• In assessing the group activities, identify the most specific and least necessary characteristics that make two triangles congruent. Ask students to identify some redundant attributes of two congruent triangles. For example, if two triangles have side-angle-side correspondence, then they also have side-side-side correspondence. The triangles will also have corresponding side and corresponding angle congruence. [IS.5 - All Students]
• Conduct observations during group activities and class discussions. Spot check selected measurements of side length and angle size. Make sure students are comparing corresponding sides and angles and point out that comparing non-corresponding sides and angles will be of no value.
• Lesson 1 Exit Ticket requires specific identification of the attributes of congruent triangles, the identification of corresponding sides and angles, and the minimal requirements for congruence (M-G-4-1_Lesson 1 Exit Ticket.doc and M-G-4-1_Lesson 1 Exit Ticket KEY.doc).

Activity 1: Exploring Perimeter and Its Relationship to Congruence [IS.3 - Struggling Learners]

Exploration: Students will discover that two triangles with equal perimeters do not necessarily mean that the triangles are congruent.

Divide students into groups of three or four. Each group will make triangles out of two identical lengths of string or ribbon.

Construct two triangles with the same perimeter. Shift the vertex of one triangle to change the position of the vertex without changing the perimeter. Straws with flexible joints or chenille sticks work well for this. Then draw one triangle. Draw a second triangle with the same perimeter but different side lengths on tracing paper. Overlay one triangle on the second and try to make an exact fit.

Give each group two identical pieces of string or ribbon whose ends have been stapled together and several pieces of paper large enough to make triangles determined by the size of the ribbon to each small group. [IS.4 - All Students] (Starting with 12½ inches of string or ribbon allows for a ¼-inch overlap to staple together; if this length is used, 8½ x 11 inch-paper is suitable.) Have three students each hold one vertex of a triangle created by the ribbon, and have the fourth student trace it onto paper. Have each group cut out their triangles and compare them with other groups to see if their triangles are congruent.

Have students record their discoveries about the triangles. Students should observe that even though the perimeters of the triangles are the same, the length of individual sides and measure of angles may be different.

Ask students what is significant about the perimeter being the same when the sides and angles are often different? Students should observe that even though the perimeter is the same, that does not indicate they are congruent triangles. Then ask the class what it would indicate if the sides and angles were the same. Students should observe that the triangles with identical sides and angles are congruent.

Activity 2: Exploring Congruence by Definition (and Implying SAS)

The following information should be displayed on the board or an overhead projector to help students draw congruent triangles on their paper:

Angle A is 90 degrees.                             

Side AB should be 9 cm.

Side AC should be 12 cm.

Then, connect B and C to create .

Have students draw a triangle identical to ΔABC, but label it DEF.

Steps to prove congruence:

1.   Compare the angle measurements for the corresponding angles in ΔABC and ΔDEF and ensure that they are identical.

2.   Compare the length of the corresponding sides in ΔABC and ΔDEF and ensure that they are identical.

3.   Use the definition of congruent triangles to state that ΔABC is congruent to ΔDEF.

Ask students to share their observations about the two triangles. Students should respond that the triangles have 3 pairs of congruent corresponding sides and 3 pairs of congruent corresponding angles.

Ask students which angle in  in ΔABC. Introduce the term corresponding to refer to  and . Ask students what side in ΔDEF is congruent to  in ΔABC. Use the term corresponding to refer to . Write an example congruence statement for the triangles: . Ask students how the statement would change if you referred to the first triangle as ΔCBA.

After discussing the importance of corresponding parts in triangle congruence statements, have students write a congruence statement of their own for the two triangles and then exchange with a partner to confirm their statement.

Remind students that they proved the two triangles were congruent by noting that each pair of corresponding angles as well as each pair of corresponding sides had the same measure. Affirm the idea of corresponding in terms of the order in which the triangle congruence statements are written.

Activity 3: Discovering SSS

Hand each group one or two sets of popsicle sticks. You can also use straws with flexible joints, chenille sticks, wood skewers. Have students construct a triangle out of each set of popsicle sticks so the sticks meet at the ends but do not overlap. Have students trace the inside edge of their triangles onto paper and then cut them out. Then students should compare their triangles to see if they are congruent by placing the triangles on top of one another.

Ask students to raise their hands if they found someone with a triangle congruent to the one they created. Point out to students that for this activity, they did not have to measure any angles, but still ended up with congruent triangles. Provide the following information for students to copy in their notes:

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

Tell students that this is a shortcut. Remind them what the definition of congruence is, and point out that if you can just show that three corresponding pairs of sides are congruent, that is sufficient to show that two triangles are congruent.

Ask students whether knowing that one pair of corresponding sides is congruent is enough to know that the triangles are congruent. Follow up by asking about one pair of corresponding angles. Use their observations to illustrate that we need to know that particular corresponding parts of two triangles are congruent in order to show congruence, and that SSS is just one of many congruence theorems.

Point out that SSS is truly a shortcut—it doesn’t require us to prove that everything in the two triangles is congruent. Students should rely on shortcuts when they can—shortcuts are a fundamental part of geometric proofs.

Extension:

• Given two right triangles, ΔQRS and ΔVRT, where  are both right angles, what other side or angle congruences are minimally necessary to prove that ?

Answer: ; point out to students that for right triangles, there is also a congruence condition known as SSA (side-side-angle) which applies only to right triangles. We can show by counterexample that for non-right triangles, SSA congruence may not be sufficient for triangle congruence. In the case of right triangles, this is known as the Hypotenuse Leg Congruence Theorem: If, in two right triangles, the hypotenuse and one leg of one are congruent to the hypotenuse and one leg of the other, then the two triangles are congruent.