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Triangles
Name:
Congruent Triangles
Define

Congruent Polygons:

Congruent Triangles:

Included Angle:

Included Side:

Reflexive Property:
Naming Congruent Parts
1. If
sides.
, name the congruent angles and
2. If
sides.
, name the congruent angles and
Identifying Congruent Triangles
Complete the congruency statements.
3.
4.
GIH
Proving Triangle Congruency
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  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 s and the
included side of 1 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 s and the nonincluded side of 1 triangle are congruent to the corresponding parts of the other triangle, then 2
triangles are congruent.
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Triangles
Name:
Examples:
5. Use the tick marks for each pair of triangles, name the method, if any, that can be used to prove the
triangles are congruent.
a.
b.
c.
d.
6. 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
Congruency Proofs
7. Given: AD  CD, B is midpoint of AC
Prove: ∆ABD  ∆CBD
Statement
1.
2.
3. AB  CB
4.
5. ∆ABD  ∆CBD
Reason
1. Given
2. Given
3.
4. Reflexive Property
5.
8. Given: T is midpoint of CH and AO
Prove: ∆CAT  ∆HOT
Statement
1. T is midpoint of CH and AO
2.
3.
4. CTA  OTH
5. ∆CAT  ∆HOT
Reason
1. Given
2. If point is midpoint,  2 
3. If point is midpoint,  2 
4.
5.
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9. Given: C  H, T is the midpoint of AO
Prove: ∆CAT  ∆HOT
Statement
1. C  H
2. T is the midpoint of AO
3.
4.
5. ∆CAT  ∆HOT
Reason
1.
2.
3. If point is midpoint,  2 
4. Vertical  
5.
10. Given 3  6, KR  PR, KRO  PRM
Prove: ∆KRM  ∆PRO
Statement
1. 3  6
2. KR  PR
3. KRO  PRM
4.
5.
6.
7.
8. KRM  PRO
9. ∆KRM  ∆PRO
Reason
1.
2.
3.
4. AFD
5. if 2  form straight  then sup
6. if 2  form straight  then sup
7. sup of   are 
8.
9.
11. Given: AB  DC, AC  DB
Prove: ∆BAC  ∆CDB
Statement
1. AB  DC
2. AC  DB
3.
5. ∆BAC  ∆CDB
Reason
1.
2.
3. Reflexive
5.
Congruency in Right Triangles
HL Postulate:
 If the hypotenuse and the leg of 1 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.
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Examples:
12. Given: BC  AC, BD  AD, AC  AD
Prove: ∆ACB  ∆ADB
Statement
1.
2. BD  AD
3. AC  AD
4. C is right angle
5.
6. ∆ACB and ∆ADB are right triangles
7.
8. ∆ACB  ∆ADB
Reason
1. Given
2.
3.
4.
5. if  then form right angle
6.
7. Reflexive
8.
13. Given: JK  KM, ML || JK, JM  KL
Prove: ∆JKM  ∆LMK
Statement
1. JK  KM
2. ML || JK
3. JM  KL
4. JKM is right angle
5. JKM  LMK
6. JMK  LKM
7. ∆JKM and ∆LMK are right triangles
8.
9. ∆JKM  ∆LMK
Reason
1.
2.
3.
4.
5.
6.
7.
8. Reflexive
9.
14. Given: FGH is a right, JHG is right 
FG  JH
Prove: ∆FGH  ∆JHG
Statement
1.
2.
3.
4. ∆FGH and ∆JHG are right triangles
5.
6. ∆FGH  ∆JHG
Reason
1.
2.
3.
4.
5. Reflexive
6.
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Triangles
Name:
CPCTC
What is CPCTC?

Corresponding Parts of Congruent Triangles are Congruent
Why would this be true?
 Def of  ∆ says every pair of corresponding parts is  .
When is it used?
 Only after 2 ∆ have been proven or stated to be  . Cannot be used to prove ∆ 
Define

Altitude:

Auxillary Lines:

Bisect:

Median:
Key Concepts/Theorems
 All radii of a circle are congruent.
Examples:
15. Given: Circle P
Prove: AB  CD
Statement
1. Circle P
2. PC  PD  PB  PA
3.
4. ∆CPD  ∆BPA
5. AB  CD
16. Given: AC  AB, AE  AD
Prove: CE  BD
Statement
1.
2.
3. A  A
4. ∆AEC  ∆ADB
5. CE  BD
17. Given: Circle O,
T is comp to MOT,
S is comp to POS
Prove: MO  PO
Statement
1.
2.
3.
4.
5. MOT  POS
6.
7. ∆MOT  ∆POS
8. MO  PO
5
Reason
1.
2.
3. Vertical angles 
4.
5.
Reason
1. Given
2. Given
3.
4.
5.
Reason
1. Given
2. Given
3. Given
4. All radii 
5.
6. Complements of
7.
8.
  are 
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Triangles
Name:
18. Given: AC  BC, AD  BD
Prove: CD bisects ACB
Statement
1. AC  BC
2. AD  BD
3. CD  CD
4.
5. ACD  BCD
6. CD bisects ACB
19. Given: CD & BE are altitudes of
∆ABC, AD  AE
Prove: DB  EC
Statement
1. CD and BE are
altitudes of ∆ABC
2. AD  AE
3.
Reason
1.
2.
3.
4. SSS
5.
6.
4.
5.
6. A  A
7. ∆AEB  ∆ADC
8. AB  AC
9.
Reason
1. Given
2.
3. if a line is an altitude, it is  and
forms right angles.
4. if a line is an altitude, it is  and
forms right angles
5. All right angles 
6.
7.
8.
9. Subtraction Property
Right Angle Theorem
Right-Angle Theorem

If 2 angles are both supplementary and  , then they are right angles.
Examples:
20. Given: AB  AC, BD  CD
Prove: AD is an altitude
Statement
1. AB  AC
2. BD  CD
3.
4. ∆ABD  ∆ACD
5. ADB  ADC
6. ADB and ADC are right angles
7.
8. AD is an altitude
Reason
1.
2.
3. Reflexive
4.
5.
6.
7. if form right angles, then 
8.
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Triangles
Name:
Isosceles Triangles
Angle-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  , then the sides opposite those angles 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.
Consequences of Angle-Side Theorems:
 Is an equiangular triangle also equilateral? Why or why not?
Examples:
A
21. Given: AC > AB, mB + mC < 180, mB = 6x-45, mC=15+x
What are the restrictions on the value of x?
B
22. Given: E  H, EF  GH
Prove: DF  DG
Statement
1. E  H
2. EF  GH
3.
4.
5. DF  DG
C
Reason
1.
2.
3. if angles then sides
4. SAS
5.
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  bis of AC
 POPBTEE: if a point is on the perpendicular bisector of a segment, then it is equidistant from the
endpoints of that segment.
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23. Given: PQ is ┴ bisector of AB
Prove: PA  PB
Statement
1. PQ is ┴ bisector of AB
2. PQA and PQB are right angles
3.
4.
5. PQ  PQ
6. ∆PQA  ∆PQB
7. PA  PB
Reason
1.
2.
3. All right angles are 
4. if bisect then divide into 2  parts
5.
6.
7.
Examples:
24. Given: 1  2, 3  4
Prove: AE bisector BD
Statement
1. 1  2
2. AB  AD
3.
4.
5. AE bisector BD
Reason
1.
2.
3. Given
4. if angles then sides
5.
25. Prove: The line joining the vertex of isosceles triangle to
midpoint of base is perpendicular to base.
Given: ∆PIE is isosceles, S is midpoint of PE
Prove: IS  PE
Statement
1. ∆PIE is isosceles
2. PI  IE
3.
4. PS  SE
5. IS  PE
Reason
1.
2.
3. Given
4.
5.
26. Given: AB  AD, BC  CD.
Prove: BE  ED
Statement
1. AB  AD
2.
3. AE  bis BD
4.
Reason
1.
2. Given
3.
4.
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Name:
Indirect Proofs
Define:

Indirect Proofs: method of proving where direct proof would be difficult to apply.
Procedures for Indirect Proof
1. List the possibilities for the conclusion.
a. Your conclusion is or is not true.
2. Assume that the negation of the desired conclusion is true.
a. So the OPPOSITE of the conclusion
3. Write a “chain of reasons” until you reach an IMPOSSIBILITY or a CONTRADICTION.
a. This will be a statement that either disputes a known theorem/definition/postulate 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
1. Given A  D, AB  DE, AC  DF
Prove: B  E.
Proof:
Either: _____B  E ______________ or _______B  E __________________
Assume:_____ B  E ________________________
From given information, ___A  D, AB  DE, _________________________
__thus ∆ABC  ∆DEF by ASA and therefore AC  DF _____________________
Not possible since _given that AC  DF ____therefore, assumption of ___B  E _ is false
and __B  E _________ is true.
2. Given: RS  PQ, PR ≠ QR
Prove: RS does not bisect PRQ
Proof:
Either: ________________________ or ________________________
Assume: ________________________
From given information, ___________________________________________________________
_______________________________________________________________________________
Not possible since __________________therefore, assumption of __________________ is false and
________________________ is true
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