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Rectangles, Rhombi and Squares

Rectangles, Rhombi and Squares. Notes 23 – Sections 6.4 & 6.5. Essential Learnings. Students will understand and be able to recognize and apply properties of rectangles, rhombi and squares. Students will be able to determine whether parallelograms are rectangles, rhombi, or squares.

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Rectangles, Rhombi and Squares

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  1. Rectangles, Rhombi and Squares Notes 23 – Sections 6.4 & 6.5

  2. Essential Learnings • Students will understand and be able to recognize and apply properties of rectangles, rhombi and squares. • Students will be able to determine whether parallelograms are rectangles, rhombi, or squares.

  3. Rectangles • A rectangle is a parallelogram with four right angles.

  4. Diagonals of a Rectangle • If a parallelogram is a rectangle, then its diagonals are congruent.

  5. Theorem • If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle.

  6. Example 1 • A rectangular garden gate is reinforced with diagonal braces to prevent it from sagging. If JK = 12 feet, and LN = 6.5 feet, find KM. K J M L

  7. Example 2 • Quadrilateral RSTU is a rectangle. If m∠RTU = 8x+4 and m∠SUR = 3x–2, find x. R S U T

  8. Example 3 • Quadrilateral WXYZ is a rectangle. If PW = 2x+3 and XP= x+4, find XZ. X Y P W Z

  9. Example 4 • Quadrilateral ABCD is a rectangle. Find each measure if m∠1 = 37 . A B D C

  10. Example 5 • Quadrilateral OMGR is a rectangle. If MG=6, RG=8, and RM=a, find OG. O M R G

  11. Rhombus • A rhombus is a parallelogram with all four sides congruent. • The plural of rhombus is rhombi.

  12. Diagonals of a Rhombus • If a parallelogram is a rhombus, then its diagonals are perpendicular.

  13. Conditions for Rhombi and Squares • If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus. A B D C

  14. Diagonals of a Rhombus • If a parallelogram is a rhombus, then each diagonal bisects a pair of opposite angles.

  15. Conditions for Rhombi and Squares • If one diagonal of a parallelogram bisects a pair of opposite angles, then the parallelogram is a rhombus. A B D C

  16. Conditions for Rhombi and Squares • If one pair of consecutive sides of a parallelogram are congruent, then the parallelogram is a rhombus. B A C D

  17. Square • A square is a parallelogram with all four congruent sides and four right angles. • All squares are rhombi.

  18. Example 6 ABCD is a rhombus. If AP = 4x and PC = 5x - 3, find AC. B A P D C

  19. Example 7 Ifm∠DAP = 6y + 7 and m∠PAB = 9y - 5, find y. B A P D C

  20. Example 8 ABCD is a rhombus. If AP = 5, BC = 13, and m∠CBD = 67º, find each measure. PB = DP = m∠BPC = m∠DCA = B A P D C

  21. Assignment p. 422: 10-13, 15, 17, 19, 26-31, 39-41 p. 431: 7-12, 23-27, 29, 31-33 Quiz - Friday Unit Study Guide 4

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