HL and More Practice with Proofs
Lesson 6 of 10
Objective: Students will be able to use hypotenuse-leg theorem and show that triangles are congruen and students will review all methods of proving triangles congruent.
Students will look at a diagram and determine how to prove these triangles congruent. The diagram allows for two different ways to prove these triangles congruent, SSS and SAS, and students can work through one of these methods using a flow proof. This is a review of SSS and SAS and can be used as a formative assessment of student learning about these topics and also writing flow proofs.
This pair-share reviews how to use the Pythagorean Theorem to find missing sides of right triangles and also introduces how this theorem can be used to prove the Hypotenuse Leg theorem. Students will be required to make sense of the pattern that they see (MP 7) to realize that whenever we have a leg and hypotenuse congruent in two right triangles, the third side must also always be congruent - hence, HL Theorem.
Additional resource (quick overview for students):
CPCTC and Proofs
CPCTC is introduced in this section of notes, and can be described by using a video from Sir Tyler, a math teacher who is very entertaining. I find it helps when I introduce a new concept during the middle of the lengthy class with a video or another person's voice. The first 4 minutes of the video are most relevant, and the proof that he discusses is also the same proof provided on student notes. You can pause the video to illicit answers from students or ask students to complete the proof themselves. The proof can be done multiple ways, and I think the way that Tyler does it is not as direct as other methods. If you have time, then this would be a great conversation to have with students to critique his method and compare/contrast with students' own work (MP3).
One tricky part of CPCTC is helping students to realize that they need to identify the congruent triangles first and then prove these congruent before using CPCTC. Thus, before starting the proof provided in notes, you may want to prompt students to highlight or color in the corresponding congruent parts in the diagram. Then, from here, students can start to identify which triangles we need to prove are congruent.
The practice/homework provides students with an opportunity to persevere through difficult proofs (MP 1) which can be proved using any of the five congruence theorems, SSS, SAS, AAS, ASA and HL.
The exit ticket give students with an opportunity to review the connection between Pythagorean Theorem and HL Theorem (as well as SSS). You can ask students to turn to a neighbor to answer this question or ask students to write their answer in words and provide a sketch which you then would collect.
Triangle Proof Project
Depending on if you have time left in this class, if you have been parsing out the unit project, you can provide students with in-person feedback on their proof or give students time to organize and plan the video shoot for their triangle proof. The materials for this project can be found here.