Triangles and Angles
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Transcript of Triangles and Angles
Triangles and AnglesTriangles and AnglesTriangles and AnglesTriangles and Angles
2
Standard/Objectives:Standard 3: Students will learn and apply
geometric concepts.Objectives:• Classify triangles by their sides and
angles.• Find angle measures in trianglesDEFINITION: A triangle is a figure formed
by three segments joining three non-collinear points.
3
Names of triangles
Equilateral—3 congruent sides
Isosceles Triangle—2 congruent sides
Scalene—no congruent sides
Triangles can be classified by the sides or by the angle
4
Acute Triangle
mCAB = 41.76 mBCA = 67.97
mABC = 70.26
B
A
C
3 acute angles
5
Equiangular triangle• 3 congruent angles. An
equiangular triangle is also acute.
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Right Triangle• 1 right angle
Obtuse Triangle
7
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Parts of a triangle• Each of the three
points joining the sides of a triangle is a vertex.(plural: vertices). A, B and C are vertices.
• Two sides sharing a common vertext are adjacent sides.
• The third is the side opposite an angle
B
C
A
adjacent
adjacent
Side opposite A
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Right Triangle• Red represents
the hypotenuse of a right triangle. The sides that form the right angle are the legs.
hypotenuseleg
leg
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• An isosceles triangle can have 3 congruent sides in which case it is equilateral. When an isosceles triangle has only two congruent sides, then these two sides are the legs of the isosceles triangle. The third is the base.
leg
leg
base
Isosceles Triangles
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Identifying the parts of an isosceles triangle
• Explain why ∆ABC is an isosceles right triangle.
• In the diagram you are given that C is a right angle. By definition, then ∆ABC is a right triangle. Because AC = 5 ft and BC = 5 ft; AC BC. By definition, ∆ABC is also an isosceles triangle.
A B
C
About 7 ft.
5 ft 5 ft
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Identifying the parts of an isosceles triangle
• Identify the legs and the hypotenuse of ∆ABC. Which side is the base of the triangle?
• Sides AC and BC are adjacent to the right angle, so they are the legs. Side AB is opposite the right angle, so it is t he hypotenuse. Because AC BC, side AB is also the base.
A B
C
About 7 ft.
5 ft 5 ftleg leg
Hypotenuse & Base
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14
A
B
C
Using Angle Measures of Triangles Smiley faces are
interior angles and hearts represent the exterior angles
Each vertex has a pair of congruent exterior angles; however it is common to show only one exterior angle at each vertex.
15
Ex. 3 Finding an Angle Measure.
65
x
Exterior Angle theorem: m1 = m A +m 1
(2x+10)
x + 65 = (2x + 10)
65 = x +10
55 = x
16
17
Finding angle measures• Corollary to the
triangle sum theorem
• The acute angles of a right triangle are complementary.
• m A + m B = 90
2x
x
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Finding angle measuresX + 2x = 903x = 90X = 30
• So m A = 30 and the m B=60
2x
xC
B
A
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Congruence and Congruence and TrianglesTriangles
2222
Standards/Objectives:Standards/Objectives:
Standard 2: Students will learn and apply Standard 2: Students will learn and apply geometric conceptsgeometric concepts
Objectives:Objectives: Identify congruent figures and Identify congruent figures and
corresponding partscorresponding parts Prove that two triangles are congruentProve that two triangles are congruent
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Identifying congruent Identifying congruent figuresfigures
Two geometric figures are congruent if Two geometric figures are congruent if they have exactly the same size and they have exactly the same size and shape. shape.
CONGRUENT
NOT CONGRUENT
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TrianglesTriangles
Corresponding anglesCorresponding angles
A A ≅ ≅ PP
B B ≅ ≅ QQ
C C ≅ ≅ RR
Corresponding SidesCorresponding Sides
AB AB ≅ PQ≅ PQ
BC ≅ QRBC ≅ QR
CA ≅ RPCA ≅ RP
A
B
C
Q
P R
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ZZ
If If ΔΔ ABC is ABC is to to ΔΔ XYZ, which XYZ, which angle is angle is to to C?C?
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Thm 4.3Thm 4.333rdrd angles thm angles thm
If 2 If 2 s of one s of one ΔΔ are are to 2 to 2 s of another s of another ΔΔ, then the , then the 3rd 3rd s are also s are also ..
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Ex: find xEx: find x
)
))
2222oo
8787oo
)
)) (4x+15)(4x+15)oo
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Ex: continuedEx: continued
22+87+4x+15=18022+87+4x+15=180
4x+15=714x+15=71
4x=564x=56
x=14x=14
3131
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Ex: ABCD is Ex: ABCD is to HGFE, find x to HGFE, find x and y.and y.
4x-3=9 5y-12=113
4x=12 5y=125
x=3 y=25
A B
D C
91°
86° 113°
9 cm
AB
DC
(5y-12)°
HG
F E
4x – 3 cm
3333
Thm 4.4Thm 4.4Props. of Props. of ΔΔss
Reflexive prop of Reflexive prop of ΔΔ - - Every Every ΔΔ is is to itself to itself ((ΔΔABC ABC ΔΔABC).ABC).
Symmetric prop of Symmetric prop of ΔΔ - - If If ΔΔABC ABC ΔΔPQR, then PQR, then ΔΔPQR PQR ΔΔABC.ABC.
Transitive prop of Transitive prop of ΔΔ - If - If ΔΔABC ABC ΔΔPQR & PQR & ΔΔPQR PQR ΔΔXYZ, then XYZ, then ΔΔABC ABC ΔΔXYZ.XYZ.
A
BB
CPP
Q
R
XY
Z
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3737
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Proving Δs are : SSS and SAS
40
Standards/Benchmarks
Standard 2: Students will learn and apply geometric concepts
Objectives:
• Prove that triangles are congruent using the SSS and SAS Congruence Postulates.
• Use congruence postulates in real life problems such as bracing a structure.
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Remember?
As of yesterday, Δs could only be if ALL sides AND angles were
NOT ANY MORE!!!!There are two short cuts to add.
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Post. 19Side-Side-Side (SSS) post
• If 3 sides of one Δ are to 3 sides of another Δ, then the Δs are .
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Meaning:
If seg AB seg ED, seg AC seg EF & seg BC seg DF, then ΔABC ΔEDF.
___
___
___
___
___
___
___
___
___
___
___
___
A
B CE
D F
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Given: seg QR seg UT, RS TS, QS=10, US=10
Prove: ΔQRS ΔUTS
Q
R S T
U
10 10
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46
47
Proof
Statements Reasons
1. 1. given
2. QS=US 2. subst. prop. =
3. Seg QS seg US 3. Def of segs.
4. Δ QRS Δ UTS 4. SSS post
48
Post. 20Side-Angle-Side post. (SAS)
• If 2 sides and the included of one Δ are to 2 sides and the included of another Δ, then the 2 Δs are .
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• If seg BC seg YX, seg AC seg ZX, and C X, then ΔABC ΔZXY.
B
AC
X
Y
Z)(
50
51
Given: seg WX seg. XY, seg VX seg ZX,
Prove: Δ VXW Δ ZXY
1 2
W
V
X
Z
Y
52
Proof
Statements Reasons
1. seg WX seg. XY 1. given seg. VX seg ZX
2. 1 2 2. vert s thm
3. Δ VXW Δ ZXY 3. SAS post
53
54
Given: seg RS seg RQ and seg ST seg QT
Prove: Δ QRT Δ SRT.Q
R
S
T
55
Proof
Statements Reasons
1. Seg RS seg RQ 1. Given seg ST seg QT
2. Seg RT seg RT 2. Reflex prop
3. Δ QRT Δ SRT 3. SSS post
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Given: seg DR seg AG and seg AR seg GR
Prove: Δ DRA Δ DRG.
D
AR
G
57
Proof
Statements
1. seg DR seg AG
Seg AR seg GR
2. seg DR Seg DR
3.DRG & DRA are rt. s
4.DRG DRA
5. Δ DRG Δ DRA
Reasons1. Given
2. reflex. Prop of 3. lines form 4 rt. s
4. Rt. s thm
5. SAS post.
58
59
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Proving Triangles are Congruent: ASA and
AAS
62
Objectives:
1. Prove that triangles are congruent using the ASA Congruence Postulate and the AAS Congruence Theorem
2. Use congruence postulates and theorems in real-life problems.
63
Postulate 21: Angle-Side-Angle (ASA) Congruence Postulate• If two angles and the
included side of one triangle are congruent to two angles and the included side of a second triangle, then the triangles are congruent.
B
C
A
F
D
E
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65
Theorem 4.5: Angle-Angle-Side (AAS) Congruence Theorem• If two angles and a
non-included side of one triangle are congruent to two angles and the corresponding non-included side of a second triangle, then the triangles are congruent.
B
C
A
F
D
E
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Theorem 4.5: Angle-Angle-Side (AAS) Congruence TheoremGiven: A D, C
F, BC EF
Prove: ∆ABC ∆DEF
B
C
A
F
D
E
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Theorem 4.5: Angle-Angle-Side (AAS) Congruence TheoremYou are given that two angles of
∆ABC are congruent to two angles of ∆DEF. By the Third Angles Theorem, the third angles are also congruent. That is, B E. Notice that BC is the side included between B and C, and EF is the side included between E and F. You can apply the ASA Congruence Postulate to conclude that ∆ABC ∆DEF.
B
C
A
F
D
E
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Ex. 1 Developing Proof
Is it possible to prove the triangles are congruent? If so, state the postulate or theorem you would use. Explain your reasoning.
G
E
JF
H
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Ex. 1 Developing Proof
A. In addition to the angles and segments that are marked, EGF JGH by the Vertical Angles Theorem. Two pairs of corresponding angles and one pair of corresponding sides are congruent. You can use the AAS Congruence Theorem to prove that ∆EFG ∆JHG.
G
E
JF
H
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Ex. 1 Developing Proof
Is it possible to prove the triangles are congruent? If so, state the postulate or theorem you would use. Explain your reasoning.
N
M
Q
P
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Ex. 1 Developing Proof
B. In addition to the congruent segments that are marked, NP NP. Two pairs of corresponding sides are congruent. This is not enough information to prove the triangles are congruent.
N
M
Q
P
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Ex. 1 Developing Proof
Is it possible to prove the triangles are congruent? If so, state the postulate or theorem you would use. Explain your reasoning.
UZ ║WX AND UW
║WX.
U
W
Z
X
12
34
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Ex. 1 Developing Proof
The two pairs of parallel sides can be used to show 1 3 and 2 4. Because the included side WZ is congruent to itself, ∆WUZ ∆ZXW by the ASA Congruence Postulate.
U
W
Z
X
12
34
75
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Ex. 2 Proving Triangles are CongruentGiven: AD ║EC, BD
BC
Prove: ∆ABD ∆EBC
Plan for proof: Notice that ABD and EBC are congruent. You are given that BD BC
. Use the fact that AD ║EC to identify a pair of congruent angles.
B
A
ED
C
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Proof:
Statements:
1. BD BC
2. AD ║ EC
3. D C
4. ABD EBC
5. ∆ABD ∆EBC
Reasons:
1.
B
A
ED
C
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Proof:
Statements:
1. BD BC
2. AD ║ EC
3. D C
4. ABD EBC
5. ∆ABD ∆EBC
Reasons:
1. Given
B
A
ED
C
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Proof:
Statements:
1. BD BC
2. AD ║ EC
3. D C
4. ABD EBC
5. ∆ABD ∆EBC
Reasons:
1. Given
2. Given
B
A
ED
C
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Proof:
Statements:
1. BD BC
2. AD ║ EC
3. D C
4. ABD EBC
5. ∆ABD ∆EBC
Reasons:
1. Given
2. Given
3. Alternate Interior Angles
B
A
ED
C
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Proof:
Statements:
1. BD BC
2. AD ║ EC
3. D C
4. ABD EBC
5. ∆ABD ∆EBC
Reasons:
1. Given
2. Given
3. Alternate Interior Angles
4. Vertical Angles Theorem
B
A
ED
C
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Proof:
Statements:1. BD BC2. AD ║ EC3. D C4. ABD EBC5. ∆ABD ∆EBC
Reasons:1. Given2. Given3. Alternate Interior
Angles4. Vertical Angles
Theorem5. ASA Congruence
Theorem
B
A
ED
C
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Note:
• You can often use more than one method to prove a statement. In Example 2, you can use the parallel segments to show that D C and A E. Then you can use the AAS Congruence Theorem to prove that the triangles are congruent.
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85
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Using Congruent Triangles
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Objectives:
Use congruent triangles to plan and write proofs.
Use congruent triangles to prove constructions are valid.
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Planning a proof
Knowing that all pairs of corresponding parts of congruent triangles are congruent can help you reach conclusions about congruent figures.
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Planning a proof
For example, suppose you want to prove that PQS ≅ RQS in the diagram shown at the right. One way to do this is to show that ∆PQS ≅ ∆RQS by the SSS Congruence Postulate. Then you can use the fact that corresponding parts of congruent triangles are congruent to conclude that PQS ≅ RQS.
Q
S
P R
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Ex. 1: Planning & Writing a Proof
Given: AB ║ CD, BC ║ DA
Prove: AB≅CDPlan for proof: Show that ∆ABD ≅ ∆CDB. Then use the fact that corresponding parts of congruent triangles are congruent.
B
A
C
D
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Ex. 1: Planning & Writing a Proof
Solution: First copy the diagram and mark it with the given information. Then mark any additional information you can deduce. Because AB and CD are parallel segments intersected by a transversal, and BC and DA are parallel segments intersected by a transversal, you can deduce that two pairs of alternate interior angles are congruent.
B
A
C
D
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Ex. 1: Paragraph Proof
Because AD ║CD, it follows from the Alternate Interior Angles Theorem that ABD ≅CDB. For the same reason, ADB ≅CBD because BC║DA. By the Reflexive property of Congruence, BD ≅ BD. You can use the ASA Congruence Postulate to conclude that ∆ABD ≅ ∆CDB. Finally because corresponding parts of congruent triangles are congruent, it follows that AB ≅ CD.
B
A
C
D
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Ex. 2: Planning & Writing a Proof
Given: A is the midpoint of MT, A is the midpoint of SR.Prove: MS ║TR.Plan for proof: Prove that ∆MAS ≅ ∆TAR. Then use the fact that corresponding parts of congruent triangles are congruent to show that M ≅ T. Because these angles are formed by two segments intersected by a transversal, you can conclude that MS ║ TR.
A
M
T
R
S
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Given: A is the midpoint of MT, A is themidpoint of SR.Prove: MS ║TR.
Statements:1. A is the midpoint of MT,
A is the midpoint of SR.2. MA ≅ TA, SA ≅ RA3. MAS ≅ TAR4. ∆MAS ≅ ∆TAR5. M ≅ T6. MS ║ TR
Reasons:1. Given
A
M
T
R
S
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Given: A is the midpoint of MT, A is themidpoint of SR.Prove: MS ║TR.
Statements:1. A is the midpoint of MT,
A is the midpoint of SR.2. MA ≅ TA, SA ≅ RA3. MAS ≅ TAR4. ∆MAS ≅ ∆TAR5. M ≅ T6. MS ║ TR
Reasons:1. Given
2. Definition of a midpoint
A
M
T
R
S
97
Given: A is the midpoint of MT, A is themidpoint of SR.Prove: MS ║TR.
Statements:1. A is the midpoint of MT,
A is the midpoint of SR.2. MA ≅ TA, SA ≅ RA3. MAS ≅ TAR4. ∆MAS ≅ ∆TAR5. M ≅ T6. MS ║ TR
Reasons:1. Given
2. Definition of a midpoint3. Vertical Angles
Theorem
A
M
T
R
S
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Given: A is the midpoint of MT, A is themidpoint of SR.Prove: MS ║TR.
Statements:1. A is the midpoint of MT,
A is the midpoint of SR.2. MA ≅ TA, SA ≅ RA3. MAS ≅ TAR4. ∆MAS ≅ ∆TAR5. M ≅ T6. MS ║ TR
Reasons:1. Given
2. Definition of a midpoint3. Vertical Angles
Theorem4. SAS Congruence
Postulate
A
M
T
R
S
99
Given: A is the midpoint of MT, A is themidpoint of SR.Prove: MS ║TR.
Statements:1. A is the midpoint of MT,
A is the midpoint of SR.2. MA ≅ TA, SA ≅ RA3. MAS ≅ TAR4. ∆MAS ≅ ∆TAR5. M ≅ T6. MS ║ TR
Reasons:1. Given
2. Definition of a midpoint3. Vertical Angles
Theorem4. SAS Congruence
Postulate5. Corres. parts of ≅ ∆’s
are ≅
A
M
T
R
S
100
Given: A is the midpoint of MT, A is themidpoint of SR.Prove: MS ║TR.
Statements:1. A is the midpoint of MT,
A is the midpoint of SR.2. MA ≅ TA, SA ≅ RA3. MAS ≅ TAR4. ∆MAS ≅ ∆TAR5. M ≅ T6. MS ║ TR
Reasons:1. Given
2. Definition of a midpoint3. Vertical Angles Theorem4. SAS Congruence
Postulate5. Corres. parts of ≅ ∆’s
are ≅6. Alternate Interior Angles
Converse.
A
M
T
R
S
101
102
EC
D
B
A
Ex. 3: Using more than one pair of triangles.
Given: 1≅2, 3≅4.Prove ∆BCE≅∆DCEPlan for proof: The only information you have about ∆BCE and ∆DCE is that 1≅2 and that CE ≅CE. Notice, however, that sides BC and DC are also sides of ∆ABC and ∆ADC. If you can prove that ∆ABC≅∆ADC, you can use the fact that corresponding parts of congruent triangles are congruent to get a third piece of information about ∆BCE and ∆DCE.
21
43
103
Given: 1≅2, 3≅4.Prove ∆BCE ∆DCE≅
Statements:1. 1≅2, 3≅42. AC ≅ AC3. ∆ABC ≅ ∆ADC4. BC ≅ DC5. CE ≅ CE6. ∆BCE≅∆DCE
Reasons:1. Given
EC
D
B
A43
21
104
Given: 1≅2, 3≅4.Prove ∆BCE ∆DCE≅
Statements:1. 1≅2, 3≅42. AC ≅ AC
3. ∆ABC ≅ ∆ADC4. BC ≅ DC5. CE ≅ CE6. ∆BCE≅∆DCE
Reasons:1. Given2. Reflexive property
of Congruence
EC
D
B
A43
21
105
Given: 1≅2, 3≅4.Prove ∆BCE ∆DCE≅
Statements:1. 1≅2, 3≅42. AC ≅ AC
3. ∆ABC ≅ ∆ADC
4. BC ≅ DC5. CE ≅ CE6. ∆BCE≅∆DCE
Reasons:1. Given2. Reflexive property
of Congruence3. ASA Congruence
Postulate
EC
D
B
A43
21
106
Given: 1≅2, 3≅4.Prove ∆BCE ∆DCE≅
Statements:1. 1≅2, 3≅42. AC ≅ AC
3. ∆ABC ≅ ∆ADC
4. BC ≅ DC5. CE ≅ CE6. ∆BCE≅∆DCE
Reasons:1. Given2. Reflexive property
of Congruence3. ASA Congruence
Postulate4. Corres. parts of ≅ ∆’s
are ≅
EC
D
B
A43
21
107
Given: 1≅2, 3≅4.Prove ∆BCE ∆DCE≅
Statements:1. 1≅2, 3≅42. AC ≅ AC
3. ∆ABC ≅ ∆ADC
4. BC ≅ DC5. CE ≅ CE6. ∆BCE≅∆DCE
Reasons:1. Given2. Reflexive property
of Congruence3. ASA Congruence
Postulate4. Corres. parts of ≅ ∆’s
are ≅5. Reflexive Property of
Congruence
EC
D
B
A43
21
108
Given: 1≅2, 3≅4.Prove ∆BCE ∆DCE≅
Statements:1. 1≅2, 3≅42. AC ≅ AC
3. ∆ABC ≅ ∆ADC
4. BC ≅ DC5. CE ≅ CE6. ∆BCE≅∆DCE
Reasons:1. Given2. Reflexive property of
Congruence3. ASA Congruence
Postulate4. Corres. parts of ≅ ∆’s are
≅5. Reflexive Property of
Congruence6. SAS Congruence Postulate
EC
D
B
A43
21
109
Ex. 4: Proving constructions are valid
In Lesson 3.5 – you learned to copy an angle using a compass and a straight edge. The construction is summarized on pg. 159 and on pg. 231.
Using the construction summarized above, you can copy CAB to form FDE. Write a proof to verify the construction is valid.
110
Plan for proof
Show that ∆CAB ≅ ∆FDE. Then use the fact that corresponding parts of congruent triangles are congruent to conclude that CAB ≅ FDE. By construction, you can assume the following statements:– AB ≅ DE Same compass
setting is used– AC ≅ DF Same compass
setting is used– BC ≅ EF Same compass
setting is usedE
A
C
B
F
D
111
Given: AB DE, AC DF, BC EF ≅ ≅ ≅Prove CAB≅FDE
Statements:1. AB ≅ DE2. AC ≅ DF3. BC ≅ EF4. ∆CAB ≅ ∆FDE5. CAB ≅ FDE
Reasons:1. Given
43
21
E
A
C
B
F
D
112
Given: AB DE, AC DF, BC EF ≅ ≅ ≅Prove CAB≅FDE
Statements:1. AB ≅ DE2. AC ≅ DF3. BC ≅ EF4. ∆CAB ≅ ∆FDE5. CAB ≅ FDE
Reasons:1. Given2. Given
43
21
E
A
C
B
F
D
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Given: AB DE, AC DF, BC EF ≅ ≅ ≅Prove CAB≅FDE
Statements:1. AB ≅ DE2. AC ≅ DF3. BC ≅ EF4. ∆CAB ≅ ∆FDE5. CAB ≅ FDE
Reasons:1. Given2. Given3. Given
43
21
E
A
C
B
F
D
114
Given: AB DE, AC DF, BC EF ≅ ≅ ≅Prove CAB≅FDE
Statements:1. AB ≅ DE2. AC ≅ DF3. BC ≅ EF4. ∆CAB ≅ ∆FDE5. CAB ≅ FDE
Reasons:1. Given2. Given3. Given4. SSS Congruence
Post
43
21
E
A
C
B
F
D
115
Given: AB DE, AC DF, BC EF ≅ ≅ ≅Prove CAB≅FDE
Statements:1. AB ≅ DE2. AC ≅ DF3. BC ≅ EF4. ∆CAB ≅ ∆FDE5. CAB ≅ FDE
Reasons:1. Given2. Given3. Given4. SSS Congruence
Post5. Corres. parts of ≅
∆’s are ≅.
43
21
E
A
C
B
F
D
116
Given: QSRP, PT RT≅ Prove PS≅ RS
Statements:1. QS RP2. PT ≅ RT
Reasons:1. Given2. Given
43
21
TP
Q
R
S
117
118
119
120
121
Isosceles, Equilateral
and Right s
Isosceles, Equilateral
and Right sPg 236
123
Standards/Objectives:
Standard 2: Students will learn and apply geometric concepts
Objectives:• Use properties of Isosceles and
equilateral triangles.• Use properties of right triangles.
124
Isosceles triangle’s special parts
A is the vertex angle (opposite the base)
B and C are base angles (adjacent to the base)
A
BC
Leg
Leg
Base
125
Thm 4.6Base s thm
• If 2 sides of a are , the the s opposite
them are .( the base s of an isosceles are )
A
B C
If seg AB seg AC, then B C
) (
126
127
Thm 4.7Converse of Base s thm
• If 2 s of a are the sides opposite them are .
) (
A
B C
If B C, then seg AB seg AC
128
Corollary to the base s thm
• If a triangle is equilateral, then it is equiangular. A
B C
If seg AB seg BC seg CA, then A B C
129
Corollary to converse of the base angles thm
• If a triangle is equiangular, then it is also equilateral.
)
)
(
A
BC
If A B C, then seg AB seg BC seg CA
130
Example: find x and y
• X=60• Y=30
X Y120
131
132
133
Thm 4.8Hypotenuse-Leg (HL) thm
• If the hypotenuse and a leg of one right are to the hypotenuse and leg of another right , then the s are .
__
_ _
A
B C
X Y
Z
If seg AC seg XZ and seg BC seg YZ, then ABC XYZ
134
Given: D is the midpt of seg CE, BCD and FED are rt s
and seg BD seg FD.Prove: BCD FED
B
CD
F
E
135
ProofStatements1. D is the midpt of
seg CE, BCD
and <FED are rt s and seg BD to seg FD
2. Seg CD seg ED3. BCD FED
Reasons1. Given
2. Def of a midpt3. HL thm
136
Are the 2 triangles
)(
(
)
((
Yes, ASA or AAS
137
Find x and y.
75
x
x
y
2x + 75=180
2x=105
x=52.5 y=75
90
x
y60
x=60 y=30
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Find x.
)
)
(
))
((
56ft
8xft
56=8x
7=x
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Triangles and Coordinate Proof
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Objectives:
1. Place geometric figures in a coordinate plane.
2. Write a coordinate proof.
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Placing Figures in a Coordinate Plane
So far, you have studied two-column proofs, paragraph proofs, and flow proofs. A COORDINATE PROOF involves placing geometric figures in a coordinate plane. Then you can use the Distance Formula (no, you never get away from using this) and the Midpoint Formula, as well as postulate and theorems to prove statements about figures.
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Ex. 1: Placing a Rectangle in a Coordinate Plane
Place a 2-unit by 6-unit rectangle in a coordinate plane.
SOLUTION: Choose a placement that makes finding distance easy (along the origin) as seen to the right.
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Ex. 1: Placing a Rectangle in a Coordinate Plane
One vertex is at the origin, and three of the vertices have at least one coordinate that is 0.
6
4
2
-2
-4
-5 5
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Ex. 1: Placing a Rectangle in a Coordinate Plane
One side is centered at the origin, and the x-coordinates are opposites.
4
2
-2
-4
-6
-5 5 10
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Note:
Once a figure has been placed in a coordinate plane, you can use the Distance Formula or the Midpoint Formula to measure distances or locate points
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Ex. 2: Using the Distance Formula
A right triangle has legs of 5 units and 12 units. Place the triangle in a coordinate plane. Label the coordinates of the vertices and find the length of the hypotenuse.
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4
2
-2
-4
-6
-8
-10
5 10 15 20
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Ex. 2: Using the Distance Formula
One possible placement is shown. Notice that one leg is vertical and the other leg is horizontal, which assures that the legs meet as right angles. Points on the same vertical segment have the same x-coordinate, and points on the same horizontal segment have the same y-coordinate.
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5 10 15 20
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Ex. 2: Using the Distance Formula
You can use the Distance Formula to find the length of the hypotenuse.
d = √(x2 – x1)2 + (y2 – y1)2
= √(12-0)2 + (5-0)2
= √169
= 13
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2
-2
-4
-6
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-10
5 10 15 20
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Ex. 3 Using the Midpoint Formula
In the diagram, ∆MLN ≅∆KLN). Find the coordinates of point L.
Solution: Because the triangles are congruent, it follows that ML ≅ KL. So, point L must be the midpoint of MK. This means you can use the Midpoint Formula to find the coordinates of point L.
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140
120
100
80
60
40
20
-20
-40
-60
-80
-100
-120
-140
-160
-180
-50 50 100 150 200 250 300 350 400 450
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Ex. 3 Using the Midpoint Formula
L (x, y) = x1 + x2, y1 +y2
2 2 Midpoint Formula
=160+0 , 0+160
2 2 Substitute values
= (80, 80) Simplify.
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80
60
40
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-40
-60
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-160
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-50 50 100 150 200 250 300 350 400 450
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Writing Coordinate Proofs
Once a figure is placed in a coordinate plane, you may be able to prove statements about the figure.
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Ex. 4: Writing a Plan for a Coordinate Proof
Write a plan to prove that SQ bisects PSR. Given: Coordinates of vertices of ∆PQS and ∆RQS. Prove SQ bisects PSR. Plan for proof: Use the Distance Formula to find the
side lengths of ∆PQS and ∆RQS. Then use the SSS Congruence Postulate to show that ∆PQS ≅ ∆RQS. Finally, use the fact that corresponding parts of congruent triangles are congruent (CPCTC) to conclude that PSQ ≅RSQ, which implies that SQ bisects PSR.
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Ex. 4: Writing a Plan for a Coordinate Proof
Given: Coordinates of vertices of ∆PQS and ∆RQS.
Prove SQ bisects PSR.
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6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
-7
-5 5 10 15RQP
S
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NOTE:
The coordinate proof in Example 4 applies to a specific triangle. When you want to prove a statement about a more general set of figures, it is helpful to use variables as coordinates.
For instance, you can use variable coordinates to duplicate the proof in Example 4. Once this is done, you can conclude that SQ bisects PSR for any triangle whose coordinates fit the given pattern.
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No coordinates – just variables
x
(-h, 0) (0, 0) (h, 0)
(0, k)
y
S
P R
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Ex. 5: Using Variables as Coordinates Right ∆QBC has leg lengths
of h units and k units. You can find the coordinates of points B and C by considering how the triangle is placed in a coordinate plane.
Point B is h units horizontally from the origin (0, 0), so its coordinates are (h, 0). Point C is h units horizontally from the origin and k units vertically from the origin, so its coordinates are (h, k). You can use the Distance Formula to find the length of the hypotenuse QC.
k units
h units
C (h, k)
B (h, 0)Q (0, 0)
hypotenuse
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Ex. 5: Using Variables as Coordinates
OC = √(x2 – x1)2 + (y2 – y1)2
= √(h-0)2 + (k - 0)2
= √h2 + k2
k units
h units
C (h, k)
B (h, 0)Q (0, 0)
hypotenuse
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Ex. 5 Writing a Coordinate Proof
Given: Coordinates of figure OTUV
Prove ∆OUT ∆UVO Coordinate proof:
Segments OV and UT have the same length.
OV = √(h-0)2 + (0 - 0)2=h UT = √(m+h-m)2 + (k - k)2=h
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4
2
-2
-4
-6
-5 5O (0, 0)
T (m, k) U (m+h, k)
V (h, 0)
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Ex. 5 Writing a Coordinate Proof
6
4
2
-2
-4
-6
-5 5O (0, 0)
T (m, k) U (m+h, k)
V (h, 0)
Horizontal segments UT and OV each have a slope of 0, which implies they are parallel. Segment OU intersects UT and OV to form congruent alternate interior angles TUO and VOU. Because OU OU, you can apply the SAS Congruence Postulate to conclude that ∆OUT ∆UVO.
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