# Using Unique Triangles to Solve Real-World and Mathematical Problems

Video solutions to help Grade 7 students learn how to use conditions that determine a unique triangle to construct viable arguments that angle measures and lengths are equal between triangles.

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Common Core For Grade 7

## New York State Common Core Math Grade 7, Module 6, Lesson 15

### Lesson 15 Student Outcomes

• Students use conditions that determine a unique triangle to construct viable arguments that angle measures and lengths are equal between triangles.

### Lesson 15 Summary

• In deciding whether two triangles are identical, examine the structure of the diagram of the two triangles to look for a relationship that might reveal information about corresponding parts of the triangles. This information may determine whether the parts of the triangle satisfy a particular condition, which might determine whether the triangles are identical.
• Be sure to identify and label all known measurements and then determine if any other measurements can be established based on knowledge of geometric relationships.

Lesson 15 Classwork

Example 1
A triangular fence with two equal angles, ∠S = ∠T, is used to enclose some sheep. A fence is constructed inside the triangle that exactly cuts the other angle into two equal angles: ∠SRW = ∠TRW. Show that the gates, represented by SW and WT, are the same width.

Example 2
In △ABC, AC = BC. John says that the triangle correspondence △ABC ↔ △BAC matches two sides and the included angle and shows that ∠A = ∠B. Is John correct?

Exercises 1–4
1. Mary puts the center of her compass at the vertex O of the angle and locates points A and B on the sides of the angle. Next, she centers her compass at each of A and B to locate point C. Finally, she constructs the ray OC. Explain why ∠BOC = ∠AOC.

2. Quadrilateral ABCD is a model of a kite. The diagonals AB and CD represent the sticks that help keep the kite rigid.
a. John says that ∠ACD = ∠BCD. Can you use identical triangles to show that John is correct?
b. Jill says that the two sticks are perpendicular to each other. Use the fact that ∠ACD = ∠BCD and what you know about identical triangles to show ∠AEC = 90°.
c. John says that Jill’s triangle correspondence that shows the sticks are perpendicular to each other also shows that the sticks cross at the midpoint of the horizontal stick. Is John correct? Explain.

3. In △ABC, ∠A = ∠B. Jill says that the triangle correspondence △ABC ↔ △BAC matches two sides and the included angle and shows that AC = BC. Is Jill correct?

4. Right triangular corner flags are used to mark a soccer field. The vinyl flags have a base of 40 cm and a height of 14 cm.
a. Mary says that the two flags can be obtained by cutting a rectangle that is 40 cm × 14 cm on the diagonal. Will that create two identical flags? Explain.
b. Will measures the two non-right angles on a flag and adds the measurements together. Can you explain, without measuring the angles, why his answer is 90°? The two non-right angles of the flags are adjacent angles that together form one angle of the four angles of the rectangle. We know that a rectangle has four right angles, so it must be that the two non-right angles of the flag together sum to 90°.

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