Geometry of shapes, parallelogram, quadrilaterals.pptx

Mathematics

1. CONDITIONS THAT MAKE A PARALLELOGRAM A QUADRILATERAL

2. TERMS TO REMEMBER: A D C B Congruent: Having the same size and shape, and denoted by the symbol  Bisect : To divide into two congruent parts . Consecutive Angles : Two angles are consecutive angles if one of the rays of the two angles are collinear

3. TERMS TO REMEMBER: A D C B Supplementary angles: Two angles whose sum measures 180 Parallel: Lines on the same plane that do not intersect. Denoted by the symbol,  

4. Parallelograms Quadrilaterals are four-sided polygons Definition: A parallelogram is a quadrilateral with both pairs of opposite sides parallel.

5. Properties of Parallelograms Theorem 79: A diagonal of a parallelogram forms two congruent triangles. Corollary 79.1: Opposite angles of a parallelogram are congruent A D C B <A  <C and <B  <D  

6. Properties of Parallelograms Theorem 81: Opposite sides of a parallelogram are congruent . Theorem 80: Consecutive angles in a parallelogram are supplementary. A D C B AD  BC and AB  DC m< A+m <B = 180° m < B+m <C = 180° m< C+m <D = 180° m< D+m <A = 180°

7. Properties of Parallelograms Diagonals of a figure : Segments that connect any two opposite vertices of a polygon Theorem 82: The diagonals of a parallelogram bisect each other. A B C D

8. Conditions for a Quadrilateral to be a Parallelograms Theorem 83 : If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. Theorem 84: If one pair of opposite angles of a quadrilateral are both congruent and parallel, 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

9. Conditions for a Quadrilateral to be a Parallelograms Theorem 85 : If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram. Theorem 86: If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. A D C B If A  C and B  D, then ABCD is a parallelogram

10. A quadrilateral is a parallelogram if ... The opposite sides are parallel. The opposite sides are congruent. The opposite angles are congruent. Consecutive angles are supplementary. Diagonals bisect each other. Both pairs of opposite sides are parallel. ( Definition)

11. Example 1: Using properties of Parallelograms FGHJ is a parallelogram. Find the unknown length. JH JK F G J H K 5 3

12. Example 1: Using properties of Parallelograms FGHJ is a parallelogram. Find the unknown length. JH JK F G J H K 5 3

13. Ex. 2: Using properties of parallelograms PQRS is a parallelogram. Find the angle measure. m R m Q P R Q 70 ° S

14. Ex. 3: Using Algebra with Parallelograms PQRS is a parallelogram. Find the value of x . S Q P R 3x ° 120 °

15. Example 4.

16. 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 .

17. 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

18. 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

19. 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 .

20. 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

21. 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

22. 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.

23. 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.