Extending Congruence to CPCTC

Objectives

Students will apply their knowledge from Lesson 1 on using the definitions to prove congruency. Students will learn the concept of Corresponding Parts of Congruent Triangles are Congruent (CPCTC) and learn to apply that concept in proofs. Students will:

apply triangle theorems to prove triangle congruence.
use CPCTC in conjunction with triangle congruence to prove other statements about triangles.

Essential Questions

How can we use congruence to extend our knowledge about parts of congruent triangles?
How does this apply to real-world problems?

Vocabulary

AAS (Angle-Angle-Side correspondence): 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. [IS.1 - All Students]
ASA (Angle-Side-Angle correspondence): If two pairs of corresponding angles have the same measure and the pair of corresponding sides has the same length, the two triangles are congruent.
Congruent: Having the same size and shape.
Corresponding Angles: Angles in the same relative position in similar or congruent figures.
Corresponding Sides: Sides in the same relative position in similar or congruent figures.
Deductive Reasoning: A method which arrives at conclusions from accepted principles; reasoning such that the conclusion necessarily follows from a set of premises.
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.
Included Angle: An angle of a triangle whose vertex is the common endpoint of two consecutive sides of a triangle.
Included Side: A side of a triangle whose endpoints are the vertices of two consecutive angles of the triangle.
Inductive Reasoning: Drawing conclusions from several known cases; reasoning from the particular to the general. The premises of an inductive logical argument indicate some degree of support for the conclusion; they suggest truth, but do not ensure it.
SAS (Side-Angle-Side Correspondence): If two pairs of corresponding sides have the same length and the pair of corresponding angles has the same measure, the two triangles are congruent.
SSS (Side-Side-Side Correspondence): If three pairs of corresponding sides have the same length of measure, the two triangles are congruent.

Duration

60–90 minutes [IS.2 - All Students]

Prerequisite Skills

Prerequisite Skills haven't been entered into the lesson plan.

Materials

paper and markers or colored pencils
graph paper
rulers
protractors
Lesson 2 Exit Ticket (M-G-4-2_Lesson 2 Exit Ticket.doc and M-G-4-2_Lesson 2 Exit Ticket KEY.doc)

Related Unit and Lesson Plans

Related Materials & Resources

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Formative Assessment

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- In Activity 3, completed in pairs, require each student to evaluate the work of his or her partner. Students should identify what each partner completed correctly, which parts were incorrect, and what understanding was necessary to complete the work correctly. Remind students to identify differences between errors of measurement and computation and errors of correspondence and alignment. [IS.5 - All Students]
- Observations of student work during group activities and class discussion is most effective when specific categories of incorrect or inaccurate work are used as examples for correction and wider understanding. Encourage students to evaluate the work of other members of the group, identify what is and is not correct, why, and what corrections are appropriate.
- Lesson 2 Exit Ticket summarizes the knowledge and skill required to identify the necessary conditions for congruence and the constraints of the remaining sides and angles after the conditions have been established (M-G-4-2_Lesson 2 Exit Ticket.doc and M-G-4-2_Lesson 2 Exit Ticket KEY.doc).

Suggested Instructional Supports

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Active Engagement, Modeling, Explicit Instruction

W:	After this lesson, students understand how to use congruence theorems (SSS, SAS, AAS, ASA) to prove that other corresponding parts of congruent triangles are congruent. Students learn how to prove geometric figures are congruent because there are real-world situations that apply these concepts. Students are able to apply deductive and inductive reasoning to hypotheses about two-dimensional objects that build real-world reasoning skills.
H:	Activity 1 begins with a measured drawing of a triangle built, successively, from side length 7 centimeters, angle measure 40 degrees, side length 8 centimeters. From these three triangle parts only, students have a visual representation of the remaining possibilities of the third side. That third side may be determined by its length, the angle measure at its left end point, or the angle measure at its right end point.
E:	The construction of the triangle in Activity 2 is begun in a similar way to the triangle in Activity 1. The difference is that the angle between the first two drawn sides is not specified; at this point, the only correspondence between the various triangles is between two adjacent sides.
R:	In Activity 3, the constructed triangle has sufficient side and angle measurements to constrain the measurements of the missing sides or angles. The activity initially invites a re-evaluation of the possibility that more than one alternate triangle may be constructed after establishing three parts of a triangle that force congruence.
E:	Each pair of students considers the possible triangles that may result from their first three measurements: three sides; one side, one included angle, one side; one angle, one included side, one included angle; and two angles, one non included side. After all three parts of the triangle have been established and drawn, the congruence relationship is established visually and logically, as no other possible sides or angles may be drawn to complete the triangle.
T:	By having students work in partners on a regular basis, students learn from and help each other. By working in pairs on a consistent basis, students learn how to work well with specific students and develop strategies to help one another. For example, students who are mathematically strong learn to help other students who may have more difficulty in carrying out particular instructions or completing activities.
O:	The first two activities have students work with partners, with you providing over-arching instruction. The partner activities allow each student time to work closely with the new topics and also a resource (the other student) to help answer exploratory questions about the new material. While the third activity is still arranged to be done with a partner, students work on their own initially to construct their triangle. This activity leads into the repetition of Activity 3 (using different triangle congruence theorems) and ultimately leads into the Exit Ticket for Lesson 2, which should be completed on an individual basis (M-G-4-2_Lesson 2 Exit Ticket.doc and M-G-4-2_Lesson 2 Exit Ticket KEY.doc).

IS.1 - All Students

Consider using graphic organizers (e.g., Frayer Model, Verbal Visual Word Association, Concept Circles) to review key vocabulary prior to or during the lesson

IS.2 - All Students

Consider preteaching the concepts critical to this lesson, including the use of hands-on materials. Throughout the lesson (based upon the results of formative assessment), consider the pacing to be flexible to the needs of the students. Also consider the need for reteaching and/or review both during and after the lesson as necessary

IS.3 - Struggling Learners

Consider providing struggling students with time to preview/review information related to triangle congruence and CPCTC at www.khanacademy.org

IS.4 - All Students

Consider cooperative learning and student engagement strategies (Think-Pair-Share, Random Reporter) as appropriate. Information on these strategies can be found at www.pdesas.org/Main/Instruction

IS.5 - All Students

Prior to teaching this lesson, consider the prior knowledge and misconceptions that students may bring to this lesson. Use that information to determine when formative assessments techniques are needed (e.g., white boards, thumbs up/thumbs down, and asking assessing and advancing questions). To learn more about assessing and advancing questions, please download Teachers’ Desk Reference: Essential Practices for Effective Mathematics Instruction at: http://www.pattan.net/category/Resources/PaTTAN%20Publications/Browse/Single/?id=4e1f51d3150ba09c384e0000

Instructional Procedures

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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:
- “Did you know that would be the same in each triangle?” (yes)
- “How did end up being the same when all students drew their triangles on their own?” (the triangles were congruent by ASA)
- “What does the definition of congruence mean about the corresponding parts of the triangles?” (the corresponding parts are congruent)
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:
- “Is the third side () of your triangle congruent to your partner’s?” (no)
- “How did the construction of this triangle differ from the construction of the last triangle, in which everyone’s was congruent?” (we got to choose whatever angle we wanted)
- “What did we know about the triangles in the first example?” (they were congruent)
- “Do we know that the triangles in this example are congruent?” (no)
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:

Extending Congruence to CPCTC

Lesson Plan