1. Prop. of Rhom., Rect., and Squares
5.10
Prop. of Rhom., Rect., and Squares
Rhombus Corollary
A quadrilateral is a rhombus if and only if it has four congruent _______.
sides A D B C
ABCD is a rhombus if and only if

2. Prop. of Rhom., Rect., and Squares
5.10
Prop. of Rhom., Rect., and Squares
Rectangle Corollary
A quadrilateral is a rectangle if and only if it has four _____________.
right angles A D B C
ABCD is a rectangle if and only if

3. Prop. of Rhom., Rect., and Squares
5.10
Prop. of Rhom., Rect., and Squares
Square Corollary
A quadrilateral is a square if and only if it is a ________ and a __________.
rhombus
rectangle A D B C
ABCD is a square if and only if

4. Prop. of Rhom., Rect., and Squares
5.10
Prop. of Rhom., Rect., and Squares
Example 1
Use properties of special quadrilaterals
For any rhombus RSTV, decide whether the statement is always or sometimes true. Draw a sketch and explain your reasoning. R V S T
By definition, a rhombus is a parallelogram with four congruent _______.
parallelogram
sides
By Theorem 5.19, opposite angles of a parallelogram are ___________.
congruent
The statement is _________ true.
always

5. Prop. of Rhom., Rect., and Squares
5.10
Prop. of Rhom., Rect., and Squares
Example 1
Use properties of special quadrilaterals
For any rhombus RSTV, decide whether the statement is always or sometimes true. Draw a sketch and explain your reasoning. R V S T
If rhombus RSTV is a _______, then all four angles are congruent right angles
square
square
squares
Because not all rhombuses are also ___________,
the statement is ___________ true.
sometimes

6. Prop. of Rhom., Rect., and Squares
5.10
Prop. of Rhom., Rect., and Squares
Example 2
Classify special quadrilaterals
Classify the special quadrilateral. Explain your reasoning.
The quadrilateral has four congruent _____.
sides
right angle
One of the angles is not a ____________,
square
so the rhombus is not also a ________.
By the Rhombus Corollary, the quadrilateral is a __________.
rhombus

7. Prop. of Rhom., Rect., and Squares
5.10
Prop. of Rhom., Rect., and Squares
Checkpoint. Complete the following exercises.
For any square CDEF, is it always or sometimes true that
? Explain your reasoning. C F D E
By the Square Corollary, all the sides are congruent.
Therefore, it is always true.

8. Prop. of Rhom., Rect., and Squares
5.10
Prop. of Rhom., Rect., and Squares
Checkpoint. Complete the following exercises.
A quadrilateral has four congruent sides and four congruent angles. Classify the quadrilateral. C F D E
By the Square Corollary, all the sides are congruent and all angles are right angles.
The quadrilateral is a square.

9. Prop. of Rhom., Rect., and Squares
5.10
Prop. of Rhom., Rect., and Squares
Theorem 5.26
A parallelogram is a rhombus if and only if its diagonals are _____________.
perpendicular A D B C
ABCD is a rhombus if and only if

10. Prop. of Rhom., Rect., and Squares
5.10
Prop. of Rhom., Rect., and Squares
Theorem 5.27
A parallelogram is a rhombus if and only if each diagonal bisects a pair of opposite sides. A D B C
ABCD is a rhombus if and only if

11. Prop. of Rhom., Rect., and Squares
5.10
Prop. of Rhom., Rect., and Squares
Theorem 5.28
A parallelogram is a rectangle if and only if its diagonals are __________.
congruent A D B C
ABCD is a rectangle if and only if

12. Prop. of Rhom., Rect., and Squares
5.10
Prop. of Rhom., Rect., and Squares
Example 3
List properties of special parallelograms F J G H
Sketch rhombus FGHJ List everything you know about it.
Solution
By definition, you need to draw a figure with the following properties:
parallelogram
The figure is a _______________.
The figure has four congruent _______.
sides
Because FGHJ is a parallelogram, it has these properties:
opposite sides are ________ and ___________.
parallel
congruent
opposite angles are __________. Consecutive angles are _________________.
congruent
supplementary
diagonals ________ each other.
bisect
perpendicular
By Theorem 5.26, the diagonals of FGHJ are _______________.
By Theorem 5.27, each diagonals bisect a pair of _____________.
opposite angles

13. Prop. of Rhom., Rect., and Squares
5.10
Prop. of Rhom., Rect., and Squares
Example 4
Solve a real-world problem
Framing You are building a frame for a painting. The measurements of the frame are shown at the right.
The frame must be a rectangle. Given the measurements in the figure, can you assume that it is? Explain.
Solution
No, you cannot. The boards on opposite sides are the same length, so they form a ______________.
parallelogram
right angles
But you do not know whether the angles are ______________.

14. Prop. of Rhom., Rect., and Squares
5.10
Prop. of Rhom., Rect., and Squares
Example 4
Solve a real-world problem
Framing You are building a frame for a painting. The measurements of the frame are shown at the right.
You measure the diagonals of the frame. The diagonals are about 25.6 inches. What can you conclude about the shape of the frame?
Solution
By Theorem 5.28, the diagonals of a rectangle are __________.
congruent
The diagonals of the frame are __________, so the frame forms a ______________.
congruent
rectangle

15. Prop. of Rhom., Rect., and Squares
5.10
Prop. of Rhom., Rect., and Squares
Checkpoint. Complete the following exercises.
Sketch rectangle WXYZ List everything you know about it. W Z X Y
The drawing of WXYZ is a parallelogram with four right angles.
It has these characteristics:
opposite sides are parallel and congruent.
opposite angles are congruent and consecutive angles are supplementary.
diagonals bisect each other and are congruent.

16. Prop. of Rhom., Rect., and Squares
5.10
Prop. of Rhom., Rect., and Squares
Checkpoint. Complete the following exercises.
Suppose the diagonals of the frame in Example 4 are not congruent. Could the frame still be a rectangle? NO
By Theorem 5.28, a rectangle must have congruent diagonals.

17. Prop. of Rhom., Rect., and Squares
5.10
Prop. of Rhom., Rect., and Squares
Pg. 331, 5.10 #2-24 even, 25