5.10

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

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

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

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

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

6. 5.10 Prop. of Rhom., Rect., and Squares Classify special quadrilaterals Example 2 Classify the special quadrilateral. Explain your reasoning. sides The quadrilateral has four congruent _____. 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. 5.10 Prop. of Rhom., Rect., and Squares D C • For any square CDEF, is it always or sometimes true that ? Explain your reasoning. F E Checkpoint. Complete the following exercises. By the Square Corollary, all the sides are congruent. Therefore, it is always true.

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

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

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

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

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

13. 5.10 Prop. of Rhom., Rect., and Squares Solve a real-world problem Example 4 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. 5.10 Prop. of Rhom., Rect., and Squares Solve a real-world problem Example 4 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 congruent By Theorem 5.28, the diagonals of a rectangle are __________. congruent The diagonals of the frame are __________, so the frame forms a ______________. rectangle

15. 5.10 Prop. of Rhom., Rect., and Squares X W Y Z Checkpoint. Complete the following exercises. • Sketch rectangle WXYZ. List everything you know about it. 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. 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. 5.10 Prop. of Rhom., Rect., and Squares Pg. 331, 5.10 #2-24 even, 25