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Tests for Parallelograms
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Proving Quadrilaterals as Parallelograms5 ways
If both pairs of opposite sides of a quadrilateral are parallel, then the quadrilateral is a parallelogram .
From definition:
H G
E F
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Proving Quadrilaterals as Parallelograms
If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram .
Theorem 1:
H G
E FIf one pair of opposite sides of a quadrilateral are both congruent and parallel, then the quadrilateral is a parallelogram .
Theorem 2:
If EF GH; FG EH, then Quad. EFGH is a parallelogram.
If EF GH and EF || HG, then Quad. EFGH is a parallelogram.
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Theorem:
If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.
Theorem 3:
If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram .
Theorem 4:
H G
EF
M
,If H F and E G
then Quad. EFGH is a parallelogram.
intIf M is themidpo of EG and FH
then Quad. EFGH is a parallelogram. EM = GM and HM = FM
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5 ways to prove that a quadrilateral is a parallelogram.
1. Show that both pairs of opposite sides are || . [definition]
2. Show that both pairs of opposite sides are .
3. Show that one pair of opposite sides are both and || .4. Show that both pairs of opposite angles are .
5. Show that the diagonals bisect each other .
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Examples ……Find the value of x and y that ensures the quadrilateral is a parallelogram.
Example 1:
6x4x+8
y+2
2y
6x = 4x+8
2x = 8
x = 4 units
2y = y+2
y = 2 unit
Example 2: Find the value of x and y that ensure the quadrilateral is a parallelogram.
120° 5y°
(2x + 8)°2x + 8 = 120
2x = 112
x = 56 units
5y + 120 = 180
5y = 60
y = 12 units