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Geometric Theorems & Proof Exercises | Isosceles Trapezoids and Kites

Learn about properties of isosceles trapezoids, midsegments, kite angles, & coordinate geometry. Practice using theorems with real-world examples. Explore the concepts in detail.

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Geometric Theorems & Proof Exercises | Isosceles Trapezoids and Kites

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  1. Splash Screen

  2. Five-Minute Check (over Lesson 6–5) CCSS 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: Standardized Test Example: 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. A. 10.25 B. 9 C. 8.375 D. 6.5 5-Minute Check 3

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

  7. QRST is a square. Find QU if QS = 16t – 14 and QU = 6t + 11. A. 9 B. 10 C. 54 D. 65 5-Minute Check 5

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

  9. Content Standards G.GPE.4 Use coordinates to prove simple geometric theorems algebraically. G.MG.3 Apply geometric methods to solve problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios). Mathematical Practices 1 Make sense of problems and persevere in solving them. 2 Reason abstractly and quantitatively. CCSS

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

  11. trapezoid • 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 LN = 3.6 feet, find mMJK. Example 1A

  15. 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 JL is 10.3 feet, find MN. Example 1B

  17. Since JKLM is an isosceles trapezoid, diagonals JL and KM are congruent. Use Properties of Isosceles Trapezoids JL= KM Definition of congruent JL = KN + MN Segment Addition 10.3= 6.7 + MN Substitution 3.6= MN Subtract 6.7 from each side. Answer:MN = 3.6 Example 1B

  18. 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. 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. 14.1 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 Answer: Exactly one pair of opposite sides are parallel, So, ABCD is a trapezoid. Isosceles Trapezoids and Coordinate Geometry Example 2

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

  26. Midsegment of a Trapezoid Read the Test 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 Test Item Trapezoid Midsegment Theorem Substitution Example 3

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

  31. 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 73 + 121 + mY + 121 = 360 Substitution mY = 45 Simplify. Answer:mXYZ= 45 Example 4A

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

  33. 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 36 + 64 = 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 Answer:NP= 10 Example 4B

  35. A. If BCDE is a kite, find mCDE. A. 28° B. 36° C. 42° D. 55° Example 4A

  36. B. If JKLM is a kite, find KL. A. 5 B. 6 C. 7 D. 8 Example 4B

  37. End of the Lesson

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