Unit 7: Modeling with Geometric Relationships Mathematics 3 Ms. C. Taylor.
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Transcript of Unit 7: Modeling with Geometric Relationships Mathematics 3 Ms. C. Taylor.
![Page 1: Unit 7: Modeling with Geometric Relationships Mathematics 3 Ms. C. Taylor.](https://reader036.fdocuments.in/reader036/viewer/2022071713/56649e315503460f94b226e6/html5/thumbnails/1.jpg)
Unit 7: Modeling with Geometric RelationshipsMathematics 3
Ms. C. Taylor
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Warm-UpA reporter wants to know the percentage of
voters in the state who support building a new highway. What is the reporter’s population?
A) the number of people who live in the state
B) the people who were interviewed in the state
C) all voters over 25 years old in the stateD) all eligible voters in the state
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Distance around an Arc
An arc of a circle is a segment of the circumference of the circle.
Arc length of a circle in radians:
Arc length of a circle in degrees:
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Constructions
https://www.youtube.com/watch?v=fVDr08YbQww
https://www.youtube.com/watch?v=il0EJrY64qE
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Warm-Up
Find the arc length of a circle if the diameter is 18 and the angle is .
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Theorems Galore!If two lines intersect to form a
linear pair of congruent angles, then the lines are perpendicular.
Proof Hints:• Write an equation based on
the angles forming a linear pair.
• Do some substitution and solve for one of the angles. It should be 90°.
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Theorems Galore!If two lines are perpendicular, then they intersect to form four right angles.
Proof Hints:• Use definition of
perpendicular lines to find one right angle.
• Use vertical and linear pairs of angles to find three more right angles.
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Theorems Galore!
If two sides of two adjacent acute angles are perpendicular, then the angles are complementary.
Proof Hints:• Use definition of perpendicular to
get the measure of ABC.• Use Angle Addition Postulate,
Substitution, and Definition of complementary angles to finish the proof.
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Perpendicular Transversal Theorem
If a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other.
Theorems Galore!
j Proof Hints:• Use definition of
perpendicular lines to find one right angle.
• Use Corresponding Angles Postulate to find a right angle on the other line.
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Theorems Galore!Lines Perpendicular to a
Transversal TheoremIn a plane, if two lines are perpendicular to the same line, then they are parallel to each other.
Proof Hints:• Use definition of
perpendicular lines to find a right angle on each parallel line.
• Use Converse of Corresponding Angles Postulate to prove the lines are parallel.
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Proving Vertical Angle TheoremVertical Angles Theorem: Vertical angles are congruent
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Proving Vertical Angle Theorem
PROVE 5 7
GIVEN 5 and 6 are a linear pair,6 and 7 are a linear pair
1
2
3
Statements Reasons
5 and 6 are a linear pair, Given6 and 7 are a linear pair
5 and 6 are supplementary, Linear Pair Postulate6 and 7 are supplementary
5 7 Congruent Supplements Theorem
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The Third Angles Theorem below follows from the Triangle Sum Theorem.
Third Angles Theorem
If two angles of one triangle are congruent to two angles of another triangle, then the third angles are also congruent.
If A D and B E, then C F.
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PROPERTIES OF PARALLEL LINESPOSTULATE 15 Corresponding Angles Postulate
1
2
1 2
If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent.
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PROPERTIES OF PARALLEL LINESTHEOREM 3.4 Alternate Interior Angles
3
4
3 4
If two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent.
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PROPERTIES OF PARALLEL LINESTHEOREM 3.5 Consecutive Interior Angles
m 5 + m 6 = 180°
5
6
If two parallel lines are cut by a transversal, then the pairs of consecutive interior angles are supplementary.
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PROPERTIES OF PARALLEL LINESTHEOREM 3.6 Alternate Exterior Angles
7
8
7 8
If two parallel lines are cut by a transversal, then the pairs of alternate exterior angles are congruent.
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PROPERTIES OF PARALLEL LINESTHEOREM 3.7 Perpendicular Transversal
j k
If a transversal is perpendicular to one of two parallellines, then it is perpendicular to the other.
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Proving the Alternate Interior Angles TheoremProve the Alternate Interior Angles Theorem.
SOLUTION
GIVEN p || q
p || q Given
Statements Reasons
1
2
3
4
PROVE 1 2
1 3 Corresponding Angles Postulate
3 2 Vertical Angles Theorem
1 2 Transitive property of Congruence
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Using Properties of Parallel Lines
SOLUTION
Given that m 5 = 65°, find each measure. Tellwhich postulate or theoremyou use.
Linear Pair Postulatem 7 = 180° – m 5 = 115°
Alternate Exterior Angles Theoremm 9 = m 7 = 115°
Corresponding Angles Postulatem 8 = m 5 = 65°
m 6 = m 5 = 65° Vertical Angles Theorem
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Using Properties of Parallel Lines
Use properties ofparallel lines to findthe value of x.
SOLUTION
Corresponding Angles Postulatem 4 = 125°
Linear Pair Postulatem 4 + (x + 15)° = 180°
Substitute.125° + (x + 15)° = 180°
Subtract.x = 40°
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Using the Third Angles TheoremFind the value of x.
SOLUTION
In the diagram, N R and L S.
From the Third Angles Theorem, you know that M T.So, m M = m T. From the Triangle Sum Theorem, m M =
180˚– 55˚ – 65˚ = 60˚.
m M = m T
60˚ = (2 x + 30)˚
30 = 2 x
15 = x
Third Angles Theorem
Substitute
Subtract 30 from each side.
Divide each side by 2.
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PROPERTIES OF SPECIAL PAIRS OF ANGLES Congruent Supplements Theorem
If two angles are supplementary to the same angle (or to congruent angles) then they are congruent.
2
3
1
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Linear Pair Postulate
If two angles for m a linear pair, then they are supplementary.
m 1 + m 2 = 180°
PROPERTIES OF SPECIAL PAIRS OF ANGLES
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CONGRUENCE OF ANGLESTHEOREM 2.2 Properties of Angle Congruence
Angle congruence is reflexive, symmetric, and transitive.Here are some examples.
TRANSITIVE If A B and B C, then A C
SYMMETRIC If A B, then B A
REFLEX IVE For any angle A, A A
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Transitive Property of Angle CongruenceProve the Transitive Property of Congruence for angles.
SOLUTION
To prove the Transitive Property of Congruence for angles, begin by drawing three congruent angles. Label the vertices as A, B, and C.
GIVEN A B, PROVE A C
A
B
C
B C
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Transitive Property of Angle CongruenceGIVEN A B,
B C
PROVE A C
Statements Reasons
1
2
3
4
m A = m B Definition of congruent angles
5 A C Definition of congruent angles
A B, Given
B C
m B = m C Definition of congruent angles
m A = m C Transitive property of equality
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Using the Transitive PropertyThis two-column proof uses the Transitive Property.
Statements Reasons
2
3
4
m 1 = m 3 Definition of congruent angles
GIVEN m 3 = 40°, 1 2, 2 3
PROVE m 1 = 40°
1
m 1 = 40° Substitution property of equality
1 3 Transitive property of Congruence
Givenm 3 = 40°, 1 2,
2 3
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Proving Right Angle Congruence Theorem Right Angle Congruence Theorem
All right angles are congruent.
You can prove Right Angle CongruenceTheorem as shown.
GIVEN 1 and 2 are right angles
PROVE 1 2
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Proving Right Angle Congruence Theorem
Statements Reasons
1
2
3
4
m 1 = 90°, m 2 = 90° Definition of right angles
1 2 Definition of congruent angles
GIVEN 1 and 2 are right angles
PROVE 1 2
1 and 2 are right angles Given
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PROPERTIES OF SPECIAL PAIRS OF ANGLES
Congruent Supplements Theorem
If two angles are supplementary to the same angle (or to congruent angles) then they are congruent.
1 23
If m 1 + m 2 = 180°
m 2 + m 3 = 180°
and
1
then
1 3
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PROPERTIES OF SPECIAL PAIRS OF ANGLES
Congruent Complements TheoremIf two angles are complementary to the same angle (or to congruent angles) then the two angles are congruent.
45
6
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PROPERTIES OF SPECIAL PAIRS OF ANGLES
Congruent Complements TheoremIf two angles are complementary to the same angle (or to congruent angles) then the two angles are congruent.
4
If m 4 + m 5 = 90°
m 5 + m 6 = 90°
and
then
4 6
56
4
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Proving Congruent Supplements Theorem
Statements Reasons
1
2
GIVEN 1 and 2 are supplements
PROVE 2 3
3 and 4 are supplements
1 4
1 and 2 are supplements Given
3 and 4 are supplements
1 4
m 1 + m 2 = 180° Definition of supplementary angles
m 3 + m 4 = 180°
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Proving Congruent Supplements Theorem
Statements Reasons
3
GIVEN 1 and 2 are supplements
PROVE 2 3
3 and 4 are supplements
1 4
4
5 m 1 + m 2 = Substitution property of equalitym 3 + m 1
m 1 + m 2 = Transitive property of equalitym 3 + m 4
m 1 = m 4 Definition of congruent angles
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Proving Congruent Supplements Theorem
Statements Reasons
GIVEN 1 and 2 are supplements
PROVE 2 3
3 and 4 are supplements
1 4
6
7
m 2 = m 3 Subtraction property of equality
2 3 Definition of congruent angles
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2
3
4
5
6 SAS Congruence Postulate DRA DRG
1
Proving Triangles Congruent
GivenDR AG
If 2 lines are , then they form 4 right angles.
DRA and DRGare right angles.
Right Angle Congruence Theorem DRA DRG
GivenRA RG
Reflexive Property of CongruenceDR DR
Statements Reasons
D
GA R
GIVEN
PROVE DRA DRG
DR AG
RA RG
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Congruent Triangles in a Coordinate Plane
sOLUTION
Use the SSS Congruence Postulate to show that ABC FGH.
AC FH
AB FGAB = 5 and FG = 5
AC = 3 and FH = 3
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Congruent Triangles in a Coordinate Plane
d = (x 2 – x1 ) 2 + ( y2 – y1 )
2
= 3 2 + 5
2
= 34
BC = (– 4 – (– 7)) 2 + (5 – 0 )
2
d = (x 2 – x1 ) 2 + ( y2 – y1 )
2
= 5 2 + 3
2
= 34
GH = (6 – 1) 2 + (5 – 2 )
2
Use the distance formula to find lengths BC and GH.
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Congruent Triangles in a Coordinate Plane
All three pairs of corresponding sides are congruent, ABC FGH by the SSS Congruence Postulate.
BC GHBC = 34 and GH = 34
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Goal 1
Identifying Congruent FiguresTwo geometric figures are congruent if they have exactly the same size and shape.
Each of the red figures is congruent to the other red figures.
None of the blue figures is congruent to another blue figure.
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Identifying Congruent FiguresWhen two figures are congruent, there is a correspondence between their angles and sides such that corresponding angles are congruent and corresponding sides are congruent.
For the triangles below, you can write , which reads “triangle ABC is congruent to triangle PQR.” The notation shows the congruence and the correspondence.
ABC PQR
There is more than one way to write a congruence statement, but it is important to list the corresponding angles in the same order. For example, you can also write .BCA QRP
Corresponding Angles Corresponding Sides
A P B Q C R
BC QR
RPCA
AB PQ
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Naming Congruent PartsThe two triangles shown below are congruent. Write a congruence statement. Identify all pairs of congruent corresponding parts.
SOLUTION
Angles:
Sides:
D R, E S, F T
, , RS DE TRFD ST EF
The diagram indicates that .
The congruent angles and sides are as follows.
DEF RST
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Example
Using Properties of Congruent FiguresIn the diagram, NPLM EFGH.
Find the value of x.
SOLUTION
You know that .GHLM
So, LM = GH.
8 = 2 x – 3
11 = 2 x
5.5 = x
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Using Properties of Congruent FiguresIn the diagram, NPLM EFGH.
SOLUTION
Find the value of x.
You know that .GHLM
So, LM = GH.
8 = 2 x – 3
11 = 2 x
5.5 = x
Find the value of y.
You know that N E.
So, m N = m E.
72˚ = (7y + 9)˚
63 = 7y
9 = y
SOLUTION
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SOLUTION
Paragraph Proof
From the diagram, you are given that all three corresponding sides are congruent.
, NQPQ ,MNRP QMQR and
Because P and N have the same measures, P N.
By the Vertical Angles Theorem, you know that PQR NQM.
By the Third Angles Theorem, R M.
Decide whether the triangles are congruent. Justify your reasoning.
So, all three pairs of corresponding sides and all three pairs of corresponding angles are congruent. By the definition of congruent triangles, .PQR NQM
Proving Triangles are Congruent
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Proving Two Triangles are Congruent
A B
C D
E
|| , DCAB ,DCAB E is the midpoint of BC and AD.
Plan for Proof Use the fact that AEB and DEC are vertical angles to show that those angles are congruent. Use the fact that BC intersects parallel segments AB and DC to identify other pairs of angles that are congruent.
GIVEN
PROVE .AEB DEC
Prove that .AEB DEC
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Proving Two Triangles are Congruent
Statements Reasons
EAB EDC, ABE DCE
AEB DEC
E is the midpoint of AD,E is the midpoint of BC
,DEAE CEBE
Given
Alternate Interior Angles Theorem
Vertical Angles Theorem
Given
Definition of congruent triangles
Definition of midpoint
|| ,DCAB DCAB
SOLUTION
AEB DEC
A B
C D
E
Prove that .AEB DEC
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You have learned to prove that two triangles are congruent by the definition of congruence – that is, by showing that all pairs of corresponding angles and corresponding sides are congruent.
Theorem 4.4 Properties of Congruent Triangles
Reflexive Property of Congruent Triangles
D
E
F
A
B
C
J K
L
Every triangle is congruent to itself.
Symmetric Property of Congruent Triangles
Transitive Property of Congruent Triangles
If , then .ABC DEF DEF ABC
If and , then .JKLABC DEF DEF ABC JKL
Proving Triangles are Congruent
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SSS AND SAS CONGRUENCE POSTULATES
If all six pairs of corresponding parts (sides and angles) are congruent, then the triangles are congruent.
and thenIf Sides are
congruent
1. AB DE
2. BC EF
3. AC DF
Angles are congruent
4. A D
5. B E
6. C F
Triangles are congruent
ABC DEF
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Using the SSS Congruence Postulate
Prove that PQW TSW.
Paragraph Proof
SOLUTION
So by the SSS Congruence Postulate, you
know that
PQW TSW.
The marks on the diagram show that PQ TS,
PW TW, and QW SW.
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SSS AND SAS CONGRUENCE POSTULATESPOSTULATE: Side-Angle-Side (SAS) Congruence Postulate
then PQS WXY
If two sides and the included angle of one triangle arecongruent to two sides and the included angle of a second triangle, then the two triangles are congruent.
If Side PQ WX
Side QS XY
Angle Q XA
S
S
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1
Using the SAS Congruence Postulate
Prove that AEB DEC.
2
3 AEB DEC SAS Congruence Postulate
21
AE DE, BE CE Given
1 2 Vertical Angles Theorem
Statements Reasons
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2
3
4
5
6 SAS Congruence Postulate DRA DRG
1
Proving Triangles Congruent
GivenDR AG
If 2 lines are , then they form 4 right angles.
DRA and DRG are right angles.
Right Angle Congruence Theorem DRA DRG
GivenRA RG
Reflexive Property of CongruenceDR DR
Statements Reasons
D
GA R
GIVEN
PROVE DRA DRG
DR AG
RA RG
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Congruent Triangles in a Coordinate Plane
AC FH
AB FGAB = 5 and FG = 5
SOLUTION
Use the SSS Congruence Postulate to show that ABC
FGH.
AC = 3 and FH = 3
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Congruent Triangles in a Coordinate Plane
d = (x 2 – x1 ) 2 + ( y2 – y1 )
2
= 3 2 + 5
2
= 34
BC = (– 4 – (– 7)) 2 + (5 – 0 )
2
d = (x 2 – x1 ) 2 + ( y2 – y1 )
2
= 5 2 + 3
2
= 34
GH = (6 – 1) 2 + (5 – 2 )
2
Use the distance formula to find lengths BC and GH.
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Congruent Triangles in a Coordinate Plane
All three pairs of corresponding sides are congruent,
ABC FGH by the SSS Congruence Postulate.
BC GHBC = √34 and GH = √34
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Theorems
Theorem 6.6: If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.
A
D
B
C
ABCD is a parallelogram.
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Theorems
Theorem 6.7: If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.
ABCD is a parallelogram.
A
D
B
C
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Theorems
Theorem 6.8: If an angle of a quadrilateral is supplementary to both of its consecutive angles, then the quadrilateral is a parallelogram.
ABCD is a parallelogram.
A
D
B
C
x°
(180 – x)° x°
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Theorems
Theorem 6.9: If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.
ABCD is a parallelogram.
A
D
B
C
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Ex. 1: Proof of Theorem 6.6
Statements:
1. AB ≅ CD, AD ≅ CB.
2. AC ≅ AC
3. ∆ABC ≅ ∆CDA
4. BAC ≅ DCA, DAC ≅ BCA
5. AB║CD, AD ║CB.
6. ABCD is a
Reasons:
1. Given
C
D
B
A
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Ex. 1: Proof of Theorem 6.6Statements:
1. AB ≅ CD, AD ≅ CB.
2. AC ≅ AC
3. ∆ABC ≅ ∆CDA
4. BAC ≅ DCA, DAC ≅ BCA
5. AB║CD, AD ║CB.
6. ABCD is a
Reasons:
1. Given
2. Reflexive Prop. of Congruence
C
D
B
A
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Ex. 1: Proof of Theorem 6.6Statements:
1. AB ≅ CD, AD ≅ CB.
2. AC ≅ AC
3. ∆ABC ≅ ∆CDA
4. BAC ≅ DCA, DAC ≅ BCA
5. AB║CD, AD ║CB.
6. ABCD is a
Reasons:
1. Given
2. Reflexive Prop. of Congruence
3. SSS Congruence Postulate
C
D
B
A
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Ex. 1: Proof of Theorem 6.6Statements:
1. AB ≅ CD, AD ≅ CB.
2. AC ≅ AC
3. ∆ABC ≅ ∆CDA
4. BAC ≅ DCA, DAC ≅ BCA
5. AB║CD, AD ║CB.
6. ABCD is a
Reasons:
1. Given
2. Reflexive Prop. of Congruence
3. SSS Congruence Postulate
4. CPCTC
C
D
B
A
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Ex. 1: Proof of Theorem 6.6Statements:
1. AB ≅ CD, AD ≅ CB.
2. AC ≅ AC
3. ∆ABC ≅ ∆CDA
4. BAC ≅ DCA, DAC ≅ BCA
5. AB║CD, AD ║CB.
6. ABCD is a
Reasons:
1. Given
2. Reflexive Prop. of Congruence
3. SSS Congruence Postulate
4. CPCTC
5. Alternate Interior s Converse
C
D
B
A
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Ex. 1: Proof of Theorem 6.6Statements:
1. AB ≅ CD, AD ≅ CB.
2. AC ≅ AC
3. ∆ABC ≅ ∆CDA
4. BAC ≅ DCA, DAC ≅ BCA
5. AB║CD, AD ║CB.
6. ABCD is a
Reasons:
1. Given
2. Reflexive Prop. of Congruence
3. SSS Congruence Postulate
4. CPCTC
5. Alternate Interior s Converse
6. Def. of a parallelogram.
C
D
B
A
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Using properties of parallelograms.
Method 1
Use the slope formula to show that opposite sides have the same slope, so they are parallel.
Method 2
Use the distance formula to show that the opposite sides have the same length.
Method 3
Use both slope and distance formula to show one pair of opposite side is congruent and parallel.
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Ex. 2: Proving Quadrilaterals are Parallelograms
As the sewing box below is opened, the trays are always parallel to each other. Why?
2.75 in. 2.75 in.
2 in.
2 in.
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Ex. 2: Proving Quadrilaterals are Parallelograms
Each pair of hinges are opposite sides of a quadrilateral. The 2.75 inch sides of the quadrilateral are opposite and congruent. The 2 inch sides are also opposite and congruent. Because opposite sides of the quadrilateral are congruent, it is a parallelogram. By the definition of a parallelogram, opposite sides are parallel, so the trays of the sewing box are always parallel.
2.75 in. 2.75 in.
2 in.
2 in.
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Another Theorem ~
Theorem 6.10—If one pair of opposite sides of a quadrilateral are congruent and parallel, then the quadrilateral is a parallelogram.
ABCD is a
parallelogram.
A
B C
D
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Ex. 3: Proof of Theorem 6.10Given: BC║DA, BC ≅ DAProve: ABCD is a
Statements:
1. BC ║DA
2. DAC ≅ BCA
3. AC ≅ AC
4. BC ≅ DA
5. ∆BAC ≅ ∆DCA
6. AB ≅ CD
7. ABCD is a
Reasons:
1. Given
C
D
B
A
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Ex. 3: Proof of Theorem 6.10Given: BC║DA, BC ≅ DAProve: ABCD is a
Statements:
1. BC ║DA
2. DAC ≅ BCA
3. AC ≅ AC
4. BC ≅ DA
5. ∆BAC ≅ ∆DCA
6. AB ≅ CD
7. ABCD is a
Reasons:
1. Given
2. Alt. Int. s Thm.
C
D
B
A
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Ex. 3: Proof of Theorem 6.10Given: BC║DA, BC ≅ DAProve: ABCD is a
Statements:
1. BC ║DA
2. DAC ≅ BCA
3. AC ≅ AC
4. BC ≅ DA
5. ∆BAC ≅ ∆DCA
6. AB ≅ CD
7. ABCD is a
Reasons:
1. Given
2. Alt. Int. s Thm.
3. Reflexive Property
C
D
B
A
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Ex. 3: Proof of Theorem 6.10Given: BC║DA, BC ≅ DAProve: ABCD is a
Statements:
1. BC ║DA
2. DAC ≅ BCA
3. AC ≅ AC
4. BC ≅ DA
5. ∆BAC ≅ ∆DCA
6. AB ≅ CD
7. ABCD is a
Reasons:
1. Given
2. Alt. Int. s Thm.
3. Reflexive Property
4. Given
C
D
B
A
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Ex. 3: Proof of Theorem 6.10Given: BC║DA, BC ≅ DAProve: ABCD is a
Statements:
1. BC ║DA
2. DAC ≅ BCA
3. AC ≅ AC
4. BC ≅ DA
5. ∆BAC ≅ ∆DCA
6. AB ≅ CD
7. ABCD is a
Reasons:
1. Given
2. Alt. Int. s Thm.
3. Reflexive Property
4. Given
5. SAS Congruence Post.
C
D
B
A
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Ex. 3: Proof of Theorem 6.10Given: BC║DA, BC ≅ DAProve: ABCD is a
Statements:
1. BC ║DA
2. DAC ≅ BCA
3. AC ≅ AC
4. BC ≅ DA
5. ∆BAC ≅ ∆DCA
6. AB ≅ CD
7. ABCD is a
Reasons:
1. Given
2. Alt. Int. s Thm.
3. Reflexive Property
4. Given
5. SAS Congruence Post.
6. CPCTC
C
D
B
A
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Ex. 3: Proof of Theorem 6.10Given: BC║DA, BC ≅ DAProve: ABCD is a
Statements:
1. BC ║DA
2. DAC ≅ BCA
3. AC ≅ AC
4. BC ≅ DA
5. ∆BAC ≅ ∆DCA
6. AB ≅ CD
7. ABCD is a
Reasons:
1. Given
2. Alt. Int. s Thm.
3. Reflexive Property
4. Given
5. SAS Congruence Post.
6. CPCTC
7. If opp. sides of a quad. are ≅, then it is a .
C
D
B
A
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Ex. 4: Using properties of parallelograms
Show that
A(2, -1), B(1, 3), C(6, 5) and D(7,1) are the vertices of a parallelogram.
6
4
2
-2
-4
5 10 15
D(7, 1)
C(6, 5)
B(1, 3)
A(2, -1)
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Ex. 4: Using properties of parallelograms Method 1—Show that opposite
sides have the same slope, so they are parallel.
Slope of AB. 3-(-1) = - 4
1 - 2 Slope of CD.
1 – 5 = - 4
7 – 6 Slope of BC.
5 – 3 = 2
6 - 1 5 Slope of DA.
- 1 – 1 = 2
2 - 7 5 AB and CD have the same
slope, so they are parallel. Similarly, BC ║ DA.
6
4
2
-2
-4
5 10 15
D(7, 1)
C(6, 5)
B(1, 3)
A(2, -1)
Because opposite sides are parallel, ABCD is a parallelogram.
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Ex. 4: Using properties of parallelograms Method 2—Show that
opposite sides have the same length.
AB=√(1 – 2)2 + [3 – (- 1)2] = √17
CD=√(7 – 6)2 + (1 - 5)2 = √17
BC=√(6 – 1)2 + (5 - 3)2 = √29
DA= √(2 – 7)2 + (-1 - 1)2 = √29
AB ≅ CD and BC ≅ DA. Because both pairs of opposites sides are congruent, ABCD is a parallelogram.
6
4
2
-2
-4
5 10 15
D(7, 1)
C(6, 5)
B(1, 3)
A(2, -1)
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Ex. 4: Using properties of parallelograms
Method 3—Show that one pair of opposite sides is congruent and parallel.
Slope of AB = Slope of CD = -4
AB=CD = √17
AB and CD are congruent and parallel, so ABCD is a parallelogram.
6
4
2
-2
-4
5 10 15
D(7, 1)
C(6, 5)
B(1, 3)
A(2, -1)
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Proving quadrilaterals are parallelograms:
Show that both pairs of opposite sides are parallel.
Show that both pairs of opposite sides are congruent.
Show that both pairs of opposite angles are congruent.
Show that one angle is supplementary to both consecutive angles.
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.. continued..
Show that the diagonals bisect each other
Show that one pair of opposite sides are congruent and parallel.
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Engineering
Deshon uses an expandable gate to keep his new puppy in the kitchen. As the gate expands or collapses, the shapes that form the gate always remain parallelograms. Explain why this is true.
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Drafting Before computer drawing program become
available, blueprints for buildings or mechanical parts were drawn by hand. One of the tools drafters used, is a parallel ruler. Holding one of the bars in the place and moving the other allowed the drafter to draw a line ll to the first in many position on the page. Why does the parallel ruler guarantee that the second line will be ll to the first?
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Arts The Navoja people are well known for their skills in
weaving. Eye- Dazzler rugs became popular with Navoja weavers in the 1880s.What types of shapes do you see most?
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Side-Side-Side (SSS) Similarity Theorem
If the corresponding side lengths of two triangles are proportional, then the triangles are similar.
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SOLUTION
Compare ABC and DEF by finding ratios of corresponding side lengths.
Shortest sides
ABDE
43
86 ==
Is either DEF or GHJ similar to ABC?
Longest sidesCAFD
43
1612 ==
Remaining sides
BCEF
43
12 9 ==
All of the ratios are equal, so ABC ~ DEF.ANSWER
Use the SSS Similarity Theorem
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Use the SSS Similarity Theorem (continued)
Longest sides
CAJG
1616 == 1
Remaining sides
BCHJ
65
1210 ==
The ratios are not all equal, so ABC and GHJ are not similar.
ANSWER
Compare ABC and GHJ by finding ratios of corresponding side lengths.
Shortest sides
ABGH
88 == 1
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Use the SSS Similarity Theorem
Which of the three triangles are similar? Write a similarity statement.
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SOLUTION
Use the SSS Similarity Theorem
Find the value of x that makes ABC ~ DEF.
STEP 1 Find the value of x that makes corresponding side lengths proportional. 4
12 = x –1 18 Write proportion.
4 18 = 12(x – 1)
72 = 12x – 12
7 = x
Cross Products Property
Simplify.
Solve for x.
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Use the SSS Similarity Theorem (continued)Check that the side lengths are
proportional when x = 7.STEP 2
BC = x – 1 = 6
618
412 =AB
DEBCEF=
?
DF = 3(x + 1) = 24
824
412 =
ABDE
ACDF=
?
When x = 7, the triangles are similar by the SSS Similarity Theorem.
ANSWER
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Use the SSS Similarity Theorem
Find the value of x that makes PQRXYZ ~
X Z
Y
P R
Q
20
12
x + 6 30
3(x – 2)
21
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Side-Angle-Side (SAS) Similarity Theorem
If an angle of one triangle is congruent to an angle of a second triangle and the lengths of the sides including these angles are proportional, then the triangles are similar.
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Use the SAS Similarity TheoremLean-to Shelter
You are building a lean-to shelter starting from a tree branch, as shown. Can you construct the right end so it is similar to the left end using the angle measure and lengths shown?
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Use the SAS Similarity Theorem (continued)
Both m A and m F equal = 53°, so A F. Next, compare the ratios of the lengths of the sides that include A and F.
~
SOLUTION
Shorter sides Longer sidesABFG
32
96 == AC
FH32
1510 ==
The lengths of the sides that include A and F are proportional.
ANSWER
So, by the SAS Similarity Theorem, ABC ~ FGH. Yes, you can make the right end similar to the left end of the shelter.
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Choose a methodTell what method you would use to show that the triangles are similar.
Find the ratios of the lengths of the corresponding sides.
Shorter sides Longer sides
SOLUTION
CACD
35
1830 ==
BCEC
35
915 ==
The corresponding side lengths are proportional. The included angles ACB and DCE are congruent because they are vertical angles. So, ACB ~ DCE by the SAS Similarity Theorem.
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Choose a method
A. SRT ~ PNQ
Explain how to show that the indicated triangles are similar.
B. XZW ~ YZX
Explain how to show that the indicated triangles are similar.
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AA Similarity (Angle-Angle)
A D
If 2 angles of one triangle are congruent to 2 angles of another triangle, then the triangles are similar.
E
DA
B
CF
B E
ABC ~ DEFConclusion:
andGiven:
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SSS Similarity (Side-Side-Side)
If the measures of the corresponding sides of 2 triangles are proportional, then the triangles are similar.
E
DA
B
CF
Given:
Conclusion:
5
11 22
8 1610
BC
EF
AB
DE
AC
DF
8
16
5
10
11
22
ABC ~ DEF
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SAS Similarity (Side-Angle-Side)
ABC ~ DEF
If the measures of 2 sides of a triangle are proportional to the measures of 2 corresponding sides of another triangle and the angles between them are congruent, then the triangles are similar.
Given:
Conclusion:
E
DA
B
CF
5
11 22
10
AB ACA D and
DE DF
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Similarity is reflexive, symmetric, and transitive.
1. Mark the Given.2. Mark …
Shared Angles or Vertical Angles3. Choose a Method. (AA, SSS , SAS)Think about what you need for the chosen method and be sure to include those parts in the proof.
Steps for proving triangles similar:
Proving Triangles Similar
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Problem #1:
Pr :
Given DE FG
ove DEC FGC
CD
E
G
F
Step 1: Mark the given … and what it implies
Step 2: Mark the vertical angles
Step 3: Choose a method: (AA,SSS,SAS)Step 4: List the Parts in the order of the method with reasons
Step 5: Is there more? Statements Reasons
Given
Alternate Interior <s
AA Similarity
Alternate Interior <s
1. DE FG2. D F 3. E G
4. DEC FGC
AA
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Problem #2
Step 1: Mark the given … and what it implies
Step 2: Choose a method: (AA,SSS,SAS)Step 4: List the Parts in the order of the method with reasons
Step 5: Is there more? Statements Reasons
Given
Division Property
SSS Similarity
Substitution
SSS
: 3 3 3
Pr :
Given IJ LN JK NP IK LP
ove IJK LNP
N
L
P
I
J K
1. IJ = 3LN ; JK = 3NP ; IK = 3LP
2. IJ
LN=3,
JK
NP=3,
IK
LP=3
3. IJ
LN=
JK
NP=
IK
LP
4. IJK~ LNP
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Problem #3
Step 1: Mark the given … and what it implies
Step 3: Choose a method: (AA,SSS,SAS)
Step 4: List the Parts in the order of the method with reasons
Next Slide………….
Step 5: Is there more?
SAS
: int
int
Pr :
Given G is the midpo of ED
H is the midpo of EF
ove EGH EDF
E
DF
G H
Step 2: Mark the reflexive angles
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Statements Reasons
1. G is the Midpoint of H is the Midpoint of
Given
2. EG = DG and EH = HF Def. of Midpoint
3. ED = EG + GD and EF = EH + HF Segment Addition Post.
4. ED = 2 EG and EF = 2 EH Substitution
Division Property
Substitution
Reflexive Property
SAS Postulate7. GEHDEF
8. EGH~ EDF
6. ED
EG=
EF
EH
5. ED
EG=2 and
EF
EH =2
ED
EF