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Properties of Parallelograms: Rectangles, Rhombuses, and Squares

Learn about the properties and characteristics of parallelograms, specifically rectangles, rhombuses, and squares. Understand their definitions, theorems about diagonals, and examples of trapezoids and kites.

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Properties of Parallelograms: Rectangles, Rhombuses, and Squares

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  1. Do Now: List all you know about the following parallelograms. 1.) Rectangle 2.) Rhombus 3.) Square

  2. Geometry 8.4: Properties of Rhombuses, Rectangles, and Squares 8.5: e Properties of Trapezoids and Kites

  3. Rectangles • Parallelogram with 4 right angles

  4. Rhombus • Parallelogram with four congruent sides.

  5. Square • A square is a parallelogram with four congruent sides and four right angles. • A square is a rhombus and a rectangle.

  6. Theorems about Diagonals • Diagonals of a rhombus are perpendicular (also true for a square- remember a square is a rhombus.)

  7. Theorems about Diagonals • Diagonals of a rhombus bisect the opposite angles (also true for a square - remember a square is a rhombus.)

  8. Theorems about Diagonals • Diagonals of a rectangle are congruent (also true for a square- remember a square is a rectangle.)

  9. Examples

  10. Examples

  11. Examples

  12. Trapezoids • A trapezoid is a quadrilateral with exactly one pair of parallel sides. (not parallelogram) • The parallel sides are called bases. The other two sides are called legs. • A trapezoid has two pairs of base angles. base A B One pair of base Angles: A & B. Another pair: D and C. leg leg C D base

  13. Isosceles Trapezoids • If the legs of a trapezoid are congruent, then the trapezoid is an isosceles trapezoid. • Theorem 8.14: If a trapezoid is isosceles, then each pair of base angles is congruent. A B D C

  14. Theorem 8.15 • If a trapezoid has a pair of congruent base angles, then it is an isosceles trapezoid. A B D C ABCD is an isosceles trapezoid. (AD is congruent to BC).

  15. Examples: Find the missing angles.

  16. Theorem 8.16 • A trapezoid is isosceles if and only if its diagonals are congruent. A B D C ABCD is an isosceles trapezoid if and only if .

  17. Midsegment of a Trapezoid • Midsegment connects the midpoints of the legs. • Theorem 8.17: The midsegment of a trapezoid is parallel to each base and its length is one half the sum of the lengths of the bases. (average) A B Midsegment M N D C

  18. Examples Find the length of the other base.

  19. Examples

  20. Kites • A kite is a quadrilateral that has two pairs of consecutive congruent sides, but opposite sides are not congruent. (not parallelogram)

  21. Kites • Theorem 8.18: If a quadrilateral is a kite, then its diagonals are perpendicular. • Theorem 8.19: If a quadrilateral is a kite, then exactly one pair of opposite angles are congruent.

  22. Examples

  23. Example: Find the side lengths of the kite.

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