March 07, 2018

Lesson 6.1.1  Are the triangles congruent? Congruent Triangles

Lesson Objective: Students will practice identifying congruent triangles by first determining that the triangles are similar and that the ratio of corresponding sides is 1.  Students will develop triangle shortcuts (such as AAS ≅) in Lesson 6.1.2.CCSS Standard(s): G-CO.6, G-CO.7, G-CO.8, G-SRT.2, G-SRT.5Mathematical Practices: Construct viable arguments as they prove triangle similarity and/or congruence Attend to precision as they give reasons in their proofs

Core Problems: 6‑1 and 6‑2

Homework: 6‑4 through 6‑10

Note:Problem 6‑5 is revisited in Lesson 6.1.2.  Be sure that you assign this problem.

Materials: None

Suggested Lesson Activity: After the introduction, assign teams problems 6‑1 through 6‑3.  6-1 requires students to review how to construct a flowchart to show that triangles are similar.  It also helps them remember that if similar shapes have a common ratio of 1, then the shapes are congruent.  Finally, problem 6‑1 also introduces the congruence symbol.  In part (d), make sure students’ congruence statements name vertices in order to show correspondence.

6-2 presents a number of challenging contexts in which students identify congruent triangles.  One stumbling block that may arise is noticing that one segment can serve as a side of two distinct triangles.  In part (a), for instance,  is a side of both ΔABD and ΔCBD.  To recognize that the ratio of corresponding sides is 1, students will have to make a ratio of BD to itself. 

Expect some confusion as students work on problem 6‑2, but prevent yourself from doing the thinking for the students.  For example, in part (b), students will need to recognize that the ratios     and   must be 1 because the sides are congruent.  Part (c) is also challenging because students will need to first recognize that alternate interior angles are congruent (since the lines cut by the transversal are parallel) in order to know that the triangles are similar.  Finally, part (d) will challenge students because they need to recognize that while the smaller triangles (ΔAED and ΔBEC) are similar, they have no information regarding the ratio of corresponding sides.  However, the larger triangles (ΔABD and ΔBAC) have a shared side and therefore are congruent.

Call the class together to discuss problem 6‑2 and ask teams to present their congruence arguments.  After each argument, ask teams who solved the problem a different way to present their thinking.  Capitalize on this opportunity for students to name the different ways that congruent parts can be “hidden” in a diagram, such as a shared side, vertical angles, angles created when parallel lines are cut by a transversal, etc.6-3  As you circulate, ask students about their triangle naming in part (b): “Does it matter if you call this triangle ΔMNR or ΔNMR?”  The goal is for students to see the symmetry in the triangles, which turns out to give them flexibility in naming.

Homework: 6‑4 through 6‑10 Note: Problem 6‑5 is revisited in Lesson 6.1.2.  Be sure to assign this problem.

March 07, 2018

HW Answers in Book ... Review?

March 07, 2018

Objective(s): You will show triangles are ≅ by proving they are ~ & exhibit a LSF of 1.

Agenda:• Review HW (Teams)• Notes on Congruence from Similarity in Flow

Charts: Note that we are still using ratios of side lengths for SSS~ & SAS~, rather than equality of side lengths, which we will use in tomorrow's triangle congruence conditions (Class, Spiral)

• Lesson 6.1.1 Activity (Teams, Spiral)> 6-2 Justify your claims, using diagrams &

naming the relevant triangle similarity conditions. Note that if two triangles share a common side, & those sides are corresponding sides, then the LSF = 1, because any # divided by itself = 1.

> 6-3 Hint: in part c, there are 12 of them! In part d, you can omit the flowchart.

• Record/Begin HW (Class/Independent):> 6-4 to 6-8, 6-10;> Add MN Congruent Shapes (p345) to INB

p92, as needed.

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P(complement) = 1 - P(event)

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CW: 6-1 to 6-3

6-1 Together; 6-2 & 6-3 in Teams.

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~ "is similar to"

≅ "is congruent to"

= "is equal to"

6-1c

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∆CAB ≅ ∆

6-1d

Write a congruence statement for △ABC and △FDE

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∆CAB ≅ ∆EFD

6-1d

Write a congruence statement for △ABC and △FDE

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CW

: 6-1

to 6

-3

any sides in common?

Which sides & vertices correspond?

Justify your claims, using diagrams & references to conditions.

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6-2

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A B

CD

A BA B

CD

A B

E

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6-2

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CW

: 6-1

to 6

-3

hint: there are 12 of them

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CPM

345

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Need 6-5 in class tomorrow

HW:• 6-4 to 6-8, 6-10;• Add MN Congruent Shapes (p345) to INB

p92, as needed.

March 07, 2018

HW✓ 6-4 to 6-10(skip 6-9)

6-4 a. Alternate Interior Angles (AIA)b. Vertical angles (VA)c. ∠u & ∠z, ∠s & ∠x, ∠v & ∠w, and ∠t & ∠y

6-5 a. They're similar by SAS~b. Yes, b/c they're similar and the corresponding sides have a ratio of 1

6-6 3x+1°+52° = 180°, x = 127/3 ≈ 42.33°

6-7 a. 8 cmb. ≈ 14.97 ftc. ≈ 15.2 in

6-8 1a & 1b: statements ii and iv2. The cupcakes are burned3. The fans will not buy the cupcakes b/c they're burned.4. The team will not have enough $$$ for uniforms.

6-10 A