Take note Theorem 6-17
Theorem
If one diagonal of a parallelogram bisects a pair of opposite angles, then the parallelogram is a rhombus.
If …
ABCD is a
Then …
ABCD is a rhombus
You will prove Theorem 6-17 in Exercise 23.
Theorem 6-18
Theorem
If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle.
If …
ABCD is a
Then …
ABCD is a rectangle
You will prove Theorem 6-18 in Exercise 24.
You can use Theorems 6-16, 6-17, and 6-18 to classify parallelograms. Notice that if a parallelogram is both a rectangle and a rhombus, then it is a square.
Problem 1 Identifying Special Parallelograms
Can you conclude that the parallelogram is a rhombus, a rectangle, or a square? Explain.
Think
How can you determine whether a figure is a special parallelogram?
See if you can satisfy every condition of a definition or theorem about rhombuses, rectangles, or squares.
-
Yes. A diagonal bisects two angles. By Theorem 6-17, this parallelogram is a rhombus.
-
Yes. The diagonals are congruent, so by Theorem 6-18, this parallelogram is a rectangle. The diagonals are perpendicular, so by Theorem 6-16, it is a rhombus. Therefore, this parallelogram is a square.
✓ Got It?
-
- A parallelogram has angle measures of 20, 160, 20, and 160. Can you conclude that it is a rhombus, a rectangle, or a square? Explain.
- Reasoning Suppose the diagonals of a quadrilateral bisect each other. Can you conclude that it is a rhombus, a rectangle, or a square? Explain.
Table of Contents
- 6-1 The Polygon Angle-Sum Theorems
- 6-2 Properties of Parallelograms
- 6-3 Proving That a Quadrilateral Is a Parallelogram
- 6-4 Properties of Rhombuses, Rectangles, and Squares
- 6-5 Conditions for Rhombuses, Rectangles, and Squares
- 6-6 Trapezoids and Kites
- 6-7 Polygons in the Coordinate Plane
- 6-8 and 6-9 Coordinate Geometry and Coordinate Proofs