Download Proving that a quadrilateral is a Parallelogram

CONDITIONS FOR
PARALLELOGRAMS
Geometry CP1 (Holt 6-3)
K.Santos
THEOREM 6-3-1
If one pair of opposite sides of a quadrilateral is
both congruent and parallel then the quadrilateral
is a parallelogram.
Q
R
P
S
Given: 𝑄𝑅 ≅ 𝑃𝑆 and 𝑄𝑅|| 𝑃𝑆
Then: quadrilateral PQRS is a parallelogram
THEOREM 6-3-2
If both pairs of opposite sides of a quadrilateral are
congruent, then the quadrilateral is a
parallelogram.
Q
R
P
S
Given: 𝑄𝑅 ≅ 𝑃𝑆 and 𝑃𝑄 ≅ 𝑆𝑅
Then: quadrilateral PQRS is a parallelogram
THEOREM 6-3-3
If both pairs of opposite angles of a quadrilateral
are congruent, then the quadrilateral is a
parallelogram.
Q
R
P
S
Given: < P ≅< R and < Q ≅ < S
Then: quadrilateral PQRS is a parallelogram
THEOREM 6-3-4
If an angle of a quadrilateral is supplementary to
both of is consecutive angles then the quadrilateral
is a parallelogram.
A
B
D
Given: <A is supplementary to <B
<A is supplementary to <D
Then: parallelogram ABCD
C
THEOREM 6-3-5
If the diagonals of a quadrilateral bisect each other,
then the quadrilateral is a parallelogram.
Q
R
P
S
Given: 𝑄𝑆 & 𝑃𝑅 bisect each other
Then: quadrilateral PQRS is a parallelogram
WAYS TO PROVE A QUADRILATERAL IS A
PARALLELOGRAM
Show one of the following is true with the quadrilateral:
--Both pairs of opposite sides are parallel (definition)
--One pair of opposite sides is both congruent and
parallel
--Both pairs of opposite sides are congruent
--Both pairs of opposite angles are congruent
--One angle is supplementary to both of its consecutive
angles
--Both diagonals bisect each other
EXAMPLE—IS IT A PARALLELOGRAM???
Can you prove the quadrilateral is a parallelogram from
what is given? Explain.
1.
75°
105°
2.
7
7
105°
Yes—both pairs of
opposite angles congruent
yes—one pair of opposite
sides parallel & congruent
EXAMPLE—IS IT A PARALLELOGRAM
Determine if each quadrilateral must be a
parallelogram. Justify your answer.
Parallelogram--The diagonals bisect each other.
No—different pairs of opposite sides congruent and
parallel
EXAMPLE—SHOW QUAD. IS PARALLELOGRAM
Show PQRS is a parallelogram for a = 2.4 and b = 9.
Q 10b-16
2a+12
P
9b+25 R
7a
S
Substitute the values in for a and b
7a= 7(2.4) = 16.8
2a + 12= 2(2.4) + 12 = 16.8 opposite sides congruent
10b – 16 = 10(9) -16 = 74°
9b + 25 = 9(9) + 25 = 106° consecutive angles supplementary (parallel
sides)
Parallelogram---one pair of opposite sides is both parallel and congruent
EXAMPLE--NUMERICAL
For what value of the variables must the quadrilateral be a parallelogram?
Q a + 40°
a° R
3c – 3
c+1
P
S
Opposite sides in parallelogram are congruent (QP = RS)
3c – 3 = c + 1
2c – 3 = 1
2c = 4
c=2
Consecutive angles in a parallelogram are supplementary (<Q & <R)
a + 40 + a = 180°
2a + 40 = 180
2a = 140
a = 70°
PROOF
Given: parallelogram ABCD
parallelogram EFGD
Prove: < B ≅ < F
Statements
1. Parallelogram ABCD
2. < B ≅ < D
3. parallelogram EFGD
4. <F ≅ < D
5. <B ≅ < F
A
B
E
F
D
G
Reasons
1. Given
2. Opposite angles of a
parallelogram are
congruent
3. Given
4. opposite angles of a
parallelogram are
congruent
5. Transitive Property of
Congruence (2,4)
C