given: wxyz is a parallelogram prove: Δ yzx Δ wxz
DESCRIPTION
WARM UP. 1) Complete the Proof. Z. Y. Given: WXYZ is a parallelogram Prove: Δ YZX Δ WXZ. W. X. 1. WXYX is a. 1. Given. 2. Opposite sides of a parallelogram are congruent. 2. WX ZY, WZ YX. 3. ZX ZX. 3. Reflexive Property. 4. Δ YZX Δ WXZ. - PowerPoint PPT PresentationTRANSCRIPT
Statements Reasons
Given: WXYZ is a parallelogram
Prove: ΔYZX ΔWXZW X
YZ1) Complete the Proof.
2. WX ZY, WZ YX
1. WXYX is a
3. ZX ZX
4. ΔYZX ΔWXZ
1. Given
3. Reflexive Property
2. Opposite sides of a parallelogram are congruent.
4. SSS Postulate
Statements Reasons
Given: BCDE is a parallelogram
AE CD
Prove: EAB EBA
B C
DE2) Complete the Proof.
2. EB DC
1. BCDE is a
3. AE CD
4. EB AE
1. Given
3. Given
2. Opposite sides of a parallelogram are congruent.
4. Substitution
A
5. EAB EBA5. If two sides of a triangle are , then the angles opposite those sides are .
Pg.1689) Def. of Parallelogram10) If lines are ||, alternate interior angles are congruent.11) Opposite angles of a parallelogram are congruent.12) Opposite sides of a parallelogram are congruent.13) Diagonals of a parallelogram bisect each other.14) Diagonals of a parallelogram bisect each other.
Pg.1695)a = 8, b = 10, x = 118, y = 626)a = 8, b = 15, x = 80, y = 707)a = 5, b = 3, x = 120, y = 228)a = 9, b = 11, x = 33, y = 279)a = 8, b = 8, x = 56, y = 6810) a = 10, b = 4, x = 90, y = 45
Pg.16911) Perimeter = 6012) ST = 14, SP = 13
Pg.17019) x = 3, y = 520) x = 7, y = 1821) x = 13, y = 5
1) Name all the properties of a parallelogram.
•
•
•
•
•
2 pairs of opposite sides are parallel2 pairs of opposite sides are congruent2 pairs of opposite angles are congruent
Consecutive angles are supplementary
Diagonals bisect each other
A B
CD
M
DAC _____
AB = _____
mBCD = _____
CM = _____
AD || _____
DA = _____
BAC _____
BM = _____
Given the below
parallelogram, complete the statements.
BCA
mDAB DCA
CD
AM
BC
BC
DM
Theorem 5-4: If both pairs of opposite sides of a quadrilateral are congruent,
then the quadrilateral is a parallelogram.E F
GH
PROOF OF THEOREM 5-4: E F
GH
12
34
Given: EF GH, FG EH
Prove: EFGH is a
6. Def. of parallelogram
3. ΔEFH ΔGHF 3. SSS Postulate
4. CPCTC4. 1 4, 2 3
2. Reflexive Property
5. If alternate interior angles are congruent, then lines are parallel.5. EF || GH, FG || HE
2. FH FH
1. EF GH; FG EH 1. Given
6. EFGH is a
Theorem 5-5: If one pair of opposite sides of a quadrilateral are both congruent and parallel, then the quadrilateral is a parallelogram.
E F
GH
PROOF OF THEOREM 5-5:
Given: EF GH, EF || GH
Prove: EFGH is a
6. If both pairs of opposite sides of a quadrilateral are congruent, then it is a parallelogram.
4. ΔEFH ΔGHF 4. SAS Postulate
5. CPCTC
2. 1 4
3. Reflexive Property
2. If lines are parallel, alternate interior angles are congruent.
5. FG HE
3. FH FH
1. EF GH; EF || GH 1. Given
6. EFGH is a
E F
GH
12
34
Theorem 5-6: If both pairs of opposite angles of a quadrilateral are
congruent, then the quadrilateral is a parallelogram.
D C
BA
PROOF OF THEOREM 5-6: D C
BA
x y
y x
Given: mA = mC = y;mB = mD = x
Prove: ABCD is a
6. Def. of parallelogram
3. x + y = 180 3. Division Property
4. Definition of supp. angles4. A and D are supp.
2. The sum of the interior angles of a quadrilateral is 360.
5. If same-side interior angles are supplementary, then lines are parallel.
5. AB || CD, AD || BC
2. 2x + 2y = 360
1. mA = mC = y;mB = mD = x
1. Given
6. ABCD is a
A and B are supp.
Theorem 5-7: If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.
Q R
ST
M
Based on the markings on each figure,
A. Decide if each figure is a parallelogram (YES or NO).
B. If yes, justify your answer. State the theorem that is supported by the figure.
If no, identify which theorem is not justified or is not met by the
diagram.
EXAMPLE 1:
EXAMPLE 2:
NO
YES
Opposite sides are not
congruent.
Both pairs of opposite sides are
parallel.
EXAMPLE 3:
EXAMPLE 4:
YES
YESDiagonals bisect
each other.
Both pairs of opposite angles are congruent.
EXAMPLE 5:
EXAMPLE 6:
NO
YES
Pair of congruent/parallel
sides is not the same pair of
sides.
One pair of sides is both congruent
and parallel.