Countdown to Congruence

Display slide 28 and tell students that they are going to only focus on AAA, AAS, and ASA during this second Explore portion of the lesson. Then move to slide 29 and again, try to get into character as a game-show host. Tell students that they are going to play the challenge rounds of “Can You Match My Triangle?” These investigations will likely take longer than the previous investigations, which is why students will be given more time.

As before, provide each pair with one mini-set of AngLegs containing three legs of each color/length and a protractor. Then have pairs form groups of four, with two mini-sets of AngLegs. Students need to have access to two protractors, which is why students are now in groups of four. If you have enough protractors for each pair to have two, consider having students work in pairs for more engagement. Have students get out a pencil and piece of scratch paper; they can use the same scratch paper from the earlier exploration.

Transition to slide 30. Begin challenge round 1 by reading the first triangle’s three properties from the slide: the measure of angle A is 30°, the measure of angle B is 60°, and the measure of angle C is 90°. Start the 5-minute timer on the slide. As students work, circulate the room to make sure students are understanding the task. This is the time for you to ask more guiding questions than give answers.

Once the timer expires, or students are done, move to slide 31 and have students hold up their triangles and see if they are all the same (congruent) or if at least one is different (a counterexample). Then ask the class which congruence possibility they just tested.

Use slide 32 or unhide slide 33 to share your triangle. If everyone’s triangles happen to be congruent, show the counterexample implying that no one was able to match your triangle. If you notice that there is at least one counterexample in the class, there is no need to unhide any slides. Students should conclude that AAA does not determine congruence.

Repeat these steps using slides 34–36 for students to test the Angle–Angle–Side (AAS) possibility for challenge round 2. Students should conclude that all of the triangles are congruent.

For challenge round 3, again repeat these steps using slides 37–39 for students to test the Angle–Side–Angle (ASA) possibility. Students should conclude that all of the triangles are congruent.

Display slide 40. Facilitate a discussion about how to complete the given conjectures. Have students jot down notes about the conjectures on the backs of their Triangle Congruence Theorems handouts. Be sure that students know what the difference between included and non-included sides is.

Move to slide 41 and share with students that, due to time, they are not going to formally prove the congruence theorems but instead appreciate that mathematicians already have proven ASA and AAS as triangle congruence theorems. Have students complete their handout by writing the theorem names, labeling the triangles, and writing the theorems in their own words. Depending on the needs of your students and time, you might consider collaborating as a class and agreeing upon the language used to write the theorems.

Transition through slides 42–43 to share with students the Hypotenuse–Leg (HL) congruence theorem in as much detail as you feel necessary. There is empty space at the bottom of the front page of their Triangle Congruence Theorems handout for notes on this theorem.

Before continuing the lesson, make sure students understand that the minimum number of parts they would need to measure to determine triangle congruence is three.