Geometry Chapter 8 8-5: USE PROPERTIES OF TRAPEZOIDS AND KITES

Transcription

1 Geometry Chapter 8 8-5: USE PROPERTIES OF TRAPEZOIDS AND KITES

2 Use Properties of Trapezoids and Kites Objective: Students will be able to identify and use properties to solve trapezoids and kites. Agenda Trapezoid Kite Examples

3 Trapezoid A quadrilateral with exactly one pair of parallel sides is called a Trapezoid. The parallel sides are called the bases. The other sides are called the legs. base leg leg base

4 Trapezoid A quadrilateral with exactly one pair of parallel sides is called a Trapezoid. The parallel sides are called the bases. The other sides are called the legs. base leg leg base base leg leg base

5 Trapezoid A quadrilateral with exactly one pair of parallel sides is called a Trapezoid. The parallel sides are called the bases. The other sides are called the legs. base leg leg base base leg leg base Knowledge Connection: What do you notice about the angles?

6 Example 1 Use the graph to show that quadrilateral QRST is a trapezoid.

7 Example 1 Use the graph to show that quadrilateral QRST is a trapezoid. To see if QRST is a trapezoid, we must show that it has only 1 pair of opposite sides that are parallel. To do that, we must find the slope of all four sides and compare them.

8 Example 1 Use the graph to show that quadrilateral QRST is a trapezoid. You can use the slope equation, or you can count the rise and run, to find the slope of each line.

9 Example 1 Use the graph to show that quadrilateral QRST is a trapezoid. Slope RS m = m = 1 2

10 Example 1 Use the graph to show that quadrilateral QRST is a trapezoid. Slope RS m = m = 1 2 Slope QT m = = 2 4 m = 1 2

11 Example 1 Use the graph to show that quadrilateral QRST is a trapezoid. Slope RS m = m = 1 2 Slope QT m = = 2 4 m = 1 2 Same slope, thus RS QT

12 Example 1 Use the graph to show that quadrilateral QRST is a trapezoid. Slope QR m = = 3 0 Slope Undefined

13 Example 1 Use the graph to show that quadrilateral QRST is a trapezoid. Slope QR m = = 3 0 Slope Undefined Slope ST m = = 2 2 m = 1

14 Example 1 Use the graph to show that quadrilateral QRST is a trapezoid. Slope QR m = = 3 0 Slope Undefined Slope ST m = = 2 2 m = 1 Slopes are not the same, thus QR ST

15 Example 1 Use the graph to show that quadrilateral QRST is a trapezoid. We showed that exactly 1 pair of parallel opposites sides. Thus, QRST is a Trapezoid.

16 Example 2 Use the graph to show that quadrilateral WXYZ is a trapezoid.

17 Example 2 Use the graph to show that quadrilateral WXYZ is a trapezoid. Slope WX m = 1 ( 3) 2 6 = 4 8 m = 1 2

18 Example 2 Use the graph to show that quadrilateral WXYZ is a trapezoid. Slope WX m = 1 ( 3) 2 6 = 4 8 Slope YZ m = ( 1) = 3 6 m = 1 2 m = 1 2

19 Example 2 Use the graph to show that quadrilateral WXYZ is a trapezoid. Slope WX m = 1 ( 3) 2 6 = 4 8 Slope YZ m = ( 1) = 3 6 m = 1 2 m = 1 2 Same slope, thus WX YZ

20 Example 2 Use the graph to show that quadrilateral WXYZ is a trapezoid. Slope XY m = (2) = 3 1 m = 3

21 Example 2 Use the graph to show that quadrilateral WXYZ is a trapezoid. Slope XY m = (2) = 3 1 m = 3 Slope WZ m = = 4 1 m = 4

22 Example 2 Use the graph to show that quadrilateral WXYZ is a trapezoid. Slope XY m = (2) = 3 1 m = 3 Slope WZ m = = 4 1 m = 4 Slopes are not the same, thus XY WZ

23 Example 2 Use the graph to show that quadrilateral WXYZ is a trapezoid. We showed had exactly 1 pair of parallel opposites sides. Thus, WXYZ is a Trapezoid.

24 Isosceles Trapezoids A Trapezoid with congruent legs is known as an Isosceles Trapezoid.

25 Theorem 8.14 Theorem 8.14: If a trapezoid is isosceles, then each pair of base angles is congruent. B C If trapezoid ABCD is isosceles, then < A < D and < B < C A D

26 Theorem 8.15 Theorem 8.15: If a trapezoid has a pair of congruent base angles, then it is an isosceles trapezoid. B C In trapezoid ABCD, If < A < D (or < B < C) Then ABCD is an isosceles trapezoid A D

27 Theorem 8.16 Theorem 8.16: A trapezoid is isosceles if and only if its diagonals are congruent. B C Trapezoid ABCD is isosceles iff AC BD A D

28 Example 3 Use Trapezoid EFGH to answer the following. E F a.) If EG = FH, is trapezoid EFGH isosceles? H G

29 Example 3 Use Trapezoid EFGH to answer the following. E F a.) Answer: Yes Since EG = FH, then EG FH, making trapezoid EFGH isosceles by thm 8-16 H G

30 Example 3 Use Trapezoid EFGH to answer the following. E F b.) If m < HEF = 70, and m < FGH = 110, is trapezoid EFGH isosceles? H G

31 Example 3 Use Trapezoid EFGH to answer the following. E 70 F b.) If m < HEF = 70, and m < FGH = 110, is trapezoid EFGH isosceles? 110 H G

32 Example 3 Use Trapezoid EFGH to answer the following. E 70 F b.) We can make m < EHG = 110 and m < EFG = 70. (How?) 110 H G

33 Example 3 Use Trapezoid EFGH to answer the following. E F b.) Answer: Yes We will have < HEF < EFG and < FGH EHG, making trapezoid EFGH isosceles by thm 8.15 H G

34 Midsegment The Midsegment of a trapezoid is the segment that connects the midpoints of its legs. Midsegment

35 Theorem 8.17 Theorem 8.17 Midsegment Theorem for Trapezoids: The midsegment of a trapezoid is parallel to each base and its length is one half the sum of the lengths of the bases. M A B N If MN is the midsegment of trapezoid ABCD, Then MN AB, MN DC, and D C MN = 1 2 (AB + CD)

36 Example 4 In the diagram, MN is the midsegment of the trapezoid PQRS. Find MN. 12 in. 28 in.

37 Example 4 In the diagram, MN is the midsegment of the trapezoid PQRS. Find MN. 12 in. MN = 1 (PQ + SR) 2 MN = 1 ( ) 2 MN = = in.

38 Example 4 In the diagram, MN is the midsegment of the trapezoid PQRS. Find MN. 12 in. MN = 1 (PQ + SR) 2 MN = 1 ( ) 2 The length of MN is 20 inches. MN = = in.

39 Example 5 In the diagram, HK is the midsegment of the trapezoid DEFG. Find HK. D 6 in. E H K G 18 in. F

40 Example 5 In the diagram, HK is the midsegment of the trapezoid DEFG. Find HK. D 6 in. E HK = 1 (DE + GF) 2 H K HK = 1 (6 + 18) 2 HK = = 12 G 18 in. F

41 Example 5 In the diagram, HK is the midsegment of the trapezoid DEFG. Find HK. D 6 in. E HK = 1 (DE + GF) 2 H K HK = 1 (6 + 18) 2 The length of HK is 12 inches. HK = = 12 G 18 in. F

42 Kite A Kite is a quadrilateral with one pair of congruent consecutive sides, but no opposite sides are congruent. C D B A

43 Theorem 8.18 Theorem 8.18: If a quadrilateral is a kite, then its diagonals are perpendicular. C If ABCD is a kite, D B then AC BD A

44 Theorem 8.19 Theorem 8.19: If a quadrilateral is a kite, then exactly one pair of opposite angles is congruent. C If ABCD is a kite, D A B then < A < C (Or < B < D)

45 Example 6 Find m < C in the kite shown

46 Example 6 Find m < C in the kite shown. 84 m < A + m < C =

47 Example 6 Find m < C in the kite shown. 84 m < A + m < C = m < C = m < C = 136 m < C =

48 Example 7 Find m < D in the kite shown.

49 Example 7 Find m < D in the kite shown. m < D + m < F = 360

50 Example 7 Find m < D in the kite shown. m < D + m < F = m < D = m < D = 172 m < D = 86