1. LESSON 6–6
Trapezoids and Kites

2. Five-Minute Check (over Lesson 6–5) TEKS Then/Now New Vocabulary
Theorems: Isosceles Trapezoids
Proof: Part of Theorem 6.23
Example 1: Real-World Example: Use Properties of Isosceles Trapezoids
Example 2: Isosceles Trapezoids and Coordinate Geometry
Theorem 6.24: Trapezoid Midsegment Theorem
Example 3: Midsegment of a Trapezoid
Theorems: Kites
Example 4: Use Properties of Kites
Lesson Menu

3. LMNO is a rhombus. Find x.
A. 5
B. 7
C. 10
D. 12
5-Minute Check 1

4. LMNO is a rhombus. Find y. A. 6.75 B. 8.625 C. 10.5 D. 12
5-Minute Check 2

5. QRST is a square. Find n if mTQR = 8n + 8.
B. 9 C
D. 6.5
5-Minute Check 3

6. QRST is a square. Find w if QR = 5w + 4 and RS = 2(4w – 7).
B. 5
C. 4
D. 3.3 _
5-Minute Check 4

7. QRST is a square. Find QU if QS = 16t – 14 and QU = 6t + 11.
B. 10
C. 54
D. 65
5-Minute Check 5

8. Which statement is true about the figure shown, whether it is a square or a rhombus?
C. JM║LM D.
5-Minute Check 6

9. Mathematical Processes G.1(F), G.1(G)
Targeted TEKS
G.2(B) Derive and use the distance, slope, and midpoint formulas to verify geometric relationships, including congruence of segments and parallelism or perpendicularity of pairs of lines.
Mathematical Processes
G.1(F), G.1(G)
TEKS

10. You used properties of special parallelograms.
Apply properties of trapezoids.
Apply properties of kites.
Then/Now

11. midsegment of a trapezoid kite
bases
legs of a trapezoid
base angles
isosceles trapezoid
midsegment of a trapezoid
kite
Vocabulary

12. Concept 1

13. Concept 2

14. Use Properties of Isosceles Trapezoids
A. BASKET Each side of the basket shown is an isosceles trapezoid. If mJML = 130, KN = 6.7 feet, and MN= 3.6 feet, find mMJK.
Example 1A

15. Since JKLM is a trapezoid, JK║LM.
Use Properties of Isosceles Trapezoids
Since JKLM is a trapezoid, JK║LM.
mJML + mMJK = 180 Consecutive Interior Angles Theorem
130 + mMJK = 180 Substitution
mMJK = 50 Subtract 130 from each side.
Answer: mMJK = 50
Example 1A

16. Use Properties of Isosceles Trapezoids
B. BASKET Each side of the basket shown is an isosceles trapezoid. If mJML = 130, KN = 6.7 feet, and MN is 10.3 feet, find JL.
Example 1B

17. JL = KM Definition of congruent JL = KN + MN Segment Addition
Use Properties of Isosceles Trapezoids
Since JKLM is an isosceles trapezoid, diagonals JL and KM are congruent.
JL = KM Definition of congruent
JL = KN + MN Segment Addition
JL = Substitution
JL = 10.3 Add.
Answer: JL= 10.3
Example 1B

18. A. Each side of the basket shown is an isosceles trapezoid
A. Each side of the basket shown is an isosceles trapezoid. If mFGH = 124, FI = 9.8 feet, and IG = 4.3 feet, find mEFG.
A. 124
B. 62
C. 56
D. 112
Example 1A

19. B. Each side of the basket shown is an isosceles trapezoid
B. Each side of the basket shown is an isosceles trapezoid. If mFGH = 124, FI = 9.8 feet, and EG = 14.1 feet, find IH.
A. 4.3 ft
B. 8.6 ft
C. 9.8 ft
D ft
Example 1B

20. Isosceles Trapezoids and Coordinate Geometry
Quadrilateral ABCD has vertices A(5, 1), B(–3, –1), C(–2, 3), and D(2, 4). Show that ABCD is a trapezoid and determine whether it is an isosceles trapezoid.
A quadrilateral is a trapezoid if exactly one pair of opposite sides are parallel. Use the Slope Formula.
Example 2

21. slope of slope of slope of
Isosceles Trapezoids and Coordinate Geometry
slope of
slope of
slope of
Answer: Exactly one pair of opposite sides are parallel, So, ABCD is a trapezoid.
Example 2

22. Use the Distance Formula to show that the legs are congruent.
Isosceles Trapezoids and Coordinate Geometry
Use the Distance Formula to show that the legs are congruent.
Answer: Since the legs are not congruent, ABCD is not an isosceles trapezoid.
Example 2

23. A. trapezoid; not isosceles B. trapezoid; isosceles C. not a trapezoid
Quadrilateral QRST has vertices Q(–1, 0), R(2, 2), S(5, 0), and T(–1, –4). Determine whether QRST is a trapezoid and if so, determine whether it is an isosceles trapezoid.
A. trapezoid; not isosceles
B. trapezoid; isosceles
C. not a trapezoid
D. cannot be determined
Example 2

24. Concept 3

25. Midsegment of a Trapezoid
In the figure, MN is the midsegment of trapezoid FGJK. What is the value of x?
Example 3

26. Trapezoid Midsegment Theorem
Midsegment of a Trapezoid
Read the Item
You are given the measure of the midsegment of a trapezoid and the measures of one of its bases. You are asked to find the measure of the other base.
Solve the Item
Trapezoid Midsegment Theorem
Substitution
Example 3

27. Subtract 20 from each side.
Midsegment of a Trapezoid
Multiply each side by 2.
Subtract 20 from each side.
Answer: x = 40
Example 3

28. WXYZ is an isosceles trapezoid with median Find XY if JK = 18 and WZ = 25.
A. XY = 32
B. XY = 25
C. XY = 21.5
D. XY = 11
Example 3

29. Concept 4

30. A. If WXYZ is a kite, find mXYZ.
Use Properties of Kites
A. If WXYZ is a kite, find mXYZ.
Example 4A

31. mW + mX + mY + mZ = 360 Polygon Interior Angles Sum Theorem
Use Properties of Kites
Since a kite only has one pair of congruent angles, which are between the two non-congruent sides, WXY  WZY. So, WZY = 121.
mW + mX + mY + mZ = 360 Polygon Interior Angles Sum Theorem
mY = 360 Substitution
mY = 45 Simplify.
Answer: mXYZ = 45
Example 4A

32. B. If MNPQ is a kite, find NP.
Use Properties of Kites
B. If MNPQ is a kite, find NP.
Example 4B

33. NR2 + MR2 = MN2 Pythagorean Theorem (6)2 + (8)2 = MN2 Substitution
Use Properties of Kites
Since the diagonals of a kite are perpendicular, they divide MNPQ into four right triangles. Use the Pythagorean Theorem to find MN, the length of the hypotenuse of right ΔMNR.
NR2 + MR2 = MN2 Pythagorean Theorem
(6)2 + (8)2 = MN2 Substitution
= MN2 Simplify.
100 = MN2 Add.
10 = MN Take the square root of each side.
Example 4B

34. Since MN  NP, MN = NP. By substitution, NP = 10.
Use Properties of Kites
Since MN  NP, MN = NP. By substitution, NP = 10.
Answer: NP = 10
Example 4B

35. A. If BCDE is a kite, find mCDE.
Example 4A

36. B. If JKLM is a kite, find KL.
C. 7
D. 8
Example 4B

37. LESSON 6–6
Trapezoids and Kites