Lesson 5 3 Triangle Congruence Cpctc Continued

Provide the following notes for students to read and/or copy: [IS.3 - Struggling Learners]

Definition of congruent triangles: Congruent triangles are triangles that have the same size and shape. They have corresponding angles that have the same measure and corresponding sides that have the same length.

Use the following to establish congruency of two triangles:

SSS : If three pairs of corresponding sides have the same length, the two triangles are congruent.

SAS: If two pairs of corresponding sides have the same length and the pair of included corresponding angles has the same measure, the two triangles are congruent.

ASA: If two pairs of corresponding angles have the same measure and the pair of included corresponding sides has the same length, the two triangles are congruent.

AAS : If two pairs of corresponding angles have the same measure and the pair of third sides (not included) has the same length, the two triangles are congruent.

Hypotenuse-Leg: In right triangles, if the hypotenuse and one leg of one triangle are congruent to the hypotenuse and another leg of a second triangle, then the two triangles are congruent.

CPCTC: Corresponding parts of congruent triangles are congruent.

Activity 1: Building of Triangles with Specific Given Information

Divide the class into groups of three or four. [IS.4 - All Students] Have each group draw two congruent triangles. Have each group start by drawing a side with length 7 cm and labeling one endpoint A and the other endpoint B. Then each group should construct with a measure of 40 degrees and use that angle to draw  with a length of 8 cm. At this point, ask students if all their pictures should be identical. Now, have them draw the third side (by connecting B and C). Again, ask students if their pictures are identical. Remind students that all the triangles which have which are 7 and 8 cm, respectively, and that all the triangles have which measures 40 degrees. Ask students, "How do we know that the triangles are congruent?" (SAS)

Now, have students measure , round to the nearest cm, and share their measurements. Students should discover that has the same measure in each triangle. Use the following questions to introduce CPCTC:

Follow the discussion by writing the following notes:

CPCTC: Corresponding Parts of Congruent Triangles are Congruent

Activity 2: Construction of Triangles with Limited Given Information

Have one member of each group construct a triangle. Side AB should be 5 centimeters, and  should be 7 centimeters. Angle A (the angle between ) can be any angle the student chooses, and should be constructed by connecting the endpoints of the 5- and 7-cm sides.

Once each student has constructed a triangle, ask him or her to measure and record the measure. Ask students to compare the length of  with their partner. Are the lengths of the same? (Students can also compare with other groups.) Use the following questions to reinforce the idea that you cannot use CPCTC until you have proven that two triangles are congruent:

Reinforce the idea that before students can apply the concept of CPCTC, they must know that the two triangles are congruent. However, once students have proven that two triangles are congruent (ideally, by using one of the short-cut theorems), then they know that all the pairs of corresponding parts of the congruent triangles are congruent.

Activity 3: Construction of Triangles with Congruence Theorems

Divide students into pairs. Have each pair select a single congruence theorem (SSS, SAS, AAS, or ASA) and decide on the lengths of each side and the measure of each angle. (For example, if they choose AAS, they should decide on the measures of the two angles and the length of the third side.) Then have students independently construct a triangle using the information they decided on and fill in the other information as they wish. Once students have constructed their triangles independently, ask if their triangle will be congruent to their partner's triangle.

Follow up by asking students, since they believe that their triangle will be congruent to their partner's, "What will be true about the sides and angles that you did not decide upon together? Will those measurements be the same? How do you know?"

Then have students measure each part of their triangle (all three sides and all three angles) and compare these to the measurements of their partner's triangle. Students should find all the measurements of the corresponding parts are congruent.

Have them list the theorem they used to prove their triangles were congruent as well as the theorem that proves the remaining pairs of corresponding parts of their triangles are also congruent.

Have each pair of students select two of the remaining triangle congruence theorems (SSS, SAS, ASA, or AAS) and repeat the above activity, constructing congruent triangles and then confirming that corresponding parts of congruent triangle are congruent.

Extension:

• Given two triangles, ∆UVW and ∆XYZ, without knowing any other facts, what is the least information required to prove that ∆UVW is not congruent to ∆XYZ?

Answer: Any ONE of the following:

spenceronsille1984.blogspot.com

Source: https://www.pdesas.org/ContentWeb/Content/Content/21056/Lesson%20Plan

Share this post