Lesson 6-2

Tests for Parallelograms

Proving Quadrilaterals as Parallelograms

Theorem 1:

If EF  If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram .

GH; FG  EH, then Quad. EFGH is a parallelogram.

H G Theorem 2: E F

If one pair of opposite sides of a quadrilateral are both congruent and parallel, then the quadrilateral is a parallelogram .

If EF  GH and EF || HG, then Quad. EFGH is a parallelogram.

Theorem:

Theorem 3:

If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.

H G

If F and G

,

then Quad. EFGH is a parallelogram.

M Theorem 4: E

If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram .

If M is the midpo

int

of EG and FH

F then Quad. EFGH is a parallelogram.

EM = GM and HM = FM

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 .

Examples ……

Example 1:

Find the value of x and y that ensures the quadrilateral is a parallelogram.

y+2

6x = 4x+8 2y = y+2

6x

2x = 8

y = 2 unit 4x+8 x = 4 units 2y Example 2:

Find the value of x and y that ensure the quadrilateral is a parallelogram.

2x + 8 = 120 5y + 120 = 180 (2x + 8) ° 2x = 112 5y = 60 120 ° 5y °

x = 56 units y = 12 units