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Transcript
Name ________________________________________ Date ___________________ Class __________________
LESSON
6-3
Reteach
Conditions for Parallelograms
You can use the following conditions to
determine whether a quadrilateral such
as PQRS is a parallelogram.
Conditions for Parallelograms
QR ≅ SP
QR & SP
QR ≅ SP
If one pair of opposite sides is
and ≅, then PQRS is a parallelogram.
.
∠P ≅ ∠R
∠Q ≅ ∠S
If both pairs of opposite angles are ≅,
then PQRS is a parallelogram.
PQ ≅ RS
If both pairs of opposite sides are ≅, then
PQRS is a parallelogram.
PT ≅ RT
QT ≅ ST
If the diagonals bisect each other, then
PQRS is a parallelogram.
A quadrilateral is also a parallelogram if one of the angles is
supplementary to both of its consecutive angles.
65° + 115° = 180°, so ∠A is supplementary to ∠B and ∠D.
Therefore, ABCD is a parallelogram.
Show that each quadrilateral is a parallelogram for the given values.
Explain.
1. Given: x = 9 and y = 4
2. Given: w = 3 and z = 31
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Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor.
6-22
Holt McDougal Geometry
Name ________________________________________ Date ___________________ Class __________________
LESSON
6-3
Reteach
Conditions for Parallelograms continued
You can show that a quadrilateral is a parallelogram by using any of the conditions
listed below.
Conditions for Parallelograms
• Both pairs of opposite sides are parallel (definition).
• One pair of opposite sides is parallel and congruent.
• Both pairs of opposite sides are congruent.
• Both pairs of opposite angles are congruent.
• The diagonals bisect each other.
• One angle is supplementary to both its consecutive angles.
EFGH must be a parallelogram
because both pairs of opposite
sides are congruent.
JKLM may not be a parallelogram
because none of the sets of conditions
for a parallelogram is met.
Determine whether each quadrilateral must be a parallelogram.
Justify your answer.
4.
3.
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6.
5.
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Show that the quadrilateral with the given vertices is a parallelogram by
using the given definition or theorem.
7. J(−2, −2), K(−3, 3), L(1, 5), M(2, 0)
Both pairs of opposite sides are parallel.
8. N(5, 1), P(2, 7), Q(6, 9), R(9, 3)
Both pairs of opposite sides are
congruent.
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Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor.
6-23
Holt McDougal Geometry
5. No, x° + x° may not be 180°.
triangles have measure 45°, so all the
angles of the parallelogram have
measure 90°.
6. slope of JK = slope of LM = 1; slope of
2
KL = slope of JM = − ; JKLM is a
3
parallelogram.
7. PQ = RS = 26 ; QR = PS = 5 2 ; PQRS
is a parallelogram.
8. Possible answer: UV = TW = 2 5 ; slope
of UV = slope of TW = 2; TUVW is a
parallelogram.
6. 90°
Reteach
1. QR = ST = 12; RS = TQ = 16; both pairs
of opp. sides are ≅.
Practice C
1. A(4, 4), B(2, 5), C(−2, −5), D(0,
2. DE = FC = 10; m∠E = 118° and m∠F =
62°, so ∠E and ∠F are supp. and
DE || FC ; one pair of opposite sides are ||
and ≅.
6)
2. Possible answer: ∠A ≅ ∠C
3. AB || CD ; possible answer: because ∠A
≅ ∠B and ∠C ≅ ∠D and the sum of the
interior angle measures of a quadrilateral
is 360°, 2m∠A + 2m∠D = 360° or 2(m∠A
+ m∠D) = 360°. Therefore m∠A + m∠D =
180°. ∠A and ∠D are supplementary, so
by the Converse of the Same-Side
Interior Angles Theorem, AB || CD
4. All four sides are congruent, and the two
pairs of opposite angles are congruent;
possible answer: because the diagonals
are perpendicular, all four angles created
by the intersecting diagonals are right
angles and therefore congruent. And
because the diagonals bisect each other,
all four of the right triangles are congruent
by SAS. By CPCTC, all four of the
parallelogram’s sides must be congruent.
The two pairs of opposite angles are
congruent as for any parallelogram.
3. Yes; one pair of opp. sides is || and ≅.
4. Yes; the diagonals bisect each other.
5. No; none of the sets of conditions for a
parallelogram is met.
6. Yes; both pairs of opp. ∠ are ≅.
S
7. slope of JK = slope of LM = −5;
slope of KL = slope of MJ =
1
2
8. NP = QR = 3 5 ; PQ =RN = 2 5
Challenge
1.
2. –11. Arrangements will vary.
Problem Solving
1. Yes; both pairs of opposite sides of
quadrilateral LMNP remain congruent, so
LMNP is always a .
2. 56°
.
3. Possible answer: m∠F = 120°
5. All four sides are congruent, and all four
angles are congruent; possible answer:
the sides are congruent for the same
reasons given in Exercise 4. But because
the diagonals are congruent and
bisected, each right triangle created by
the diagonals is an isosceles right
triangle. The acute angles of these
4. Possible answer: y = −x + 1; both pairs of
opposite sides have the same slope, so
they are parallel.
5. C
6. H
Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor.
A62
Holt McDougal Geometry