Download Aim #31: How do we prove a quadrilateral is a parallelogram? Do

Aim #31: How do we prove a quadrilateral is a parallelogram?
CC Geometry R
f
Do Now:
Name the figure as specifically as possible based on the given characteristics:
1. a quadrilateral with both pairs of opposite sides equal
2. a quadrilateral with that has both pairs of opposite sides parallel and one
right angle
3. a parallelogram with equal diagonals
4. a quadrilateral that is equilateral
Square
5. a regular quadrilateral
To prove a quadrilateral is a parallelogram, prove any one of the following:
Both pairs of opposite sides are parallel
1. _______________________________________________
Both pairs of opposite sides congruent
2. _______________________________________________
ONE pair of opposite sides both parallel and congruent
3. _______________________________________________
Both pairs of opposite angles congruent
4. _______________________________________________
Diagonals bisect each other
5. _______________________________________________
No, Only one pair of opposite sides congruent
Yes, both pairs of opposite sides congruent
No, Opposite sides are not congruent
No, only one pair of opposite sides parallel
Yes, Both pairs of opposite sides parallel
No, one pair of opposite sides congruent, the other parallel
Yes, one pair of opposite sides both parallel and congruent
No, only one pair opposite sides congruent
Yes, diagonals bisect each other
Yes, both pairs opposite angles congruent
D
1. Given: DB bisects AC, ≮1 ≅ ≮2
Prove: ABCD is a
E
.
1
A
Statements
C
2
Reasons
B
1) DB bisects AC, <1 = <2
1) Given
2) AE = EC
2) bisectors divide a segment into 2 congruent
segments
3) <AEB = <DEC
3) Vertical angles are congruent
4) tri AEB = Tri DEC
4)ASA
5) DE = EB
5) CPCTC
6) ABCD is a parallelogram
6) When Diagonals bisect each other, a
Quadrilateral is a parallelogram
D
2. Given: C
Prove: ABCD is a parallelogram
Statements
A
Reasons
B
1) AB = CD, <BDC = <DBA
1) Given
2) DC ll AB
2) When alternate interior angles are congruent
3) ABCD is a prallelogram
3) When one pair of opposite sides are
Lines cut by a transversal are parallel
parallel and congruent, a quadrilateral is a
parallelogram
4
3.
~
2
1
3
Statements
1) AD ll CB, AE = FC , AE
Reasons
DB, FC
DB 1) Given
2) <1 = <2, Both are right <s
2)
3) <3 = <4
3) When ll lines are cut by a transversal,
Lines form = Right <s
alternate interior angles are congruent
4) Tri AED = Tri CFB
4) AAS
5) AD = BC
5) CPCTC
6) ABCD is a parallelogram
6) When one pair of opposite sides is both
congruent and parallel, the quadrilateral is
a parallelogram
4. Given: AD || BC
E is the midpoint of AC. 1
4
Prove: ABCD is a Parallelogram
Statements
3
2
Reasons
1) AD ll BC, E is the midpoint of AC
1) Given
2) AE = EC
2) A midpoint Divides a segment into 2
congruent segments
3) <1=<2, <3=<4
3) When ll lines are cut by a transversal
Alternate interior <s are congruent
4) Tri AED = Tri BEC
4) AAS
5) AD = BC
5) CPCTC
6) ABCD is a parallelogram
6) When one pair of opposite sides are
parallel and congruent, a quadrilateral is a
parallelogram
5.
~
~
Statements
Reasons
1) <ABD = < BDC, <DAC = < ACB
1) Given
2) AD ll BC, AB ll DC
2) When alternate interior angles are congruent
lines cut by a transversal are parallel
3) ABCD is a parallelogram
3) When Both pairs of opposite sides are
parallel, a Quadrilateral is a parallelogram
6. Given: A
bisects
1
3
B
E
Prove: ABCD is a parallelogram
2
4
C
D
Statements
Reasons
1) AC bisects DB, AB ll DC
1) Given
2) DE = EB
2) A bisector Divides a segment into 2
congruent segments
3) <1=<2, <3=<4
3) when parallel lines are cut by a transversal,
Alternate interior angles are congruent
4) Tri AEB = Tri DEC
4) AAS
5) AB = DC
5) CPCTC
6) ABCD is a parallelogram
6) When one pair of opposite sides is parallel
and congruent, a quad is a parallelogram
Exercises:
7. Given: ABCD is a parallelogram,
R
A
B
RD bisects ≮ADC, SB bisects ≮CBA
D
Prove: DRBS is a parallelogram
S
C
Reasons
Statements
1. ABCD is a parallelogram,
RD bisects ≮ADC, SB bisects ≮CBA
1. Given
2. AD ≅ CB
2. Opposite sides of a parallelogram are
congruent
3. ≮A ≅ ≮C, ≮ABC ≅ ≮CDA
3. Opposite <s of a parallelogram are
congruent
4. ≮RDA ≅ ≮RDC, ≮SBC ≅ ≮SBA
4. An angle bisector divides an angle into 2
5. ≮RDA ≅ ≮SBC, ≮RDC ≅ ≮SBA
5. 1/2s of equals are equal
6. ΔDAR ≅ ΔBCS
6. ASA
7. ≮ARD ≅ ≮CSB
7. CPCTC
8. ≮DRB ≅ ≮BSD
8. Congruent <s have congruent
congruent angles
Supplements
9. DRBS is a parallelogram.
9. When both pairs of opposite angles are
congruent, a Quad is a parallelogram