TODAY > 8.2 Properties of Parallelograms By definition, a parallelogram is a quadrilateral with 2 pairs of parallel sides

Given: GEOM is a parallelogram. Prove: (i.e. opposite sides are  ) G E O M

Given: GEOM is a parallelogram. Prove: a)  G and  E are supplementary.  E and  O are supplementary.  O and  M are supplementary.  M and  G are supplementary. (i.e. consecutive angles are supplementary) b)  G   O,  M   E (i.e. opposite angles are congruent) G E O M

Given: GEOM is a parallelogram. Prove: Diagonals bisect each other. G E O M T

8.2 //ogram Properties  2 pairs of opposite sides are // (by defn.)  2 pairs of consecutive interior  s are supplementary  2 pairs of opposite  s are   2 pairs of opposite sides are   The diagonals bisect each other. Exercises: p. 512 #8, 11, 15, 23 – 28, 33, 36

TODAY > 8.3 Proving Parallelograms Aside from using the definition of a parallelogram (opposite sides are parallel), there are five (5) other ways to prove that a quadrilateral is a parallelogram.

Given: Quadrilateral GEOM  G and  E are supplementary.  E and  O are supplementary.  O and  M are supplementary.  M and  G are supplementary. Prove: GEOM is a parallelogram. G E O M

Given: Quadrilateral GEOM  M   E and  G   O Prove: GEOM is a parallelogram. G E O M b b a a

Given: Quadrilateral GEOM Diagonals bisect each other at T. Prove: GEOM is a parallelogram. G E O M T

Given: Quadrilateral GEOM Prove: GEOM is a parallelogram. G E O M

Given: Quadrilateral GEOM Prove: GEOM is a parallelogram. G E O M

A quadrilateral is a parallelogram if:  2 pairs of opposite sides are // (by defn.)  2 pairs of consecutive interior  s are supplementary  2 pairs of opposite  s are   2 pairs of opposite sides are   The diagonals bisect each other.  One pair of opposite sides are // and .

8.3 Proving Parallelograms 1. Given:  ABCD is a parallelogram &. Prove:  AECF is a parallelogram. Warm-up: p. 521 #15 – 18

8.3 Proving Parallelograms 3. Given:  ABCD is a parallelogram. E and F are midpoints. Prove: EFCD is a parallelogram. AB C D E F

8.3 Proving Parallelograms 4. Given:  JOHN is a parallelogram. Prove: JBHD is a parallelogram. J O H N B D

8.3 Proving Parallelograms

TODAY > 8.4 Special Parallelograms

Rhombus Properties  The diagonals are  bisectors of each other.  The diagonals bisect the angles of the rhombus. Remember your P.T. & Special Right  s.

Rectangle Properties  The measure of each  of a rectangle is 90 o.  The diagonals of a rectangle are  and bisect each other. How many  Isosceles  s are there?

Square Properties  The diagonals of a square are ,  and bisect each other. Exercises: p. 531 How many  isosceles RIGHT  s are there?

A-S-N (True) 1. The diagonals of a parallelogram are congruent. 2. The consecutive angles of a rectangle are congruent and supplementary. 3. The diagonals of a rectangle bisect each other. 4. The diagonals of a rectangle bisect the angles. 5. The diagonals of a square are perpendicular bisectors of each other. 6. The diagonals of a square divides it into 4 isosceles right triangles. 7. Opposite angles in a parallelogram are congruent. 8. Consecutive angles in a parallelogram are congruent.

SUMMARY ParallelogramRhombusRectangleSquare Opp sides are // Opp sides are  Opp  s are  Diagonals bisect each other Diagonals are  Diagonals are  Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y YY Y Y

Proving 1. Given:  MPQS is a rhombus. G, H, I and K are midpoints. Prove:  GHIK is a rectangle. S B G K Q M P I H