Unit 6: Lesson #4 Warm-Up: Pick up the Lesson #4 Example sheet and complete the Warm-Up on the back.

Properties of Rhombi, Rectangles, and Squares Unit 6: Lesson #4 Properties of Rhombi, Rectangles, and Squares (Textbook Section 6-4)

Properties of Rhombi and Squares

Statements Reasons Warm-Up: E H G F Given: EFGH is a rhombus Prove: Diagonal EG is the angle bisector of E and G Statements Reasons EFGH is a rhombus Given EF  FG  GH  HE Definition of a rhombus EG  EG Reflexive Property EFG  EHG SSS GEF  GEH and EGF  EGH CPCTC Diagonal EG is the angle bisector of E and G Definition of an Angle Bisector

1) Could we also prove that diagonal FH is also the angle bisector of F and H? Explain. 2) Using what we know about isosceles triangles, what can we conclude about the angle at which the two diagonals meet each other? Explain. E H G F Yes, the proof we just completed would have also worked for the diagonal FH. E H G F The diagonals are each acting as the angle bisectors of the vertex angle of the isosceles triangles. Therefore, they must be perpendicular to each other.

Diagonals of a Rhombus Theorem A parallelogram is a rhombus if and only if the diagonals … a) bisect the angles of the rhombus. b) are perpendicular.

NOTE: Because a rhombus is a parallelogram, the opposite angles are congruent. Therefore, when the diagonals bisect A and C, it creates FOUR congruent angles. The same thing happens at B and D.

Example #1 12 78 12 90 28 90 12 90 28 12 78 124 The following figures are rhombuses. Find the measures of the numbered angles. 78 12 124° 1 2 3 78° 4 1 12 90 28 90 12 90 28 3 12 78 2 124 78 + 90 = 168 180 – 168 = 12 180 – 124 = 56 56 / 2 = 28

Properties of Rectangles and Squares

Statements Reasons Warm-Up: J M K L Given: JKLM is a rectangle Prove: Diagonal JL  Diagonal KM Statements Reasons JKLM is a rectangle Given JM  KL Opposite sides of a parallelogram are congruent ML  ML Reflexive Property M and L are right angles. Definition of a rectangle. M  L All right angles are congruent KJM  MLK SAS JL  KM CPCTC

Diagonals of a Rectangle Theorem: A parallelogram is a rectangle if and only if its diagonals are congruent. NOTE: Because a rectangle is a parallelogram, the diagonals bisect each other. Therefore, each half of the two diagonals are all congruent to each other.

Example #2A A) 1 = ________ BD = ________ 68.5 A D B C 20 137 M 43 20 180 – 137 = 43 AC = BD 180 – 43 = 137 AC = 2 CM = 20 BD = 20 137/2 = 68.5

Example #2B B) Find the length of the diagonals of rectangle QRST if QS = 5y – 9 and RT = y + 5. T Q R S 5y – 9 = y + 5 4y – 9 = 5 4y = 14 y = 3.5

Properties of Squares

Example #3 4) The figure is a square. EP = 10a – 22 a = ______ FP = 48 y = ______ EP = 10a – 22 FP = 48 7 45° x y E G F H P 90° (Diagonals of a rhombus are perpendicular) 10a – 22 = 48 10a = 70 a = 7 x = 90 / 2 = 45 (In a rectangle the angles are 90 at the vertices) (In a rhombus, the diagonals bisect the angles at the vertices)

Homework Time!