Lesson 7
From Parallelograms to Triangles
Let’s compare parallelograms and triangles.
Problem 1
To decompose a quadrilateral into two identical shapes, Clare drew a dashed line as shown in the diagram.
![Trapezoid base 1, 6 units, base 2, 2 units, height 4 units.](https://cms-im.s3.amazonaws.com/KzBGN8Px8at5Lvees9j5RBBX?response-content-disposition=inline%3B%20filename%3D%226-6.1.C.PP_Image_5.1.png%22%3B%20filename%2A%3DUTF-8%27%276-6.1.C.PP_Image_5.1.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240726%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240726T232407Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=6829b488911cae71d856fab8bc15134f61ff68f3e6e4a6fe95eba475a48ae67d)
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She said the that two resulting shapes have the same area. Do you agree? Explain your reasoning.
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Did Clare partition the figure into two identical shapes? Explain your reasoning.
Problem 2
Triangle R is a right triangle. Can we use two copies of Triangle R to compose a parallelogram that is not a square?
![2 identical triangles labeled R](https://cms-im.s3.amazonaws.com/FpYzLvDFcV8sAVwXZGPrraZp?response-content-disposition=inline%3B%20filename%3D%226-6.1.C1.Image.09.R.png%22%3B%20filename%2A%3DUTF-8%27%276-6.1.C1.Image.09.R.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240726%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240726T232407Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=51060273654c92885acf6e4869f4c590810db384d476552cb4cf7b2ceeef1768)
If so, explain how or sketch a solution. If not, explain why not.
Problem 3
Two copies of this triangle are used to compose a parallelogram. Which parallelogram cannot be a result of the composition? If you get stuck, consider using tracing paper.
![A three sided figure in a grid. Vertices located at two units right, 1 unit up; 7 units right, 2 units up; and 1 unit right, 3 units up.](https://cms-im.s3.amazonaws.com/VmjzaFPtL3B9MDRKts6bcxN4?response-content-disposition=inline%3B%20filename%3D%226-6.1.C.PP.New.Image.02.png%22%3B%20filename%2A%3DUTF-8%27%276-6.1.C.PP.New.Image.02.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240726%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240726T232407Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=caf3551a466a6887000b3b5e5ba20497e30c96db86a471fbf7b199f200da47d1)
![Four parallelograms labeled A, B, C, and D.](https://cms-im.s3.amazonaws.com/Gt95bSgxHEEJGwNXNBLCM9o9?response-content-disposition=inline%3B%20filename%3D%226-6.1.C.PP.New.Image.02a.png%22%3B%20filename%2A%3DUTF-8%27%276-6.1.C.PP.New.Image.02a.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240726%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240726T232407Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=3c63e26931370c1d8fe69785dec1a2f6dc5f227b0817941a8e3577b1aab16f4e)
A
B
C
D
Problem 4
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On the grid, draw at least three different quadrilaterals that can each be decomposed into two identical triangles with a single cut (show the cut line). One or more of the quadrilaterals should have non-right angles.
- Identify the type of each quadrilateral.
Problem 5
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A parallelogram has a base of 9 units and a corresponding height of \(\frac23\) units. What is its area?
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A parallelogram has a base of 9 units and an area of 12 square units. What is the corresponding height for that base?
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A parallelogram has an area of 7 square units. If the height that corresponds to a base is \(\frac14\) unit, what is the base?
Problem 6
Select all the segments that could represent the height if side \(n\) is the base.
![A parallelogram with a bottom side labeled m and a right side labeled n. Dashed lines e, f, j, and k are drawn perpendicular to side m, and dashed lines g and h are drawn perpendicular to side n.](https://cms-im.s3.amazonaws.com/tFqfcBp8q4QSRV2SEENK4hVi?response-content-disposition=inline%3B%20filename%3D%226-6.1.B.PP.New.Image.10.png%22%3B%20filename%2A%3DUTF-8%27%276-6.1.B.PP.New.Image.10.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240726%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240726T232407Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=3d2007edda3cebf42152b1be8b57de61d31d20e1a0406c07adb2d0a2577c2e56)
\(e\)
\(f\)
\(g\)
\(h\)
\(m\)
\(n\)
\(j\)
\(k\)