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Triangles
Congruent Triangles
Define
• Congruent Polygons:
•
Congruent Triangles:
•
Reflection:
•
Reflexive Property:
•
Rotate:
•
Slide:
Naming Congruent Parts
1. If IiHIJ L,.PQR, name the congruent angles and sides.
Q
if
I
2. If tDEF
AABC, name the congruent angles and sides.
R
D
Identifying Congruent Triangles
Complete the congruency statements.
3.
A
4.
a
D
H
Z
L
?
D
L\GIHL
&‘DE
Transformations
Movement
Flip
Slide
Turn
?
Name
Reflection
Translation
Rotation
Reflection: When done on the y-axis, the x-sign changes. When done on the x-axis, the y-sign changes.
Translation: can slide up or down, right or left or diagonally.
• To right: Add number of units to x value
• To left: Subtract number of units from x value
• Up: Add number of units to the y value
J -. Down: Subtract number of units from y value
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Triangles
Proving Triangle Congruency
Define
•
•
Included Angle:
Included Side:
Postulates:
•
Postulate A: SSS
If there exists a correspondence between the vertices of 2 triangles such that 3 sides of 1
triangle are congruent to 3 sides of other triangle, then the 2 triangles are congruent.
•
Postulate B: SAS
If there exists a correspondence between the vertices of 2 triangles such that 2 sides and the
included Z of 1 triangle are congruent to corresponding parts of the other triangle, then 2
triangles are congruent.
•
Postulate C: ASA
If there exists a correspondence between the vertices of 2 triangles such that 2 Zs and the
included side of I triangle are congruent to the corresponding parts of the other triangle, then 2
triangles are congruent.
•
Postulate D: AAS
If there exists a correspondence between the vertices of 2 triangles such that 2 Zs and the nonincluded side of 1 triangle are congruent to the corresponding parts of the other triangle, then 2
triangles are congruent.
Examples:
1. Use the tick marks for each pair of triangles, name the method, if any, that can be used to prove the
triangles are congruent.
C.
b.
d.
a.
g
U
2. Name the additional congruent parts needed so that the triangles are congruent by the specified
method.
b. by AAS
a. by SSS
c. by ASA
y
E
A
Z
2
F
A
C
R
T
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Triangles
Congruency Proofs:
1. Given: AD
CD, B is midpoint of AC
Prove: tXABD ACBD
Reason
1. Given
2. Given
3.
4. Reflexive Property
5.
Statement
1.
3.ABECB
4.
5.AABDACBD
2. Given: T is midpoint of CH, AO
Prove: ACAT
AHOT
Reason
1. Given
2. If point is midpoint, ÷ 2
3.Ifpointismidpoint, ÷2
4.
5.
Statement
1. T is midpoint of CH, AO
2.
3.
4.ZCTAZOTH
AHOT
5. ACAT
ZH, T is the midpoint of AO
3. Given: rc
Prove: ACAT
AHOT
Reason
1.
2.
3. If point is midpoint, ÷ 2
4.Vertica1/
5.
Statement
1.ZCZH
2. T is the midpoint of AO
3.
4.
5.ACATAHOT
4. Given /3 /6, KR PR, ZKRO
Prove: AKRM APRO
/PRM
Reason
1.
2.
3.
4.AFD
5. if 2 / form straight / then sup
6. if 2 / form straight / then sup
7. sup ofE / are
8.
9.
Statement
1./3Z6
2.KRPR
3.ZKRO/PRM
4.
5.
6.
7.
8./KRM/PRO
9. AKRM APRO
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Triangles
Congruency in Right Triangles
ilL Postulate:
• If the hypotenuse and the leg of I right triangle are congruent to the hypotenuse and corresponding leg
of another right triangle, then triangles are congruent.
HA Theorem:
• If the hypotenuse and an acute angle of 1 right triangle are congruent to the hypotenuse and
corresponding acute angle of another right triangle, then triangles are congruent.
LA Theorem:
• If 1 leg and an acute angle of 1 right triangle are congruent to corresponding leg and acute angle of
another right triangle, then triangles are congruent.
LL Theorem:
• If legs of 1 right triangle are congruent to corresponding legs of another right triangle, then triangles
are congruent.
Examples:
5. Given:BC±AC,BD±AD,AC
Prove: AACB AADB
AD
Reason
1. Given
2.
3.
4.
5.Defofl
6.
7. Reflexive
8.
Statement
1.
2.BDIAD
3.ACAD
4. ZCisrightangle
5.
6. AACB and AADB are right triangles
7.
8.AACBAADB
6. Given:JK±KM,MLIIJK,JM KL
Prove: AJKM ALMK
Reason
1.
2.
3.
4.
5.
6.
7.
8. Reflexive
9.
Statement
1.JK±KM
2.MLIIJK
3.JMKL
4. LJKM is right angle
5. ZJKM ZLMK
6. ZJMK ZLKM
7. AJKM and ALMK are right triangles
8.
9. AJKM ALMK
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Triangles
CPCTC
What is CPCTC?
•
Corresponding Parts of Congruent Triangles are Congruent
Why would this be true?
• Def of A says every pair of corresponding parts is
When is it used?
• Only after 2 A have been proven or stated to be
.
Cannot be used to prove A
Define
•
Altitude:
•
Auxiliary Lines:
•
Bisect:
•
Median:
Key Concepts/Theorems
• All radii of a circle are congruent.
Examples:
7. Given: Circle P
Prove: AB CD
Statement
l.CircleP
2,PCPDPBPA
3.
4.ACPDABPA
5.ABCD
Reason
1.
2.
3. Vertical angles
4.
5.
8. Given:AC AB,AEAD
Prove: CE
BD
Statement
1.
2.
3.ZAXA
4.AAECAADB
5.CEBD
Reason
1. Given
2. Given
3.
4.
5.
CB
9. Given: Circle 0,
ZT is comp to ZMOT,
ZS is comp to ZPOS
Prove: MO
P0
Statement
1.
2.
3.
4.
5.ZMOTZPOS
6.
APOS
7. AMOT
8,MOPO
5
Reason
1. Given
2. Given
3. Given
4. All radii
5.
6. Complements of
7.
8.
Z are
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Triangles
10. Given: AC BC, AD
BD
Prove: CD bisects /ACB
Statement
l.ACBC
2.ADBD
3.CDCD
4.
5./ACDZBCD
6. CD bisects ZACB
AAB
11. Given: CD & BE are altitudes of
AABC,AD AE
Prove: DB
EC
Statement
1. CD and BE are
altitudes_of AABC
2.ADAE
3.
4.
5.
6.rAzA
7. AAEB
AADC
8.ABAC
9.
Reason
1.
2.
3.
4. SSS
5.
6.
Reason
1. Given
2.
3. if a line is an altitude, it is I and
forms right angles.
4. if a line is an altitude, it is I and
forms right angles
5. All right angles
6.
7.
8.
9. Subtraction Property
Isosceles Triangles
An1e-Side Theorems
• If sides then angles: If 2 sides of a triangle are ,then the angles opposite those sides are
•
If angles then sides: If 2 angles of a triangle are
•
If 2 sides of a triangle are not congruent, then the angles opposite those sides are not congruent and the
larger angle is opposite the longer side.
•
If 2 angles of a triangle are not congruent, then the sides opposite those angles are not congruent and
the longer side is opposite the larger angle.
,
then the sides opposite those angles are
Consequences of Angle-Side Theorems:
• Is an equiangular triangle also equilateral? Why or why not?
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Triangles
Examples:
12. Given: AC > AB, mZB + mZC < 180, mLB
What are the restrictions on the value of x?
=
6x-45, mLC=15+x
15
Statement
1.ZEZH
2.EFGH
3.
4.
5.DFDG
13. Given: ZE ZH, EF GH
DG
Prove: DF
Er
U
Reason
1.
2.
3. if angles then sides
4. SAS
5.
Right Angle Theorem
Right-Ane Theorem
If 2 angles are both supplementary and
,
then they are right angles.
Examples:
14. Given: AB AC, BD CD
Prove: AD is an altitude
Statement
1. ABAC
2.BDCD
3.
4.txABD AACD
5.ZADB ZADC
6. XADB and ZADC are right angles
7.
8. AD is an altitude
Reason
2.
3. Reflexive
4.
5.
6.
7. if form right angles, then I
8.
Equidistance Theorems
Define:
•
Distance:
•
Equidistant:
•
Perpendicular Bisector:
Theorems/Postulates:
•
TPEEEDPB: if 2 points are each equidistant from the endpoints of a segment, then the 2 points
determine the perpendicular bisector of that segment
BD is I bis of AC
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Triangles
•
POPBTEE: if a point is on the perpendicular bisector of a segment, then it is equidistant from the
endpoints of that segment.
15. Given: PQ is bisector of AB
Prove: PAPB
Statement
1.PQis±bisectorofAB
2. /PQA and ZPQB are right angles
3.
4.
5. PQ PQ
6. APQA APQB
7.PAPB
Reason
1.
2.
3. All right angles are
4. Definition of bisect
5.
6.
7.
Examples:
16. Given: Z1
/2, /3
Z4
Prove: AE Ibisector BD
Statement
1./lZ2
2. AB AD
3.
4.
5. AE ±bisector BD
3. Given
4. if angles then sides
5.
17. Prove: The line joining the vertex of isosceles triangle to
midpoint of base is perpendicular to base.
Given: APIE is isosceles, S is midpoint of PE
Prove: IS I PE
Statement
1. APTE is isosceles
2.PIIE
3.
4.PSSE
5. ISIPE
Reason
1.
2.
3. Given
4.
5.
Reason
1.
18. Given: AB AD, BCCD.
Prove: BE ED
Statement
1.ABAD
Reason
1.
2. Given
3.
4.
3.AEIbisBD
4.
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_________________________
_____________________________
______________________
_________________therefore,
___________________________
________________________________
______LB
________________
____________
____________________
_______________
Triangles
Indirect Proofs
Define:
•
Indirect Proofs: method of proving where direct proof would be difficult to apply.
Procedures for Indirect Proof
1.
2.
3.
List the possibilities for the conclusion.
a. Your conclusion is or is not true.
Assume that the negation of the desired conclusion is true.
a. So the OPPOSITE of the conclusion
Write a “chain of reasons” until you reach an IMPOSSIBILITY or a CONTRADICTION.
a. This will be a statement that either disputes a known theoremldefinitionlpostulate or your given
information.
4. State that what you assumed to start was WRONG and that the desired conclusion then must be true.
Examples: Paragraph Proofs
I.
Given LA LD, AB
Prove: LB LE.
DE, AC
DF
Proof:
Either:
ZB
Assume:_____ LB
Since,
thus
LA
AABC
ZE
or
ZE
ZE
Li), A13
DL
ADEF by ASA and therefore AC
Not possible since contradicts given that AC
Therefore, assumption of/B
LE
—
DF
1W
is false and
LB
LE
is true
2. Given: RS L PQ, PR QR
Prove: RS does not bisect LPRQ
Proof:
Either:
or
Assume:
Since,
Not possible since contradicts
and
is true
assumption of
9
is false
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