We are asked to prove Theorem 6.22. Let's copy the diagram from the book indicating the parallel bases and the congruent angles. To show that the legs of trapezoid KLMP are congruent, let's draw a line through M that is parallel to KL and extend this line until the opposite base.
To show that KLMP is an isosceles trapezoid, we will show that TM is congruent to both KL and PM. First, let's focus on quadrilateral KLMT.
By construction, the opposite sides are parallel, so KLMT is a parallelogram. According to Theorem 6.3 and Theorem 6.4, the opposite sides and angles are congruent.
TM≅KL
∠ L≅ ∠ MTK
Let's now concentrate on triangle △ MPT and focus on the angles at vertices T and P.
We know that segments KP and LM are parallel and ∠ M and ∠ MPT are consecutive interior angles. According to the Consecutive Interior Angles Theorem, these angles are supplementary.
m ∠ M+m ∠ MPT=180
We can see two angles that form a linear pair at point T. Since angles ∠ MTK and ∠ L are congruent, this means that angles ∠ MTP and ∠ L are supplementary.
m ∠ L+m ∠ MTP=180
Since angles ∠ MPT and ∠ MTP are supplementary to the congruent angles ∠ M and ∠ L, these are also congruent.
∠ MPT≅ ∠ MTP
According to the Converse of the Isosceles Triangle Theorem, this means that the sides opposite to these angles in triangle △ MPT are congruent.
TM≅PM
We now know that TM is congruent to both KL and PM. Using the Transitive Property of Congruence, this means that these two segments are congruent.
KL≅PM
These segments are the two legs of the trapezoid, so by definition, KLMP is isosceles. Let's summarize the steps above in a paragraph proof.
Completed Proof
2
&Given:&& KLMP is a trapezoid
& && ∠ L≅∠ M
&Prove:&& KLMP is isosceles
Proof:
Let's pick point T on KP so that TM is parallel to KL. Quadrilateral KLMP has two pairs of parallel sides, so it is a parallelogram. According to Theorem 6.3 and Theorem 6.4, opposite sides and angles are congruent, so TM≅KL and ∠ L≅∠ MTK. Using this angle congruence and that ∠ MTK and ∠ MTP form a linear pair, we can conclude that ∠ L and ∠ MTP are supplementary. Angles ∠ M and ∠ MPT are also supplementary, because they are consecutive interior angles between parallel lines. Since angles ∠ MPT and ∠ MTP are supplementary to the congruent angles ∠ M and ∠ L, angles ∠ MPT and ∠ MTP are also congruent. According to the converse of the Isosceles Triangle Theorem this means that TM≅PM. We now know that TM is congruent to both KL and PM, so KL and PM are congruent. These segments are the two legs of the trapezoid, so by definition, KLMP is isosceles.
