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Assignment. P. 537-540: 1, 2, 3-48 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. Rhombuses Or Rhombi. What makes a quadrilateral a rhombus?. Rhombuses Or Rhombi. A rhombus is an equilateral parallelogram. - PowerPoint PPT PresentationTRANSCRIPT
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Assignment• P. 537-540: 1, 2, 3-
48 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 RhombiWhat makes a quadrilateral a rhombus?
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Rhombuses Or RhombiA rhombus is an
equilateral parallelogram.– All sides are
congruent
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Rhombus CorollaryA quadrilateral is a
rhombus if and only if it has four congruent sides.
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RectanglesWhat makes a quadrilateral a rectangle?
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RectanglesA rectangle is an
equiangular parallelogram.
• All angles are congruent
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Example 1What must each angle of a rectangle
measure?
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Rectangle CorollaryA quadrilateral is a
rectangle if and only if it has four right angles.
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SquaresWhat makes a quadrilateral a square?
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SquaresA square is a regular
parallelogram.• All angles are
congruent• All sides are
congruent
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Square CorollaryA 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 squares2. To find the area of rhombuses, rectangles,
and squares
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Example 2Below 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 2Below 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 3For any rhombus QRST, decide whether the
statement is always or sometimes true. Draw a sketch and explain your reasoning.
1. Q S2. Q R
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Example 4For any rectangle ABCD, decide whether the
statement is always or sometimes true. Draw a sketch and explain your reasoning.
1. AB CD2. AB BC
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Example 5Classify the special quadrilateral. Explain
your reasoning.
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Investigation 1We 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 1A parallelogram is a rectangle if and only if
its diagonals are congruent.
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Example 6The 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 7You’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 2A parallelogram is a rhombus if and only if its
diagonals are perpendicular.
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Diagonal Theorem 3A parallelogram is a rhombus if and only if
each diagonal bisects a pair of opposite angles.
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Example 8Prove 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: SATIn 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 10In the diagram below, MRVU SPTV. Let the area of MRVU equal A. Show that A = bh.
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Rhombus AreaSince a rhombus is a
parallelogram, we could find its area by multiplying the base and the height.
A b h
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Rhombus AreaHowever, you’re not
always given the base and height, so let’s look at the two diagonals. Notice that d1 divides the rhombus into 2 congruent triangles.
Ah, there’s a couple of triangles in there.
12
A b h
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Rhombus AreaSo find the area of
one triangle, and then double the result.
12
A b h
122
A b h
1 21 122 2
A d d
1 2124
A d d
1 212d d
1 212
A d d
Ah, there’s a couple of triangles in there.
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Polygon Area Formulas
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Exercise 11Find the area of the shaded region.1. 2. 3.
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Exercise 12If the length of each diagonal of a rhombus is
doubled, how is the area of the rhombus affected?
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Assignment• P. 537-540: 1, 2, 3-
48 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