Download Definitions: Congruent triangle: all pairs of corresponding parts are

3.1 Notes
Wednesday, October 01, 2008
8:33 PM
Definitions:
1. Congruent triangle: all pairs of corresponding parts are congruent.
2. Congruent Polygons: all pairs of corresponding parts are congruent.
Postulate: Any segment or angle is congruent to itself. (Reflexive Property)
Notes Page 1
3.2 notes
Wednesday, October 01, 2008
8:43 PM
Four ways to prove triangles congruent
1. SSS Postulate: If there exists a correspondence between the
vertices of two triangles such that three sides of one triangle
are congruent to the corresponding sides of the other, the two
triangles are congruent. (SSS).
D
A
F
C
B
E
2. SAS Postulate: If there exists a correspondence between the
vertices of two triangles such that two sides and the included
angle of one triangle are congruent to the corresponding parts
of the other triangle, the two triangles are congruent. (SAS)
D
A
F
C
B
E
3. ASA Postulate: If there exists a correspondence between the
vertices of two triangles such that two angles and the included
side of one triangle are congruent to the corresponding parts of
the other triangle, the two triangles are congruent. (ASA)
D
A
F
C
B
E
Notes Page 2
4. AAS Postulate: If there exists a correspondence between the
vertices of two triangles such that two angles and the adjacent
side of one triangle are congruent to the corresponding parts of
the other triangle, the two triangles are congruent.
D
A
F
C
B
E
Example 1:
D
Given : AB  BC
ABD  CBD
Pr ove : ABD  CBD
A
C
B
Statements
1.
AB  BC
Reasons
1. Given
ABD  CBD
2. Reflexive property
2. DB  DB
3. ABD  CBD 3. SAS(1, 1, 2)
Example 2:
R
Given : 3  6
KR  PR
KRO  PRM
Pr ove : KRM  PRO
3
A
Notes Page 3
K
5
4
M
O
6
P
B
Example 2:
R
Given : 3  6
KR  PR
KRO  PRM
Pr ove : KRM  PRO
3
A
Statements
K
5
4
M
O
6
P
Reasons
1. Given
3  6
1. KR  PR
KRO  PRM
2. AKM and BPO are straight angles 2. Assumed from diagram
3. 3 is supp. to 4
3. Def. of supp. 's
5 is supp. to 6
4. 4 5
4. Supp. to  's
5. KRM PRO
5. Subtraction prop.
6. KRM  PRO
6. ASA(4, 1, 5)
Notes Page 4
B
3.3 notes
Monday, October 06, 2008
8:46 AM
CPCTC: Corresponding parts of congruent triangles are congruent.
Introduction to circles:
Point P is the center of the circle shown. By definition, every point of a
circle is the same distance from the center. The center, however, is not an
element of the circle; the circle consists only of the "rim". A circle is named
by its center: this circle is called circle P. Points A, B, and C lie on circle P.
A
P
B
C
Theorem 19: All radii of a circle are congruent.
Example 1:
T
Given :
O
T is comp. to MOT
S is comp. to POS
O
K
P
M
Pr ove : MO  PO
S
Statements
Reasons
O
1. Given
1. T is comp. to MOT
S is comp. to POS
2. OT  OS
2. Radii of a circle are 
3. MOT  POS
3. Vertical 's are 
T  S
Comp. to  's
Notes Page 5
R
4. T  S
4. Comp. to  's
MOT  POS
5. ASA(3, 2, 4)
6. MO  PO
6. CPCTC
Notes Page 6
3.4 Notes
Monday, October 06, 2008
9:17 AM
Beyond CPCTC
Median: is a line segment drawn from any vertex of the
triangle to the midpoint of the opposite side.
E
B
A
D
F
C
Altitude: is a line segment drawn from any vertex of the
triangle to the opposite side, extended if necessary, and
perpendicular to that side.
Case 1: Acute triangle
A
F
E
G
B
C
D
Case 2: Right triangle
EAF
G
B
C
D
Case 3: obtuse triangle
G
J
Notes Page 7
G
H
B
A
J
D
C
Postulate: Two points determine a line, ray, or segment.
Note: Many proofs involve lines, rays or segments that do not
appear in the original figure. These additions to the diagram
are called auxiliary lines.
Example 1:
E
Given : G is the midpo int of FH
EF  EH
Pr ove : 1  2
Statements
1.
G is the midpo int of FH
2
1
A
H
G
F
B
Reasons
1. Given
EF  EH
2. HG  GF
2. By def. of midpoint
3. AHG and GFB are straight 's 3. Assumed from diagram
4. EG  EG
4. Reflexive property
5. EGH  EGF
5. SSS(1, 2, 4)
6. EHG  EFG
6. CPCTC
7. EHG is supp. to 1
7. Def. of supp. 's
EFG is supp. to 2
8. 1  2
8. Supp. to  's
Notes Page 8
A
Example 2:
Given : CD and BE are altitudes of
D
ABC
E
AD  AE
Pr ove : DB  EC
B
Statements
1.
Reasons
ABC 1. Given
CD and BE are altitudes of
AD  AE
2. ADC and AEB are right 's
2. By def. of altitude
3. ADC  AEB
3. Right 's are 
4. A  A
4. Reflexive property
5. ADC  AEB
5. ASA(3, 1, 4)
6. AB  AC
6. CPCTC
7. DB  EC
7. Subtraction property
Notes Page 9
C
3.5 Notes
Monday, October 06, 2008
10:16 AM
Example 1:
A
Given : AC  AB
E
D
DC  EB
Pr ove : CE  BD
C
Statements
1.
AC  AB
B
Reasons
1. Given
DC  EB
2. AD  AE
2. Subtraction prop.
3. A  A
3. Reflexive prop.
4. AEC  ADB 4. SAS(1, 3, 2)
5. CE  BD
5. CPCTC
Try this one!!
Example 2:
N
Given : NR  NV
P and Q are midpo int s
R  V
Q
P
PX  QX
X
Pr ove : XS  XT
R
Notes Page 10
S
T
V
Statements
Reasons
1. Given
NR  NV
P and Q are midpo int s
1. R  V
PX  QX
2. NX  NX
2. Reflexive prop.
3. NP  NQ and PR  QV
3. Division prop.
4. NPX  NQX
4. SSS(1, 2, 3)
5. NPX  NQX
5. CPCTC
6. NPR and NQV are straight 's 6. Assumed from diagram
7. NPX is supp. to XPR
7. Def. of supp. 's
NQX is supp. to XQV
8. XPR  XQV
8. Supp. to  's
9. PRT  QVS
9. ASA(1, 3, 8)
10. QS  PT
10. CPCTC
11. XS  XT
11. Subtraction prop.
Notes Page 11
3.6 Notes
Monday, October 06, 2008
12:29 PM
Definitions:
1. A scalene triangle is a triangle in which no two sides are
congruent.
A
B
C
2. An isosceles triangle is a triangle in which at least two sides are
congruent.
A
B
C
• Segment AB and segment AC are called legs of the isosceles
triangle. Segment BC is called the base of the isosceles triangle.
B and C are called the base angles, and A is called the
vertex angle.
3. An equilateral triangle is a triangle in which all sides are
congruent.
A
B
C
Notes Page 12
A
B
C
4. An equiangular triangle is a triangle in which all angles are
congruent.
5. An acute triangle is a triangle in which all angles are acute.
A
B
C
6. A right triangle is a triangle in which one of the angles is a right
angle.
A
B
C
7. An obtuse triangle is a triangle in which one of the angles is an
obtuse angle.
A
110
B
C
• Draw and label a diagram. List, in terms of the diagram, what is
given and what is to be proved. Then write a two column
Notes Page 13
given and what is to be proved. Then write a two column
proof.
1. In an isosceles triangle, if the vertex angle is bisected, then two
congruent triangles are formed.
2. In an isosceles triangle; if a segment is drawn from the vertex
angle to the midpoint of the base, then congruent triangles are
formed.
1. Given: ABC is isosceles with base 's B
and C
AC is bisected
A
Prove: ADC  ADB
B
D
Statements
Reasons
1. ABC is isosceles base 's B and C 1. Given
AC is bisected
2. BAD  CAD
2. By def. of  bisector
3. AB  AC
3. By def. isos. 
4. AD  AD
4. Reflexive prop.
Notes Page 14
C
4.
5. SAS(3, 2, 4)
ADC  ADB
1. Given: ABC is isosceles base 's B and C
Point D is a midpoint of segment BC
Prove: ADC  ADB
A
B
Statements
1. ABC is isosceles base 's B and C
Point D is a midpoint of segment BC
D
Reasons
1. Given
2. BD  CD
2. By def. of midpoint
3. AB  AC
3. By def. isos. 
4. AD  AD
4. Reflexive prop.
ADC  ADB
5. SSS(2, 3, 4)
Notes Page 15
C
3.7 Notes
Friday, October 10, 2008
12:17 PM
Theorem 20: If two sides of a triangle are congruent, then the
angles opposite the sides are congruent.
Theorem 21: If two angles of a triangle are congruent, then the
sides opposite the angles are congruent.
Theorem: If two sides of a triangle are not congruent, then the
angles opposite them are not congruent, and the larger angle is
opposite the longer side.
Theorem: If two angles of a triangle are not congruent, then
the sides opposite them are not congruent, and the longer side
is opposite the larger angle.
Ways to prove that a triangle is isosceles.
1. If at least two sides of a triangle are congruent, then the
triangle is isosceles. (by definition of isosceles triangle)
2. If at least two angles of a triangle are congruent, then the
triangle is isosceles. (base angles are congruent)
Notes Page 16
T
Example 1:
given : 3  4
BX  AY
B
BW  AZ
1
Pr ove : WTZ is isosceles
W
Statements
3  4
3
X
Reasons
1. Given
1. BX  AY
BW  AZ
2. WBT and ZAT are straight
angles
3. 3 is supp. to 1
2. Assumed from the
diagram
3. By def. of supp. 's
4 is supp. to 2
4. 1  2
4. Supp. to 's
5.  BWX   AZY
5. SAS(1, 4, 1)
6.  W  Z
6. CPCTC
7.  WTZ is isos.
7. Base 's are 
Notes Page 17
A
4
2
Y
Z
3.8 Notes
Monday, October 13, 2008
7:19 AM
The HL Postulate: If there exists a correspondence between the
vertices of two right triangles such that the hypotenuse and a
leg of one triangle are congruent to the corresponding parts of
the other triangle, the two triangles are congruent.
B
C
A
E
F
D
Example 1:
Given : OF is an altitude of
O
EOG
O
Pr ove : EF  FG
E
Statements
1.
OF is an altitude of
EOG
F
G
Reasons
1. Given
O
2. OFE and OFG are rt.  's
2. By def. of altitude
3. OE  OG
3. All radii of a circle are 
4. OF  OF
4. Reflexive prop.
5. OEF  OGF
5. HL(2, 3, 4)
6. EF  FG
6. CPCTC
Notes Page 18
Notes Page 19