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Geometry of Kites and Trapezoids: Properties and Conjectures

Explore the characteristics, angles, and diagonals of kites and trapezoids, along with midsegments of triangles and trapezoids. Learn through examples and exercises from the book "Discovering Geometry" by Michael Serra.

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Geometry of Kites and Trapezoids: Properties and Conjectures

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  1. Kites, Trapezoids, Midsegments Geometry Regular Program SY 2015-2016 Source: Discovering Geometry (2008) by Michael Serra

  2. Definitions Imagine 2 adjacent isosceles triangles.

  3. Kite Properties Kite Angles Conjecture: The non-vertex angles of a kite are congruent. Kite Diagonals Conjecture: The diagonals of a kite are perpendicular. A M T H

  4. Kite Properties Kite Angle Bisector Conjecture: The vertex angles of a kite are bisected by a diagonal. A M T H

  5. Kite Properties Kite Diagonal Bisector Conjecture: The diagonal connecting the vertex angles of a kite is the perpendicular bisector of the other diagonal. A M T H

  6. Definitions

  7. Trapezoid Properties Trapezoid Consecutive Angles Conjecture: In a trapezoid, the consecutive angles between the bases are supplementary.

  8. Trapezoid Properties Isosceles Trapezoid Conjecture: In an isosceles trapezoid, the base angles are congruent. *Converse of Isosceles Trapezoid Conjecture: In a trapezoid, if the base angles are congruent, then the trapezoid is isosceles.

  9. Trapezoid Properties Isosceles Trapezoid Diagonals Conjecture: In an isosceles trapezoid, the diagonals are congruent.

  10. Real Life Connection

  11. Real Life Connection

  12. Book Exercises

  13. Book Exercises Answer the following: p. 271

  14. Book Exercises Answer the following: p. 271

  15. Book Exercises Answer the following: p. 271

  16. Book Exercises Answer the following: p. 271

  17. Book Exercises Answer the following: p. 271

  18. Book Exercises Answer the following: p. 271

  19. Book Exercises Answer the following: p. 271

  20. Book Exercises Answer the following: p. 271

  21. Definitions What is a midsegment of a triangle ?

  22. EXAMPLES NON-EXAMPLES

  23. Definitions What is a midsegment of a triangle ? A midsegment of a triangle is… a segment whose endpoints are the midpoints of two sides of a triangle. “3rd side” – side not connected to midsegment

  24. Definitions What is a midsegment of a triangle ?

  25. Definitions What is a midsegment of a trapezoid ?

  26. Definitions What is a midsegment of a trapezoid ? A midsegment of a trapezoid is… a segment whose endpoints are the midpoints of the non-parallel sides (legs) of a trapezoid. Can you draw non-examples of a midsegment of a trapezoid?

  27. Midsegment Properties Triangle Midsegment Conjecture: In a triangle, the midsegment is parallel to the third side, and measures half the length of the third side. Trapezoid Midsegment Conjecture: In a trapezoid, the midsegment is parallel to the bases, and measures half the sum of the lengths of the bases.

  28. Book Exercises Answer the following: p. 277

  29. Book Exercises Answer the following: p. 277

  30. Book Exercises Answer the following: p. 277

  31. Book Exercises Answer the following: p. 277

  32. Book Exercises Answer the following: p. 277

  33. Book Exercises Answer the following: p. 277

  34. Book Exercises Answer the following: p. 277

  35. Book Exercises Answer the following: p. 277

  36. Book Exercises Answer the following: p. 277

  37. MORE Exercises Answer the following: p. 304

  38. MORE Exercises Answer the following: p. 304

  39. MORE Exercises Answer the following: p. 304

  40. MORE Exercises Answer the following: p. 304

  41. MORE Exercises Answer the following: p. 304

  42. MORE Exercises Answer the following: p. 304

  43. MORE Exercises Answer the following: p. 304

  44. MORE Exercises • ALWAYS. SOMETIMES. NEVER. • The diagonals of a kite are congruent. S • Consecutive angles of a kite are supplementary. N • The diagonal connecting the vertex angles of a kite divides the kite into two congruent triangles. A • The diagonals of a trapezoid bisect each other. N • The three midsegments of a triangle divide the triangle into 4 congruent triangles. A • The midsegment of a trapezoid is perpendicular to a leg of the trapezoid. S

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