Quadrilaterals are four-sided polygons
Parallelogram: is a quadrilateral with both pairs of opposite sides parallel.
Theorem 6.1 : Opposite sides of a parallelograms are congruent
Theorem 6.2: Opposite angles of a parallelogram are congruent
Theorem 6.3: Consecutive angles in a parallelogram
AD BC and AB DC
<A <C and <B <D
m<A+m<B = 180°
m <B+m<C = 180°
m<C+m<D = 180°
m<D+m<A = 180°
Diagonals of a figure: Segments that connect any to vertices of a polygon
Theorem 6.4: The diagonals of a parallelogram bisect each other.
Draw a parallelogram : ABCD on a piece of construction paper.
Cut the parallelogram.
Fold the paper and make a crease from A to C and from B to D.
Fold the paper so A lies on C. What do you observe?
Fold the paper so B lies on D. What do you observe?
What theorem is confirmed by these Observations?
Theorem 6.5 :If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.
Theorem 6.6: If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.
If AD BC and AB DC, then ABCD is a parallelogram
If <A <C and <B <D, then ABCD is a parallelogram
Tests for Parallelograms 2
Theorem 6.7: If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram
Theorem 6.8: If one pair of opposite sides of a quadrilateral is both parallel and congruent, then the quadrilateral is a parallelogram.
Diagonals bisect each other. (Theorem 6.7)
A pair of opposite sides is both parallel and congruent. (Theorem 6.8)
Both pairs of opposite sides are congruent. (Theorem 6.5)
Both pairs of opposite angles are congruent. (Theorem 6.6)
Both pairs of opposite sides are parallel. (Definition)