Assignment
description
Transcript of Assignment
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Assignment• P. 526-529: 1-11,
15-21, 33-36, 38, 41, 43
• Challenge Problems
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Proving Lines Parallel
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Proving Triangles Congruent
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Proving Triangles Congruent
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Four Window FoldableStart 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.
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Four Window FoldableNow 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.
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Four Window FoldableTo finish your foldable,
cut the two vertical fold lines to create four windows.
Outside: Property 1-4Inside Flap: IllustrationInside: Theorem
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Investigation 1In 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?
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8.3 Show a Quadrilateral is a Parallelogram
Objectives:1. To use properties to identify
parallelograms
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Property 1We 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
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Property 1Step 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
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Property 1Step 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
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Property 1Step 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 ||
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Property 1Step 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 ||
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Property 1If both pairs of opposite sides of a
quadrilateral are congruent, then the quadrilateral is a parallelogram.
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Property 2We 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
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Property 2Step 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
Byx
xy
D
A C
B
360yxyx 36022 yx 180yx
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Property 2Step 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 ||
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Property 2Step 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 ||
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Property 2If both pairs of opposite angles of a
quadrilateral are congruent, then the quadrilateral is a parallelogram.
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Property 3We 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
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Property 3Step 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
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Property 3Step 3: Now what
must be true about AB and CD?
E
D
A C
B
by SAS and CPCTCCDAB
E
D
A C
B
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Property 3Step 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
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Property 3Step 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
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Property 3If the diagonals of a quadrilateral bisect each
other, then the quadrilateral is a parallelogram.
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Property 4The 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
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Property 4Step 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
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Property 4Step 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
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Property 4Step 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
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Property 4If one pair of opposite sides of a quadrilateral
are congruent and parallel, then the quadrilateral is a parallelogram.
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Example 1In quadrilateral WXYZ, mW = 42°, mX =
138°, and mY = 42°. Find mZ. Is WXYZ a parallelogram? Explain your reasoning.
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Example 2For what value of x is the quadrilateral below
a parallelogram?
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Example 3Determine whether the following
quadrilaterals are parallelograms.
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Example 4Construct 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
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Summary
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Assignment• P. 526-529: 1-11,
15-21, 33-36, 38, 41, 43
• Challenge Problems