1. 6.5 Trapezoids and Kites
Geometry
Ms. Reser

2. Objectives:
Use properties of trapezoids.
Use properties of kites.

3. Assignment:
pp #2-33

4. Using properties of trapezoids
A trapezoid is a quadrilateral with exactly one pair of parallel sides. The parallel sides are the bases. A trapezoid has two pairs of base angles. For instance in trapezoid ABCD D and C are one pair of base angles. The other pair is A and B. The nonparallel sides are the legs of the trapezoid.

5. Using properties of trapezoids
If the legs of a trapezoid are congruent, then the trapezoid is an isosceles trapezoid.

6. Trapezoid Theorems Theorem 6.14
If a trapezoid is isosceles, then each pair of base angles is congruent.
A ≅ B, C ≅ D

7. Trapezoid Theorems Theorem 6.15
If a trapezoid has a pair of congruent base angles, then it is an isosceles trapezoid.
ABCD is an isosceles trapezoid

8. Trapezoid Theorems Theorem 6.16
A trapezoid is isosceles if and only if its diagonals are congruent.
ABCD is isosceles if and only if AC ≅ BD.

9. Ex. 1: Using properties of Isosceles Trapezoids
PQRS is an isosceles trapezoid. Find mP, mQ, mR.
PQRS is an isosceles trapezoid, so mR = mS = 50°. Because S and P are consecutive interior angles formed by parallel lines, they are supplementary. So mP = 180°- 50° = 130°, and mQ = mP = 130°
50°
You could also add 50 and 50, get 100 and subtract it from 360°. This would leave you 260/2 or 130°.

10. Ex. 2: Using properties of trapezoids
Show that ABCD is a trapezoid.
Compare the slopes of opposite sides.
The slope of AB = 5 – 0 = 5 = - 1
0 –
The slope of CD = 4 – 7 = -3 = - 1
7 –
The slopes of AB and CD are equal, so AB ║ CD.
The slope of BC = 7 – 5 = 2 = 1
4 –
The slope of AD = 4 – 0 = 4 = 2
7 –
The slopes of BC and AD are not equal, so BC is not parallel to AD.
So, because AB ║ CD and BC is not parallel to AD, ABCD is a trapezoid.

11. Midsegment of a trapezoid
The midsegment of a trapezoid is the segment that connects the midpoints of its legs. Theorem 6.17 is similar to the Midsegment Theorem for triangles.

12. Theorem 6.17: Midsegment of a trapezoid
The midsegment of a trapezoid is parallel to each base and its length is one half the sums of the lengths of the bases.
MN║AD, MN║BC
MN = ½ (AD + BC)

13. Ex. 3: Finding Midsegment lengths of trapezoids
LAYER CAKE A baker is making a cake like the one at the right. The top layer has a diameter of 8 inches and the bottom layer has a diameter of 20 inches. How big should the middle layer be?

14. Ex. 3: Finding Midsegment lengths of trapezoids
Use the midsegment theorem for trapezoids.
DG = ½(EF + CH)=
½ (8 + 20) = 14” F D G D C

15. Using properties of kites
A kite is a quadrilateral that has two pairs of consecutive congruent sides, but opposite sides are not congruent.

16. Kite theorems
Theorem 6.18
If a quadrilateral is a kite, then its diagonals are perpendicular.
AC  BD

17. Kite theorems
Theorem 6.19
If a quadrilateral is a kite, then exactly one pair of opposite angles is congruent.
A ≅ C, B ≅ D

18. Ex. 4: Using the diagonals of a kite
WXYZ is a kite so the diagonals are perpendicular. You can use the Pythagorean Theorem to find the side lengths.
WX = √ ≈ 23.32
XY = √ ≈ 16.97
Because WXYZ is a kite, WZ = WX ≈ 23.32, and ZY = XY ≈ 16.97

19. Ex. 5: Angles of a kite Find mG and mJ in the diagram at the right.
SOLUTION:
GHJK is a kite, so G ≅ J and mG = mJ.
2(mG) + 132° + 60° = 360°Sum of measures of int. s of a quad. is 360°
2(mG) = 168°Simplify
mG = 84° Divide each side by 2.
So, mJ = mG = 84°
132°
60°

20. Reminder:
Quiz after this section