Chapter 6 Lesson 4 Objective: Objective: To use properties of diagonals of rhombuses and rectangles.

Rhombuses

Theorem 6-9 Theorem 6-9 Each diagonal of a rhombus bisects two angles of the rhombus.

Theorem 6-10 Theorem 6-10 The diagonals of a rhombus are perpendicular.

Example 1: Finding Angle Measures MNPQ is a rhombus and m N = 120. Find the measures of the numbered angles. Isosceles ∆ Theorem ∆ Angle-Sum Theorem

Example 2: Finding Angle Measures Find the measures of the numbered angles in the rhombus. Theorem 6-10 Theorem 6-9

Rectangles

Theorem 6-11 Theorem 6-11 The diagonals of a rectangle are congruent.

Example 3: Finding the Lengths of Diagonals Find the length of the diagonals of rectangle GFED if FD = 2y + 4 and GE = 6y − 5. Theorem 6-11

Example 4: Finding the Lengths of Diagonals Find the length of the diagonals of GFED if FD = 5y – 9 and GE = y + 5. Theorem 6-11

Is the parallelogram a rhombus or a rectangle?

Theorem 6-12 Theorem 6-12 If one diagonal of a parallelogram bisects two angles of the parallelogram, then the parallelogram is a rhombus. Theorem 6-13 Theorem 6-13 If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus. Theorem 6-14 Theorem 6-14 If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle.

Example 5: Recognizing Special Parallelograms Determine whether the quadrilateral can be a parallelogram. If not, write impossible.

Example 6: Recognizing Special Parallelograms A diagonal of a parallelogram bisects two angles of the parallelogram. Is it possible for the parallelogram to have sides of length 5, 6, 5, and 6? No; if one diagonal bisects two angles, then the figure is a rhombus and cannot have two non-congruent sides.

Assignment Pg.315 #1-21(odd); 48-50; 57-60