Assignment
37
Assignment • P. 526-529: 1- 11, 15-21, 33- 36, 38, 41, 43 • Challenge Problems
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Assignment. P. 526-529: 1-11, 15-21, 33-36, 38, 41, 43 Challenge Problems. Proving Lines Parallel. Proving Triangles Congruent. Proving Triangles Congruent. Four Window Foldable. - PowerPoint PPT Presentation
Transcript of Assignment
Assignment• P. 526-529: 1-11,
15-21, 33-36, 38, 41, 43
• Challenge Problems
Proving Lines Parallel
Proving Triangles Congruent
Proving Triangles Congruent
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.
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.
Four Window FoldableTo finish your foldable,
cut the two vertical fold lines to create four windows.
Outside: Property 1-4Inside Flap: IllustrationInside: Theorem
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?
8.3 Show a Quadrilateral is a Parallelogram
Objectives:1. To use properties to identify
parallelograms
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
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
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
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 ||
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 ||
Property 1If both pairs of opposite sides of a
quadrilateral are congruent, then the quadrilateral is a parallelogram.
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
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
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 ||
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 ||
Property 2If both pairs of opposite angles of a
quadrilateral are congruent, then the quadrilateral is a parallelogram.
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
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
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
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
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
Property 3If the diagonals of a quadrilateral bisect each
other, then the quadrilateral is a parallelogram.
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
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
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
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
Property 4If one pair of opposite sides of a quadrilateral
are congruent and parallel, then the quadrilateral is a parallelogram.
Example 1In quadrilateral WXYZ, mW = 42°, mX =
138°, and mY = 42°. Find mZ. Is WXYZ a parallelogram? Explain your reasoning.
Example 2For what value of x is the quadrilateral below
a parallelogram?
Example 3Determine whether the following
quadrilaterals are parallelograms.
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
Summary
Assignment• P. 526-529: 1-11,
15-21, 33-36, 38, 41, 43
• Challenge Problems
