Question

9. Use Patterns and Structure Follow the steps to

prove the HL Theorem.

Given: Right triangles AMNL and
APQR, MN PQ, ML = PR
Prove: AMNL APQR
Show that there is a rigid motion that
maps L to R and M to P so that N' and
Q are on opposite sides of PR.
. Then show that APQN' is isosceles.
• Show that AM'N'L' APQR,
so AMNL APQR.

Answer 1

The HL Theorem can be proved using a rigid motion that maps corresponding parts of triangles AMNL and APQR. We can use a translation and a rotation to show that the triangles are congruent.

Step-by-step explanation:

To prove the HL Theorem, we are given right triangles AMNL and APQR, where MN = PQ and ML = PR.

We need to show that there is a rigid motion that maps L to R and M to P so that N' and Q are on opposite sides of PR. Then, we need to show that APQN' is isosceles, and finally, we can conclude that AMNL is congruent to APQR.

1. From the given information, we can use a translation to move point L to point R. This does not change the lengths of any sides because the translation preserves distance.
2. Next, we rotate triangle AMNL by 180 degrees around point N. This rotation maps M to P and keeps MN and ML equal to PQ and PR, respectively.
3. We can observe that angle APQ is congruent to angle N'PR because the rotation maps angle MNL to angle PQR, and these angles are corresponding angles. Therefore, triangle APQN' is isosceles with AP = AQ.
4. Since we have shown that there is a rigid motion that maps all corresponding parts of triangles AMNL and APQR, we can conclude that they are congruent.