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Assignment P : 1, 2, M3, 49, 52, 55, Pick one (56, 60, 61, 63) P. 723: 5, 18, 25, 27, 40 P. 732: 8, 11, 15, 20, 28, 36 Challenge Problems

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Rhombuses Or Rhombi What makes a quadrilateral a rhombus?

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Rhombuses Or Rhombi rhombus A rhombus is an equilateral parallelogram. –All sides are congruent

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Rhombus Corollary A quadrilateral is a rhombus if and only if it has four congruent sides.

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Rectangles What makes a quadrilateral a rectangle?

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Rectangles rectangle A rectangle is an equiangular parallelogram. All angles are congruent

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Example 1 What must each angle of a rectangle measure?

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Rectangle Corollary A quadrilateral is a rectangle if and only if it has four right angles.

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Squares What makes a quadrilateral a square?

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Squares square A square is a regular parallelogram. All angles are congruent All sides are congruent

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Square Corollary A quadrilateral is a square if and only if it is a rhombus and a rectangle.

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8.4 Properties of Rhombuses, Rectangles, and Squares Objectives: 1.To discover and use properties of rhombuses, rectangles, and squares 2.To find the area of rhombuses, rectangles, and squares

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Example 2 Venn Diagram Below is a concept map showing the relationships between some members of the parallelogram family. This type of concept map is known as a Venn Diagram. Fill in the missing names.

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Example 2 Venn Diagram Below is a concept map showing the relationships between some members of the parallelogram family. This type of concept map is known as a Venn Diagram.

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Example 3 For any rhombus QRST, decide whether the statement is always or sometimes true. Draw a sketch and explain your reasoning. 1. Q S 2. Q R

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Example 4 For any rectangle ABCD, decide whether the statement is always or sometimes true. Draw a sketch and explain your reasoning. 1. AB CD 2. AB BC

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Example 5 Classify the special quadrilateral. Explain your reasoning.

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Investigation 1 We know that the diagonals of parallelograms bisect each other. The diagonal of rectangles and rhombuses have a few other properties we will discover using GSP.

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Diagonal Theorem 1 A parallelogram is a rectangle if and only if its diagonals are congruent.

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Example 6 The previous theorem is a biconditional. Write the two conditional statements that must be proved separately to prove the entire theorem.

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Example 7 You’ve just had a new door installed, but it doesn’t seem to fit into the door jamb properly. What could you do to determine if your new door is rectangular?

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Diagonal Theorem 2 A parallelogram is a rhombus if and only if its diagonals are perpendicular.

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Diagonal Theorem 3 A parallelogram is a rhombus if and only if each diagonal bisects a pair of opposite angles.

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Example 8 Prove that if a parallelogram has perpendicular diagonals, then it is a rhombus. Given: ABCD is a parallelogram; AC BD Prove: ABCD is a rhombus

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Example 9: SAT In the figure, a small square is inside a larger square. What is the area, in terms of x, of the shaded region?

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Example 10 In the diagram below, MRVU SPTV. Let the area of MRVU equal A. Show that A = bh.

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Rhombus Area Since a rhombus is a parallelogram, we could find its area by multiplying the base and the height.

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Rhombus Area However, you’re not always given the base and height, so let’s look at the two diagonals. Notice that d 1 divides the rhombus into 2 congruent triangles. Ah, there’s a couple of triangles in there.

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Rhombus Area So find the area of one triangle, and then double the result. Ah, there’s a couple of triangles in there.

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Polygon Area Formulas

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Exercise 11 Find the area of the shaded region

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Exercise 12 If the length of each diagonal of a rhombus is doubled, how is the area of the rhombus affected?

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Assignment P : 1, 2, M3, 49, 52, 55, Pick one (56, 60, 61, 63) P. 723: 5, 18, 25, 27, 40 P. 732: 8, 11, 15, 20, 28, 36 Challenge Problems

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