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Identical Triangles





 

Video solutions to help Grade 7 students learn how to use a triangle correspondence to recognize when two triangles match identically.

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Common Core For Grade 7

New York State Common Core Math Grade 7, Module 6, Lesson 5


Lesson 5 Student Outcomes


• Students learn how to use a triangle correspondence to recognize when two triangles match identically.
• Students use notation to denote a triangle correspondence and use the triangle correspondence to talk about corresponding angles and sides.
• Students are able to label equal angles and sides of triangles with multiple arcs or tic marks.

Lesson 5 Summary


• Two triangles and their respective parts can be compared once a correspondence has been assigned to the two triangles. Once a correspondence is selected, corresponding sides and corresponding angles can also be determined.
• Corresponding vertices are notated by double arrows; triangle correspondences can also be notated with double arrows.
• Triangles are identical if there is a correspondence so that corresponding sides and angles are equal.
• An equal number of tick marks on two different sides indicates the sides are equal in measurement. An equal number of arcs on two different angles indicates the angles are equal in measurement.

Lesson 5 Problem Set Classwork

Opening
When studying triangles, it is essential to be able to communicate about the parts of a triangle without any confusion. The following terms are used to identify particular angles or sides:
• between
• adjacent to
• opposite to
• the included [side/angle]

Opening Exercises 1–7
Use the figure ABC to fill in the following blanks.
Now that we know what to call the parts within a triangle, we consider how to discuss two triangles. We need to compare the parts of the triangles in a way that is easy to understand. To establish some alignment between the triangles, the vertices of the two triangles are paired up. This is called a correspondence. Specifically, a correspondence between two triangles is a pairing of each vertex of one triangle with one (and only one) vertex of the other triangle. A correspondence provides a systematic way to compare parts of two triangles.
We notate correspondence with double-arrows. A simpler way to indicate the triangle correspondences is to let the order of the vertices define the correspondence; i.e., the first corresponds to the first, the second to the second, and the third to the third.
With a correspondence in place, comparisons can be made about corresponding sides and corresponding angles.


 
Example 1
Given the following triangle correspondences, use double arrows to show the correspondence between vertices, angles, and sides. other. Sometimes this may require turning the triangle over. Examine Figure 2. By simply looking, it is impossible to tell the two triangles apart unless they are labeled. They look exactly the same (just as identical twins look the same). One triangle could be picked up and placed on top of the other.
Two triangles are identical if there is a triangle correspondence so that corresponding sides and angles of each triangle is equal in measurement. In discussing identical triangles, it is useful to have a way to indicate those sides and angles that are equal. We mark sides with tick marks and angles with arcs if we want to draw attention to them. If two angles or two sides have the same number of marks, it means they are equal.

Example 2
Two identical triangles are shown below. Give a triangle correspondence that matches equal sides and equal angles.
Exercise 8
8. Sketch two triangles that have a correspondence. Describe the correspondence in symbols or words. Have a partner check your work.


 
Lesson 5 Problem Set Sample Solutions
Given the following triangle correspondences, use double arrows to show the correspondence between vertices, angles, and sides.
Name the angle pairs and side pairs to find a triangle correspondence that matches sides of equal length and angles of equal angles measurements.


 

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