Parallelograms: Parallelograms Quadrilaterals are four-sided polygons Parallelogram : is a quadrilateral with both pairs of opposite sides parallel.
Parallelograms (2): Parallelograms (2) 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 are supplementary. A D C B 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°
Parallelograms (3): Parallelograms (3) Diagonals of a figure : Segments that connect any to vertices of a polygon Theorem 6.4: The diagonals of a parallelogram bisect each other. A B C D
Parallelograms (4): Parallelograms (4) 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?
Tests for Parallelograms: Tests for Parallelograms 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. A D C B 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: Tests for Parallelograms 2 Theorem 6.7 : If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram A D C B Theorem 6.8: If one pair of opposite sides of a quadrilateral is both parallel and congruent, then the quadrilateral is a parallelogram.
A quadrilateral is a parallelogram if...: A quadrilateral is a parallelogram if... 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)
Area of a parallelogram: Area of a parallelogram If a parallelogram has an area of A square units, a base of b units and a height of h units, then A = bh. (Do example 1 p. 530) The area of a region is the sum of the areas of all its non-overlapping parts . (Do example 3 p. 531) b h
Rectangles: Rectangles A rectangle is a quadrilateral with four right angles. Theorem 6-9 : If a parallelogram is a rectangle, then its diagonals are congruent . Opp. angles in rectangles are congruent (they are right angles) therefore rectangles are parallelograms with all their properties. Theorem 6-10 : If the diagonals of a parallelogrma are congruent then the parallelogram is a rectangle .
Rectangles (2): Rectangles (2) If a quadrilateral is a rectangle, then the following properties hold true: Opp. Sides are congruent and parallel Opp. Angles are congruent Consecutive angles are supplementary Diagonals are congruent and bisect each other All four angles are right angles
Squares and Rhombi: Squares and Rhombi A rhombus is a quadrilateral with four congruent sides. Since opp. sides are , a rhombus is a parallelogram with all its properties. Special facts about rhombi Theorem 6.11 : The diagonals of a rhombus are perpendicular. Theorem 6.12: If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus. Theorem 6.13: Each diagonal of a rhombus bisects a pair of opp. angles C
Squares and Rhombi(2): Squares and Rhombi(2) If a quadrilateral is both, a rhombus and a rectangle, is a square If a rhombus has an area of A square units and diagonals of d 1 and d 2 units, then A = ½ d 1 d 2 .
Area of a triangle:: Area of a triangle: If a triangle has an area of A square units a base of b units and corresponding height of h units, then A = ½bh. h b Congruent figures have equal areas.
Trapezoids: Trapezoids A trapezoid is a quadrilateral with exactly one pair of parallel sides. The parallel sides are called bases . The nonparallel sides are called legs . At each side of a base there is a pair of base angles . C
Trapezoids (2): Trapezoids (2) C A C D B AB = base CD = base AC = leg BD = leg AB CD AC & BD are non parallel <A & <B = pair of base angles <C & <D = pair of base angles
Trapezoids (3): Trapezoids (3) Isosceles trapezoid : A trapezoid with congruent legs. Theorem 6-14 : Both pairs of base angles of an isosceles trapezoid are congruent. Theorem 6-15 : The diagonals of an isosceles trapezoid are congruent.
Trapezoids (4): Trapezoids (4) C A C D B The median of a trapezoid is the segment that joints the midpoints of the legs (PQ). Q P Theorem 6-16: The median of a trapezoid is parallel to the bases, and its measure is one-half the sum of the measures of its bases.
Area of Trapezoids: Area of Trapezoids C A C D B Area of a trapezoid : If a trapezoid has an area of A square units, bases of b 1 and b 2 units and height of h units, then A = ½(b 1 + b 2 )h. h