6.3 Proving Quadrilaterals are Parallelograms Geometry Handbook

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6.3 Proving Quadrilaterals are Parallelograms
Objective:
Prove that a given quadrilateral is a
parallelogram.
Handbook, p. 19
6.3 Proving Quadrilaterals are Parallelograms
If…
then...
both pairs of opposite sides are
parallel (definition),
>>
one pair of opposite sides are
BOTH parallel and congruent (6.3.1),
>>
>
>
it’s a parallelogram.
it’s a parallelogram.
both pairs of opposite sides are
congruent (6.3.2),
it’s a parallelogram.
both pairs of opposite angles are
congruent (6.3.3),
it’s a parallelogram.
one angle is supplementary to 1
both consecutive angles (6.3.4),2
3
the diagonals bisect each other (6.3.5),
m 2  m 1  180
m 2  m 3  180
it’s a parallelogram.
it’s a parallelogram.
6.3 Proving Quadrilaterals are Parallelograms
Example 1:
Determine if the quadrilateral
must be a parallelogram. Justify
your answer.
Definition: Both pairs of opposite sides are parallel.
6-3-1: One pair of opposite sides are parallel and congruent.
6-3-2: Both pairs of opposite sides are congruent
6-3-3: Both pairs of opposite angles are congruent.
6-3-4: One angle is supplementary to both consecutive angles.
6-3-5: The diagonals bisect each other.
By 6-3-4, the quadrilateral must be a parallelogram.
6.3 Proving Quadrilaterals are Parallelograms
Example 2:
Determine if the quadrilateral
must be a parallelogram.
Justify your answer.
Definition: Both pairs of opposite sides are parallel.
6-3-1: One pair of opposite sides are parallel and congruent.
6-3-2: Both pairs of opposite sides are congruent
6-3-3: Both pairs of opposite angles are congruent.
6-3-4: One angle is supplementary to both consecutive angles.
6-3-5: The diagonals bisect each other.
Only one pair of opposite angles are congruent, so not
enough information.
6.3 Proving Quadrilaterals are Parallelograms
Example 3:
Determine if the quadrilateral must
be a parallelogram. Justify your
answer.
Is there anything else we can add to our picture?
Definition: Both pairs of opposite sides are parallel.
6-3-1: One pair of opposite sides are parallel and congruent.
6-3-2: Both pairs of opposite sides are congruent
6-3-3: Both pairs of opposite angles are congruent.
6-3-4: One angle is supplementary to both consecutive angles.
6-3-5: The diagonals bisect each other.
Both pairs of opposite angles are congruent, so the
quadrilateral must be a parallelogram.
6.3 Proving Quadrilaterals are Parallelograms
Example 4:
Determine if the quadrilateral must
be a parallelogram. Justify your
answer.
Definition: Both pairs of opposite sides are parallel.
6-3-1: One pair of opposite sides are parallel and congruent.
6-3-2: Both pairs of opposite sides are congruent
6-3-3: Both pairs of opposite angles are congruent.
6-3-4: One angle is supplementary to both consecutive angles.
6-3-5: The diagonals bisect each other.
Consecutive sides, not opposite sides are marked
congruent, so the quadrilateral IS NOT a parallelogram.
6.3 Proving Quadrilaterals are Parallelograms
Example 5:
Show that JKLM is a
parallelogram for a = 3
and b = 9.
Definition: Both pairs of opposite sides are parallel.
6-3-1: One pair of opposite sides are parallel and congruent.
6-3-2: Both pairs of opposite sides are congruent
6-3-3: Both pairs of opposite angles are congruent.
6-3-4: One angle is supplementary to both consecutive angles.
6-3-5: The diagonals bisect each other.
15(3)  11  10(3)  4
34  34
5(9)  6  8(9)  21
51  51
Since both pairs of opposite sides are congruent JKLM is a
parallelogram.
6.3 Proving Quadrilaterals are Parallelograms
Example 6:
Show that PQRS is a
parallelogram for a = 2.4
and b = 9.
What will a and b let me find?
7(2.4)  2(2.4)  12
16.8  16.8
10(9)  16  9(9)  25  180
90  16  81  25  180
180  180
Definition: Both pairs of opposite sides are parallel.
6-3-1: One pair of opposite sides are parallel and congruent.
6-3-2: Both pairs of opposite sides are congruent
6-3-3: Both pairs of opposite angles are congruent.
6-3-4: One angle is supplementary to both consecutive angles.
6-3-5: The diagonals bisect each other.
Since one pair of opposite sides are both parallel and
congruent, PQRS is a parallelogram.
6.3 Proving Quadrilaterals are Parallelograms
Example 7:
Show that quadrilateral JKLM is a parallelogram by using the
definition of parallelogram. J(–1, –6), K(–4, –1), L(4, 5), M(7, 0).
Definition: Opposite sides are parallel, so we need to show:
slope JK = slope LM
1  6
4  1
5
3
05
74
5
3
slope KL = slope JM
5  1
4  4
6
8
0  6
7  1
6
8
Since slopes of opposite sides are the same, the opposite
sides are parallel.
6.3 Proving Quadrilaterals are Parallelograms
Example 8:
Show that quadrilateral ABCD is a parallelogram by using Theorem
6-3-1 if A(2, 3), B(6, 2), C(5, 0), and D(1, 1).
6.3 Assignment
p. 402: 9-23, 26

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