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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
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.
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.
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
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?
8.3 Show a Quadrilateral is a Parallelogram
Objectives:
1. To use properties to identify parallelograms
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
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
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
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 ||
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 ||
Property 1
If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.
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
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
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 ||
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 ||
Property 2
If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.
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
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
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
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
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
Property 3
If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.
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
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
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
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
Property 4
If one pair of opposite sides of a quadrilateral are congruent and parallel, then the quadrilateral is a parallelogram.
Example 1
In quadrilateral WXYZ, mW = 42°, mX = 138°, and mY = 42°. Find mZ. Is WXYZ a parallelogram? Explain your reasoning.
Example 2
For what value of x is the quadrilateral below a parallelogram?
Example 3
Determine whether the following quadrilaterals are parallelograms.
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
Summary
Assignment
• P. 526-529: 1-11, 15-21, 33-36, 38, 41, 43
• Challenge Problems
