1. Do Now: List all you know about the following parallelograms.
1.) Rectangle
2.) Rhombus
3.) Square

2. Geometry 8.4: Properties of Rhombuses, Rectangles, and Squares
8.5: e Properties of Trapezoids and Kites

3. Rectangles
Parallelogram with 4 right angles

4. Rhombus
Parallelogram with four congruent sides.

5. Square
A square is a parallelogram with four congruent sides and four right angles.
A square is a rhombus and a rectangle.

6. Theorems about Diagonals
Diagonals of a rhombus are perpendicular
(also true for a square- remember a square is a rhombus.)

7. Theorems about Diagonals
Diagonals of a rhombus bisect the opposite angles
(also true for a square - remember a square is a rhombus.)

8. Theorems about Diagonals
Diagonals of a rectangle are congruent
(also true for a square- remember a square is a rectangle.)

9. Examples

10. Examples

12. Examples

13. Trapezoids
A trapezoid is a quadrilateral with exactly one pair of parallel sides. (not parallelogram)
The parallel sides are called bases. The other two sides are called legs.
A trapezoid has two pairs of base angles.
base A B
One pair of base
Angles: A & B.
Another pair:
D and C.
leg
leg C D
base

14. Isosceles Trapezoids
If the legs of a trapezoid are congruent, then the trapezoid is an isosceles trapezoid.
Theorem 8.14: If a trapezoid is isosceles, then each pair of base angles is congruent. A B D C

15. ABCD is an isosceles trapezoid.
Theorem 8.15
If a trapezoid has a pair of congruent base angles, then it is an isosceles trapezoid. A B D C
ABCD is an isosceles trapezoid.
(AD is congruent to BC).

16. Examples: Find the missing angles.

17. Theorem 8.16
A trapezoid is isosceles if and only if its diagonals are congruent. A B D C
ABCD is an isosceles trapezoid
if and only if

18. Midsegment of a Trapezoid
Midsegment connects the midpoints of the legs.
Theorem 8.17: The midsegment of a trapezoid is parallel to each base and its length is one half the sum of the lengths of the bases. (average) A B
Midsegment M N D C

19. Examples
Find the length of the other base.

20. Examples

21. Kites
A kite is a quadrilateral that has two pairs of consecutive congruent sides, but opposite sides are not congruent. (not parallelogram)

22. Kites
Theorem 8.18: If a quadrilateral is a kite, then its diagonals are perpendicular.
Theorem 8.19: If a quadrilateral is a kite, then exactly one pair of opposite angles are congruent.

23. Examples

24. Example: Find the side lengths of the kite.