special parallelograms
DESCRIPTION
Special Parallelograms. Warm Up What is the sum of the interior angles of 11-gon. Given Parallelogram ABCD, what is the value of y? Explain why the quadrilateral , JKLM, is a Parallelogram. Objectives. Prove and apply properties of rectangles, rhombuses, and squares. - PowerPoint PPT PresentationTRANSCRIPT
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Special Parallelograms
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Warm Up
1.What is the sum of the interior angles of 11-gon.
2.Given Parallelogram ABCD, what is the value of y?
3.Explain why the quadrilateral, JKLM, is a Parallelogram
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Prove and apply properties of rectangles, rhombuses, and squares.Use properties of rectangles, rhombuses, and squares to solve problems.
Objectives
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A second type of special quadrilateral is a rectangle. A rectangle is a quadrilateral with four right angles.
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Since a rectangle is a parallelogram by Theorem 6-4-1, a rectangle “inherits” all the properties of parallelograms that you learned in Lesson 6-2.
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Example 1: Craft ApplicationA woodworker constructs a rectangular picture frame so that JK = 50 cm and JL = 86 cm. Find HM.
Rect. diags.
Def. of segs.
Substitute and simplify.
KM = JL = 86
diags. bisect each other
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Check It Out! Example 1a
Carpentry The rectangular gate has diagonal braces. Find HJ.
Def. of segs.
Rect. diags.
HJ = GK = 48
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Check It Out! Example 1b
Carpentry The rectangular gate has diagonal braces. Find HK.
Def. of segs.
Rect. diags.
JL = LG
JG = 2JL = 2(30.8) = 61.6 Substitute and simplify.
Rect. diagonals bisect each other
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When you are given a parallelogram with certainproperties, you can use the theorems below to determine whether the parallelogram is a rectangle.
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When you are given a parallelogram with certainproperties, you can use the theorems below to determine whether the parallelogram is a rectangle.
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Example 1: Carpentry Application
A manufacture builds a mold for a desktop so that , , and mABC = 90°. Why must ABCD be a rectangle?
Both pairs of opposites sides of ABCD are congruent, so ABCD is a . Since mABC = 90°, one angle ABCD is a right angle. ABCD is a rectangle by Theorem 6-5-1.
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A rhombus is another special quadrilateral. A rhombus is a quadrilateral with four congruent sides.
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Like a rectangle, a rhombus is a parallelogram. So you can apply the properties of parallelograms to rhombuses.
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Example 2A: Using Properties of Rhombuses to Find Measures
TVWX is a rhombus. Find TV.
Def. of rhombusSubstitute given values.Subtract 3b from both sides and add 9 to both sides.
Divide both sides by 10.
WV = XT13b – 9 = 3b + 4
10b = 13
b = 1.3
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Example 2A Continued
Def. of rhombus
Substitute 3b + 4 for XT.
Substitute 1.3 for b and simplify.
TV = XT
TV = 3b + 4TV = 3(1.3) + 4 = 7.9
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Rhombus diag.
Example 2B: Using Properties of Rhombuses to Find Measures
TVWX is a rhombus. Find mVTZ.
Substitute 14a + 20 for mVTZ.
Subtract 20 from both sides and divide both sides by 14.
mVZT = 90°14a + 20 = 90°
a = 5
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Example 2B Continued
Rhombus each diag. bisects opp. s
Substitute 5a – 5 for mVTZ.
Substitute 5 for a and simplify.
mVTZ = mZTX
mVTZ = (5a – 5)°
mVTZ = [5(5) – 5)]° = 20°
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Check It Out! Example 2a CDFG is a rhombus. Find CD.
Def. of rhombus
SubstituteSimplifySubstitute
Def. of rhombusSubstitute
CG = GF
5a = 3a + 17a = 8.5
GF = 3a + 17 = 42.5CD = GFCD = 42.5
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Check It Out! Example 2b
CDFG is a rhombus. Find the measure.mGCH if mGCD = (b + 3)°and mCDF = (6b – 40)°
mGCD + mCDF = 180°
b + 3 + 6b – 40 = 180°
7b = 217°
b = 31°
Def. of rhombus
Substitute.
Simplify.
Divide both sides by 7.
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Check It Out! Example 2b Continued
mGCH + mHCD = mGCD
2mGCH = mGCDRhombus each diag. bisects opp. s
2mGCH = (b + 3)2mGCH = (31 + 3)
mGCH = 17°
Substitute.Substitute.
Simplify and divide both sides by 2.
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Below are some conditions you can use to determine whether a parallelogram is a rhombus.
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Example 2A: Applying Conditions for Special ParallelogramsDetermine if the conclusion is valid. If not, tell what additional information is needed to make it valid.
Given:Conclusion: EFGH is a rhombus.The conclusion is not valid. By Theorem 6-5-3, if one pair of consecutive sides of a parallelogram are congruent, then the parallelogram is a rhombus. By Theorem 6-5-4, if the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus. To apply either theorem, you must first know that ABCD is a parallelogram.
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A square is a quadrilateral with four right angles and four congruent sides. In the exercises, you will show that a square is a parallelogram, a rectangle, and a rhombus. So a square has the properties of all three.
One special characteristic of a square is that the diagonals are•Congruent•Perpendicular•Bisect one another
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Example 3: Verifying Properties of Squares
Show that the diagonals of square EFGH are congruent perpendicular bisectors of each other.
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Example 3 Continued
Step 1 Show that EG and FH are congruent.
Since EG = FH,
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Example 3 Continued
Step 2 Show that EG and FH are perpendicular.
Since ,
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The diagonals are congruent perpendicular bisectors of each other.
Example 3 Continued
Step 3 Show that EG and FH are bisect each other.
Since EG and FH have the same midpoint, they bisect each other.
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Check It Out! Example 3
The vertices of square STVW are S(–5, –4), T(0, 2), V(6, –3) , and W(1, –9) . Show that the diagonals of square STVW are congruent perpendicular bisectors of each other.
111slope of SV =
slope of TW = –11
SV TW
SV = TW = 122 so, SV TW .
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Step 1 Show that SV and TW are congruent.
Check It Out! Example 3 Continued
Since SV = TW,
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Step 2 Show that SV and TW are perpendicular.
Check It Out! Example 3 Continued
Since
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The diagonals are congruent perpendicular bisectors of each other.
Step 3 Show that SV and TW bisect each other.
Since SV and TW have the same midpoint, they bisect each other.
Check It Out! Example 3 Continued
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In order to apply Theorems 6-5-1 through 6-5-5, the quadrilateral must be a parallelogram.
Caution
To prove that a given quadrilateral is a square, it is sufficient to show that the figure is both a rectangle and a rhombus. You will explain why this is true in Exercise 43.
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Example 2B: Applying Conditions for Special Parallelograms
Determine if the conclusion is valid. If not, tell what additional information is needed to make it valid.
Given:
Conclusion: EFGH is a square.Step 1 Determine if EFGH is a parallelogram.
Given
EFGH is a parallelogram. Quad. with diags. bisecting each other
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Example 2B Continued
Step 2 Determine if EFGH is a rectangle.
Given.
EFGH is a rectangle. Step 3 Determine if EFGH is a rhombus.
EFGH is a rhombus.
with diags. rect.
with one pair of cons. sides rhombus
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Example 2B Continued
Step 4 Determine is EFGH is a square.
Since EFGH is a rectangle and a rhombus, it has four right angles and four congruent sides. So EFGH is a square by definition.
The conclusion is valid.
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Check It Out! Example 2
Determine if the conclusion is valid. If not, tell what additional information is needed to make it valid.Given: ABC is a right angle.Conclusion: ABCD is a rectangle.
The conclusion is not valid. By Theorem 6-5-1, if one angle of a parallelogram is a right angle, then the parallelogram is a rectangle. To apply this theorem, you need to know that ABCD is a parallelogram .
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Example 3A: Identifying Special Parallelograms in the Coordinate Plane
Use the diagonals to determine whether a parallelogram with the given vertices is a rectangle, rhombus, or square. Give all the names that apply.
P(–1, 4), Q(2, 6), R(4, 3), S(1, 1)
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Example 3A Continued
Step 1 Graph PQRS.
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Step 2 Find PR and QS to determine is PQRS is a rectangle.
Example 3A Continued
Since , the diagonals are congruent. PQRS is a rectangle.
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Step 3 Determine if PQRS is a rhombus.
Step 4 Determine if PQRS is a square.
Since PQRS is a rectangle and a rhombus, it has four right angles and four congruent sides. So PQRS is a square by definition.
Example 3A Continued
Since , PQRS is a rhombus.
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Check It Out! Example 4
Given: PQTS is a rhombus with diagonalProve:
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Check It Out! Example 4 Continued
Statements Reasons1. PQTS is a rhombus. 1. Given.
2. Rhombus → eachdiag. bisects opp. s
3. QPR SPR 3. Def. of bisector.4. Def. of rhombus.5. Reflex. Prop. of 6. SAS7. CPCTC
2.
4.5.
7.6.
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Lesson Quiz: Part I
A slab of concrete is poured with diagonal spacers. In rectangle CNRT, CN = 35 ft, and NT = 58 ft. Find each length.
1. TR 2. CE
35 ft 29 ft
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Lesson Quiz: Part II
PQRS is a rhombus. Find each measure.
3. QP 4. mQRP
42 51°
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Lesson Quiz: Part III
5. The vertices of square ABCD are A(1, 3), B(3, 2), C(4, 4), and D(2, 5). Show that its diagonals are congruent perpendicular bisectors of each other.
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Lesson Quiz: Part IV
ABE CDF
6. Given: ABCD is a rhombus. Prove: