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Chapter 4 Notes
Mrs. Myers – Geometry
Name ______________________________
Period ______
4.1 Triangles and Angles

Triangle = is a figure formed by ____ segments joining three noncollinear points.
o A triangle can be classified by its sides and by its angles.

Classification by Sides:

Classification by Angles:
Ex. 1 Classifying Triangles
A)
B)
60
51
60
49
60
60

Vertex = all three _____________ joining the sides of a triangle.

Adjacent Sides = in a triangle, two sides sharing a common _______________.
* Right Triangle:
* Isosceles Triangle:
* Leg = 2 sides that form the __________ angle
* Leg = 2 _____ sides
* Hypotenuse = ______________ side of the right angle
* Base = 3rd side
Ex. 2
K
10
6
L
8
J
A) Explain why this is a scalene right triangle.
B) Why is there no base in the triangle?

Interior Angles: the three original angles (_______________ the triangle).

Exterior Angles: the angles that are adjacent to the interior angles (___________
the triangle) – when the sides are extended.
* Theorem 4.1: Triangle Sum Theorem = the sum of the measures of the interior angles
of a triangle is ___________.
B
A
C
m A  m B  m C  _________
* Theorem 4.2: Exterior Angle Theorem = the measure of an exterior angle of a
triangle is equal to the _________ of the measure of the two nonadjacent interior angles.
m 1  ____________________
Ex. 3 Find the value of x. Then find the measure of the exterior angle.
A)
x
 2x - 11 
72
B)
110
x
 4x - 7
* Corollary to the Triangle Sum Theorem: the acute angles of a right triangle are
_____________________________.
m A  m B  ________
Ex. 4 The measure of one acute angle of a right triangle is five times the measure of the
other acute angle. Find the measure of each acute angle.
4.2 Congruence and Triangles

Two geometric figures are congruent if they have exactly the same size and shape.
o Corresponding Angles = the angles in the same position are __________.
o Corresponding Sides = the sides in the same position are ___________.
Ex. 1
XYZ  HKJ
Corresponding Angles
Corresponding Sides
1.
1.
2.
2.
3.
3.
Ex. 2 In the diagram, ABCD  KJHL . Find the value of x and y.
9 cm
A
B
H
L
91
86
D
 5y - 12
113
6 cm
C
 4x - 3 cm
K
J
* Theorem 4.3: Third Angles Theorem = if two angles of one triangle are congruent to
two angles of another triangle, then the third angles are also __________________.
If
A
D and
B
E , then ___________
Ex. 3 Find the value of x.
Ex. 4 Decide whether the triangles are congruent. Justify your reasoning.
* Theorem 4.4: Properties of Congruent Triangles =
* Reflexive Prop. of  Triangles: every triangle is congruent to itself.
* Symmetric Prop. of  Triangles: if ABC  DEF , then DEF  ABC .
* Transitive Prop. of  Triangles:
if ABC  DEF and DEF  JKL, then ABC  JKL
Ex. 5
MN  QP
Given: MN PQ
O is the midpo int of MQ and PN
Prove:
MNO  QPO
Statements
Reasons
1. MN  QP
1. Given
2. MN
2. Given
PQ
3.
1
2
3.
4.
3
4
4.
5.
5
6
5.
6. O is the midpo int of MQ and PN
6. Given
7. MO  QO & NO  PO
7.
8.
MNO  QPO
8.
4.3 Proving Triangles are Congruent: SSS and SAS
* Postulate 19: Side-Side-Side (SSS) Congruence Postulates = if _________ sides of
one triangle are congruent to ___________ sides of a second triangle, then the two
triangles are congruent.
If MN  QR , NP  RS , and PM  SQ , then ____________________
* Postulate 20: Side-Angle-Side (SAS) Congruence Postulate = if ________ sides and
the angle inbetween of one triangle are congruent to ________ sides and the angle
inbetween of a second triangle, then the two triangles are _________________.
If PQ  WX ,
Q
X and QS  XY , then ______________________
Ex. 1 Prove KLM  NMP
Statements
Reasons
1. KL  NM
1. Given
2. LM  MP
2. Given
3. KM  NP
3. Given
4.
KLM  NMP
4.
Ex. 2 Prove VWZ  YWX
Statements
Reasons
1. VW  YW
1. Given
2. ZW  XW
3. 1  2
4. VWZ  YWX
2. Given
3. Vertical angles theorem
4.
Ex. 3 Is enough information given to prove that LMP  NPM
Ex. 4
Given:
AB  PB
MB  AP
Prove: MBA  MBP
Statements
Reasons
1. AB  PB
1. Given
2. MB  MB
2.
3. MB  AP
3. Given
4.
1&
2 are right angles
5.
6.
4.
5. Right angles theorem
MBA  MBP
6.
4.4 Proving Triangles are Congruent: ASA and AAS
* Postulate 21: Angle-Side-Angle (ASA) Congruence Postulates = if ________ angles
and the side inbetween of one triangle are congruent to _________ angles and the side
inbetween of a second triangle, then the two triangles are ______________.
If
A
D , AC  DF and
C
F , then _________________________
* Theorem 4.5: Angle-Angle-Side (AAS) Congruence Theorem = if ________ angles
and a nonincluding ____________ are congruent to two angles and the corresponding
nonincluded side of another triangle, then the two triangles are ___________________.
If
A
D,
C
F and BC  EF , then _____________________
Ex. 1 Is it possible to prove the triangles are congruent? If so, state the postulate or
theorem you would use.
A)
B)
C)
Ex. 2
Given:
B
C,
D
F
M is the midpo int of DF
Prove: BDM  CFM
B
C
D
1.
B
Statements
C, D  F
F
Reasons
1. Given
M is the midpo int of DF
2. DM  FM
3.
M
2.
BDM  CFM
3.
Ex. 3
Given:
AD
EC
BD  BC
Prove: ABD  EBC
C
A
B
D
Statements
1.
AD
EC
Reasons
1. Given
BD  BC
2.
D
3.
1
2
3.
4.
ABD  EBC
4.
C
E
2.
4.5 Using Congruent Triangles
CPCTC
 
Corresponding
Parts
Of
Congruent
Triangles
Are
Congruent
* Recall:
if  : then
* all 3 corresponding sides are congruent
* all 3 corresponding angles are congruent
* If you want to prove that one part of 2 triangles are congruent (angles or sides), then
you prove that 2 triangles are ______________ and therefore by _________________,
the parts are ________________.
Ex. 1
Given:
MS TR
MS  TR
Prove: A is the midpo int of MT
Statements
1.
Reasons
1. Given
MS TR
MS  TR
2. S  R
3.
M 
4.
MAS  TAR
T
2.
3.
4.
5. MA  TA
5.
6. A is the midpo int of MT
6. Definition of midpoint
Ex. 2
A is the midpo int of MT
Given: A is the midpo int of SR
Prove: MS TR
Statements
1. A is the midpo int of MT
1. Given
Reasons
2.
2. Definition of midpoint
3. A is the midpo int of SR
3. Given
4. SA  RA
4.
5.
1
6.
MAS  TAR
6.
7.
S 
7.
2
R
8. MS TR
5. Vertical angles theorem
8.
4.6 Isosceles, Equilateral, and Right Triangles

Isosceles Triangles:
* Theorem 4.6: Base Angles Theorem = if two ___________ of a triangle are
congruent, then the ______________ opposite them are _______________.
If BA  CA , then ___________________
* Theorem 4.7: Converse of the Base Angles Theorem = if two ___________ of a
triangle are congruent, then the ____________ opposite them are __________________.
If

B
C , then _________________
Equilateral Triangle:
* Corollary to Theorem 4.6: if a triangle is equilateral, then it is __________________.
* Corollary to Theorem 4.7: if a triangle is equiangular, then it is _________________.
* Theorem 4.8: Hypotenuse-Leg (HL) Congruence Theorem = if the hypotenuse and a
leg of a _____________ triangle are ____________ to the hypotenuse and a leg of a
second right triangle, then the two triangles are ________________.
Ex. 1 Find the values of x and y.
A)
B)
50
Y
X