Assignment P. 526-529: 1-11, 15-21, 33-36, 38, 41, 43 Challenge Problems.
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Transcript of Assignment P. 526-529: 1-11, 15-21, 33-36, 38, 41, 43 Challenge Problems.
![Page 1: Assignment P. 526-529: 1-11, 15-21, 33-36, 38, 41, 43 Challenge Problems.](https://reader030.fdocuments.in/reader030/viewer/2022032722/56649cee5503460f949bc64c/html5/thumbnails/1.jpg)
Assignment
• P. 526-529: 1-11, 15-21, 33-36, 38, 41, 43
• Challenge Problems
![Page 2: Assignment P. 526-529: 1-11, 15-21, 33-36, 38, 41, 43 Challenge Problems.](https://reader030.fdocuments.in/reader030/viewer/2022032722/56649cee5503460f949bc64c/html5/thumbnails/2.jpg)
Proving Lines Parallel
![Page 3: Assignment P. 526-529: 1-11, 15-21, 33-36, 38, 41, 43 Challenge Problems.](https://reader030.fdocuments.in/reader030/viewer/2022032722/56649cee5503460f949bc64c/html5/thumbnails/3.jpg)
Proving Triangles Congruent
![Page 4: Assignment P. 526-529: 1-11, 15-21, 33-36, 38, 41, 43 Challenge Problems.](https://reader030.fdocuments.in/reader030/viewer/2022032722/56649cee5503460f949bc64c/html5/thumbnails/4.jpg)
Proving Triangles Congruent
![Page 5: Assignment P. 526-529: 1-11, 15-21, 33-36, 38, 41, 43 Challenge Problems.](https://reader030.fdocuments.in/reader030/viewer/2022032722/56649cee5503460f949bc64c/html5/thumbnails/5.jpg)
Four Window Foldable
Start by folding a blank piece of paper in half lengthwise, and then folding it in half in the opposite direction. Now fold it in half one more time in the same direction.
![Page 6: Assignment P. 526-529: 1-11, 15-21, 33-36, 38, 41, 43 Challenge Problems.](https://reader030.fdocuments.in/reader030/viewer/2022032722/56649cee5503460f949bc64c/html5/thumbnails/6.jpg)
Four Window Foldable
Now unfold the paper, and then while holding the paper vertically, fold down the top one-fourth to meet the middle. Do the same with the bottom one-fourth.
![Page 7: Assignment P. 526-529: 1-11, 15-21, 33-36, 38, 41, 43 Challenge Problems.](https://reader030.fdocuments.in/reader030/viewer/2022032722/56649cee5503460f949bc64c/html5/thumbnails/7.jpg)
Four Window Foldable
To finish your foldable, cut the two vertical fold lines to create four windows.
Outside: Property 1-4
Inside Flap: Illustration
Inside: Theorem
![Page 8: Assignment P. 526-529: 1-11, 15-21, 33-36, 38, 41, 43 Challenge Problems.](https://reader030.fdocuments.in/reader030/viewer/2022032722/56649cee5503460f949bc64c/html5/thumbnails/8.jpg)
Investigation 1
In this lesson, we will find ways to show that a quadrilateral is a parallelogram. Obviously, if the opposite sides are parallel, then the quadrilateral is a parallelogram. But could we use other properties besides the definition to see if a shape is a parallelogram?
![Page 9: Assignment P. 526-529: 1-11, 15-21, 33-36, 38, 41, 43 Challenge Problems.](https://reader030.fdocuments.in/reader030/viewer/2022032722/56649cee5503460f949bc64c/html5/thumbnails/9.jpg)
8.3 Show a Quadrilateral is a Parallelogram
Objectives:
1. To use properties to identify parallelograms
![Page 10: Assignment P. 526-529: 1-11, 15-21, 33-36, 38, 41, 43 Challenge Problems.](https://reader030.fdocuments.in/reader030/viewer/2022032722/56649cee5503460f949bc64c/html5/thumbnails/10.jpg)
Property 1
We know that the opposite sides of a parallelogram are congruent. What about the converse? If we had a quadrilateral whose opposite sides are congruent, then is it also a parallelogram?
Step 1: Draw a quadrilateral with congruent opposite sides.
D
A C
B
![Page 11: Assignment P. 526-529: 1-11, 15-21, 33-36, 38, 41, 43 Challenge Problems.](https://reader030.fdocuments.in/reader030/viewer/2022032722/56649cee5503460f949bc64c/html5/thumbnails/11.jpg)
Property 1
Step 2: Draw diagonal AD. Notice this creates two triangles. What kind of triangles are they?
D
A C
B D
A C
B D
A C
B
by SSS DCAABD
![Page 12: Assignment P. 526-529: 1-11, 15-21, 33-36, 38, 41, 43 Challenge Problems.](https://reader030.fdocuments.in/reader030/viewer/2022032722/56649cee5503460f949bc64c/html5/thumbnails/12.jpg)
Property 1
Step 3: Since the two triangles are congruent, what must be true about BDA and CAD?
D
A C
B D
A C
B
by CPCTCCADBDA
![Page 13: Assignment P. 526-529: 1-11, 15-21, 33-36, 38, 41, 43 Challenge Problems.](https://reader030.fdocuments.in/reader030/viewer/2022032722/56649cee5503460f949bc64c/html5/thumbnails/13.jpg)
Property 1
Step 4: Now consider AD to be a transversal. What must be true about BD and AC?
D
A C
B
by Converse of Alternate Interior Angles Theorem
ACBD ||
![Page 14: Assignment P. 526-529: 1-11, 15-21, 33-36, 38, 41, 43 Challenge Problems.](https://reader030.fdocuments.in/reader030/viewer/2022032722/56649cee5503460f949bc64c/html5/thumbnails/14.jpg)
Property 1
Step 5: By a similar argument, what must be true about AB and CD?
D
A C
B D
A C
B
by Converse of Alternate Interior Angles Theorem
CDAB ||
![Page 15: Assignment P. 526-529: 1-11, 15-21, 33-36, 38, 41, 43 Challenge Problems.](https://reader030.fdocuments.in/reader030/viewer/2022032722/56649cee5503460f949bc64c/html5/thumbnails/15.jpg)
Property 1
If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.
![Page 16: Assignment P. 526-529: 1-11, 15-21, 33-36, 38, 41, 43 Challenge Problems.](https://reader030.fdocuments.in/reader030/viewer/2022032722/56649cee5503460f949bc64c/html5/thumbnails/16.jpg)
Property 2
We know that the opposite angles of a parallelogram are congruent. What about the converse? If we had a quadrilateral whose opposite angles are congruent, then is it also a parallelogram?
Step 1: Draw a quadrilateral with congruent opposite angles.
D
A C
B
![Page 17: Assignment P. 526-529: 1-11, 15-21, 33-36, 38, 41, 43 Challenge Problems.](https://reader030.fdocuments.in/reader030/viewer/2022032722/56649cee5503460f949bc64c/html5/thumbnails/17.jpg)
Property 2
Step 2: Now assign the congruent angles variables x and y. What is the sum of all the angles? What is the sum of x and y?
D
A C
B
yx
xy
D
A C
B
360yxyx 36022 yx 180yx
![Page 18: Assignment P. 526-529: 1-11, 15-21, 33-36, 38, 41, 43 Challenge Problems.](https://reader030.fdocuments.in/reader030/viewer/2022032722/56649cee5503460f949bc64c/html5/thumbnails/18.jpg)
Property 2
Step 3: Consider AB to be a transversal. Since x and y are supplementary, what must be true about BD and AC?
yx
xy
D
A C
B
by Converse of Consecutive Interior Angles Theorem
ACBD ||
![Page 19: Assignment P. 526-529: 1-11, 15-21, 33-36, 38, 41, 43 Challenge Problems.](https://reader030.fdocuments.in/reader030/viewer/2022032722/56649cee5503460f949bc64c/html5/thumbnails/19.jpg)
Property 2
Step 4: By a similar argument, what must be true about AB and CD?
yx
xy
D
A C
B
by Converse of Consecutive Interior Angles Theorem
CDAB ||
![Page 20: Assignment P. 526-529: 1-11, 15-21, 33-36, 38, 41, 43 Challenge Problems.](https://reader030.fdocuments.in/reader030/viewer/2022032722/56649cee5503460f949bc64c/html5/thumbnails/20.jpg)
Property 2
If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.
![Page 21: Assignment P. 526-529: 1-11, 15-21, 33-36, 38, 41, 43 Challenge Problems.](https://reader030.fdocuments.in/reader030/viewer/2022032722/56649cee5503460f949bc64c/html5/thumbnails/21.jpg)
Property 3
We know that the diagonals of a parallelogram bisect each other. What about the converse? If we had a quadrilateral whose diagonals bisect each other, then is it also a parallelogram?
Step 1: Draw a quadrilateral with diagonals that bisect each other.
E
D
A C
B
![Page 22: Assignment P. 526-529: 1-11, 15-21, 33-36, 38, 41, 43 Challenge Problems.](https://reader030.fdocuments.in/reader030/viewer/2022032722/56649cee5503460f949bc64c/html5/thumbnails/22.jpg)
Property 3
Step 2: What kind of angles are BEA and CED? So what must be true about them? E
D
A C
B
E
D
A C
B
by Vertical Angles Congruence Theorem
CEDBEA
![Page 23: Assignment P. 526-529: 1-11, 15-21, 33-36, 38, 41, 43 Challenge Problems.](https://reader030.fdocuments.in/reader030/viewer/2022032722/56649cee5503460f949bc64c/html5/thumbnails/23.jpg)
Property 3
Step 3: Now what must be true about AB and CD?
E
D
A C
B
by SAS and CPCTCCDAB
E
D
A C
B
![Page 24: Assignment P. 526-529: 1-11, 15-21, 33-36, 38, 41, 43 Challenge Problems.](https://reader030.fdocuments.in/reader030/viewer/2022032722/56649cee5503460f949bc64c/html5/thumbnails/24.jpg)
Property 3
Step 4: By a similar argument, what must be true about BD and AC?
E
D
A C
B
by SAS and CPCTCACBD
E
D
A C
B
E
D
A C
B
E
D
A C
B
![Page 25: Assignment P. 526-529: 1-11, 15-21, 33-36, 38, 41, 43 Challenge Problems.](https://reader030.fdocuments.in/reader030/viewer/2022032722/56649cee5503460f949bc64c/html5/thumbnails/25.jpg)
Property 3
Step 5: Finally, if the opposite sides of our quadrilateral are congruent, what must be true about our quadrilateral?
E
D
A C
B
ABDC is a parallelogram by Property 1
E
D
A C
B
E
D
A C
B
E
D
A C
B
![Page 26: Assignment P. 526-529: 1-11, 15-21, 33-36, 38, 41, 43 Challenge Problems.](https://reader030.fdocuments.in/reader030/viewer/2022032722/56649cee5503460f949bc64c/html5/thumbnails/26.jpg)
Property 3
If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.
![Page 27: Assignment P. 526-529: 1-11, 15-21, 33-36, 38, 41, 43 Challenge Problems.](https://reader030.fdocuments.in/reader030/viewer/2022032722/56649cee5503460f949bc64c/html5/thumbnails/27.jpg)
Property 4
The last property is not a converse, and it is not obvious. The question is, if we had a quadrilateral with one pair of sides that are congruent and parallel, then is it also a parallelogram?
Step 1: Draw a quadrilateral with one pair of parallel and congruent sides.
D
A C
B
![Page 28: Assignment P. 526-529: 1-11, 15-21, 33-36, 38, 41, 43 Challenge Problems.](https://reader030.fdocuments.in/reader030/viewer/2022032722/56649cee5503460f949bc64c/html5/thumbnails/28.jpg)
Property 4
Step 2: Now draw in diagonal AD. Consider AD to be a transversal. What must be true about BDA and CAD?
D
A C
B D
A C
B D
A C
B
by Alternate Interior Angles Theorem
CADBDA
![Page 29: Assignment P. 526-529: 1-11, 15-21, 33-36, 38, 41, 43 Challenge Problems.](https://reader030.fdocuments.in/reader030/viewer/2022032722/56649cee5503460f949bc64c/html5/thumbnails/29.jpg)
Property 4
Step 3: What must be true about ABD and DCA? What must be true about AB and CD?
D
A C
B D
A C
B D
A C
B D
A C
B D
A C
B
by SAS and CPCTC
CDAB
![Page 30: Assignment P. 526-529: 1-11, 15-21, 33-36, 38, 41, 43 Challenge Problems.](https://reader030.fdocuments.in/reader030/viewer/2022032722/56649cee5503460f949bc64c/html5/thumbnails/30.jpg)
Property 4
Step 4: Finally, since the opposite sides of our quadrilateral are congruent, what must be true about our quadrilateral?
D
A C
B D
A C
B D
A C
B D
A C
B D
A C
B
ABDC is a parallelogram by Property 1
![Page 31: Assignment P. 526-529: 1-11, 15-21, 33-36, 38, 41, 43 Challenge Problems.](https://reader030.fdocuments.in/reader030/viewer/2022032722/56649cee5503460f949bc64c/html5/thumbnails/31.jpg)
Property 4
If one pair of opposite sides of a quadrilateral are congruent and parallel, then the quadrilateral is a parallelogram.
![Page 32: Assignment P. 526-529: 1-11, 15-21, 33-36, 38, 41, 43 Challenge Problems.](https://reader030.fdocuments.in/reader030/viewer/2022032722/56649cee5503460f949bc64c/html5/thumbnails/32.jpg)
Example 1
In quadrilateral WXYZ, mW = 42°, mX = 138°, and mY = 42°. Find mZ. Is WXYZ a parallelogram? Explain your reasoning.
![Page 33: Assignment P. 526-529: 1-11, 15-21, 33-36, 38, 41, 43 Challenge Problems.](https://reader030.fdocuments.in/reader030/viewer/2022032722/56649cee5503460f949bc64c/html5/thumbnails/33.jpg)
Example 2
For what value of x is the quadrilateral below a parallelogram?
![Page 34: Assignment P. 526-529: 1-11, 15-21, 33-36, 38, 41, 43 Challenge Problems.](https://reader030.fdocuments.in/reader030/viewer/2022032722/56649cee5503460f949bc64c/html5/thumbnails/34.jpg)
Example 3
Determine whether the following quadrilaterals are parallelograms.
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Example 4
Construct a flowchart to prove that if a quadrilateral has congruent opposite sides, then it is a parallelogram.
Given: AB CD BC ADProve: ABCD is a
parallelogram
CB
DA
CB
DA
![Page 36: Assignment P. 526-529: 1-11, 15-21, 33-36, 38, 41, 43 Challenge Problems.](https://reader030.fdocuments.in/reader030/viewer/2022032722/56649cee5503460f949bc64c/html5/thumbnails/36.jpg)
Summary
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Assignment
• P. 526-529: 1-11, 15-21, 33-36, 38, 41, 43
• Challenge Problems