallel and congruent, then the quadrilateral is a parallelogram. Let's draw the diagonals and highlight two of the triangles. Let's also label some angles of these highlighted triangles.
Let's think about how these triangles can help in proving that quadrilateral ABCD is a parallelogram.
The highlighted triangles appear to be congruent. Let's prove this first.
The diagonals appear to bisect each other. Let's use the congruence of the triangles to prove this.
We can finish our proof using Theorem 6.11, which guarantees that a quadrilateral is a parallelogram when the diagonals bisect each other.
Let's carry out this plan.
Triangles △ ABO and △ CDO
To show that triangles △ ABO and △ CDO are congruent, let's compare their angles.
Line AC is a transversal to lines AB and DC, and angles ∠ 1 and ∠ 4 are alternate interior angles.
It is given that AB and DC are parallel, so the Alternate Interior Angles Theorem implies that ∠ 1 and ∠ 4 are congruent.
∠ 1≅∠ 4
We can also focus on the angles on the other diagonal.
These are also alternate interior angles, so they are also congruent.
∠ 2≅∠ 3
So far we have only used that segments AB and DC are parallel. It is also given that these segments are congruent, so let's use this information now.
Let's summarize what we know about triangles △ ABO and △ CDO.
∠ 1&≅∠ 4
∠ 2&≅∠ 3
AB&≅CD
Two angles and the included side of triangle △ ABO are congruent to two angles and the included side of triangle △ CDO. According to the Angle-Side-Angle (ASA) Congruence Postulate, this means that the two triangles are congruent.
△ ABO≅ △ CDO
Diagonals AC and BD
Let's now focus on the sides of these congruent triangles that form the diagonals of the quadrilateral.
Corresponding sides of congruent triangles are congruent.
AO&≅CO
BO&≅DO
This means that O is the midpoint of both diagonals.
Finishing the Proof
Since O is the midpoint of both AC and BD, the diagonals of quadrilateral ABCD bisect each other.
According to Theorem 6.11, this means that quadrilateral ABCD is a parallelogram. We can summarize the process above in a two-column proof.
Completed Proof
2
&Given:&& AB∥DC, AB≅DC
&Prove:&& ABCDis a parallelogram.
Proof:
