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

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

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

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

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 140 100 leg leg 40 80 base base leg 70 110 90 90 leg base

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 140 100 leg leg 40 80 base base leg 70 110 90 90 leg base Knowledge Connection: What do you notice about the angles?

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

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.

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.

Example 1 Use the graph to show that quadrilateral QRST is a trapezoid. Slope RS m = 4 3 2 0 m = 1 2

Example 1 Use the graph to show that quadrilateral QRST is a trapezoid. Slope RS m = 4 3 2 0 m = 1 2 Slope QT m = 2 0 4 0 = 2 4 m = 1 2

Example 1 Use the graph to show that quadrilateral QRST is a trapezoid. Slope RS m = 4 3 2 0 m = 1 2 Slope QT m = 2 0 4 0 = 2 4 m = 1 2 Same slope, thus RS QT

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

Example 1 Use the graph to show that quadrilateral QRST is a trapezoid. Slope QR m = 3 0 0 0 = 3 0 Slope Undefined Slope ST m = 4 2 2 4 = 2 2 m = 1

Example 1 Use the graph to show that quadrilateral QRST is a trapezoid. Slope QR m = 3 0 0 0 = 3 0 Slope Undefined Slope ST m = 4 2 2 4 = 2 2 m = 1 Slopes are not the same, thus QR ST

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.

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

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

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 4 5 ( 1) = 3 6 m = 1 2 m = 1 2

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 4 5 ( 1) = 3 6 m = 1 2 m = 1 2 Same slope, thus WX YZ

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

Example 2 Use the graph to show that quadrilateral WXYZ is a trapezoid. Slope XY m = 4 1 1 (2) = 3 1 m = 3 Slope WZ m = 3 1 6 5 = 4 1 m = 4

Example 2 Use the graph to show that quadrilateral WXYZ is a trapezoid. Slope XY m = 4 1 1 (2) = 3 1 m = 3 Slope WZ m = 3 1 6 5 = 4 1 m = 4 Slopes are not the same, thus XY WZ

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.

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

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

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

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

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

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

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

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

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

Example 3 Use Trapezoid EFGH to answer the following. E 70 110 F b.) Answer: Yes We will have < HEF < EFG and < FGH EHG, making trapezoid EFGH isosceles by thm 8.15 H G

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

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)

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

Example 4 In the diagram, MN is the midsegment of the trapezoid PQRS. Find MN. 12 in. MN = 1 (PQ + SR) 2 MN = 1 (12 + 28) 2 MN = 1 2 40 = 20 28 in.

Example 4 In the diagram, MN is the midsegment of the trapezoid PQRS. Find MN. 12 in. MN = 1 (PQ + SR) 2 MN = 1 (12 + 28) 2 The length of MN is 20 inches. MN = 1 2 40 = 20 28 in.

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

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 = 1 2 24 = 12 G 18 in. F

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 = 1 2 24 = 12 G 18 in. F

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

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

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)

Example 6 Find m < C in the kite shown. 84 140

Example 6 Find m < C in the kite shown. 84 m < A + m < C + 84 + 140 = 360 140

Example 6 Find m < C in the kite shown. 84 m < A + m < C + 84 + 140 = 360 2 m < C + 224 = 360 2 m < C = 136 m < C = 68 140

Example 7 Find m < D in the kite shown.

Example 7 Find m < D in the kite shown. m < D + m < F + 73 + 115 = 360

Example 7 Find m < D in the kite shown. m < D + m < F + 73 + 115 = 360 2 m < D + 188 = 360 2 m < D = 172 m < D = 86