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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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Holt McDougal Geometry
Name _______________________________________ Date ___________________ Class __________________
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.
3.
4.
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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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created by the diagonals is an isosceles
right triangle. The acute angles of these
triangles have measure 45°, so all the
angles of the parallelogram have
measure 90°.
5.No, x  x may not be 180.
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.
Practice C
1. A(4, 4), B(2, 5), C(2, 5), D(0, 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.
6. 90°
Reteach
1. QR  ST  12; RS  TQ  16; both pairs
of opp. sides are .
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 .
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
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
 3. Possible answer: mF  120
  Possible answer: y  x  1; both pairs
of opposite sides have the same slope,
so they are parallel.
5. C
6. H
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