Download 4-2 Angles in a Triangle

4-2 Angles in a Triangle
Mr. Dorn
Chapter 4
4-2 Angles in a Triangle
Angle Sum Theorem:
The sum of the angles in a triangle is 180.
y
x + y + z = 180
x
z
4-2 Angles in a Triangle
Third Angle Theorem:
If two angles of a triangle are congruent to two
angles of another triangle, then the third angles are
congruent.
A
Y
Given:
A  Y
B  Z
C
Conclusion:
X  C
X
Z
B
4-2 Angles in a Triangle
Exterior Angle Theorem:
The measure of an exterior angle of a triangle is
equal to the sum of the measures of the two remote
interior angles.
y
m=x+y
x
z
m
4-3 Congruent
Triangles
4-3 Congruent Triangles
CPCTC:
Corresponding Parts of Congruent
Triangles are Congruent.
A
X
ABC  XYZ
B
C
Y
Z
Example 1
Complete the sentence.
LMN    
Q
QPR
P
N
L
M
R
Example 2
Given: CA = 14, AT = 18, TC = 21, and DG
= 2x + 7
CAT  DOG
Find x.
A
14
21 = 2x + 7
14 = 2x
7=x
O
18
C
T
21
G
D
2x + 7
Example 3
Given: AC = 7, BC = 10, DF = 2x + 4, and
DE = 4x.
ABC  DEF
Find x and AB.
B
7 = 2x + 4
3 = 2x
1.5 = x
E
10
A
4x
C
7
AB = 4x
AB = 4(1.5)
AB = 6
F
D
2x + 4
4-4 Proving Triangles
Congruent
4-4 Proving Congruent Triangles
N
SSS Postulate:
L
(Side-Side-Side)
Q
If the sides of one triangle are
congruent to the sides of a
second triangle, then the
triangles are congruent.
M
P
R
4-4 Proving Congruent Triangles
A
SAS Postulate:
(Side-Angle-Side)
If two sides of one triangle
and the included angle
are congruent to two
sides and the included
angle of a second
triangle, then the
triangles are congruent.
Y
B
C
X
Z
4-4 Proving Congruent Triangles
A
ASA Postulate:
(Angle-Side-Angle)
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
X
Y
Z
4-4 Proving Congruent Triangles
A
AAS Postulate:
(Angle-Angle-Side)
If two angles and a
nonincluded side of one
triangle are congruent to
two angles and a
nonincluded side of a
second triangle, then the
triangles are congruent.
B
C
X
Y
Z
4-4 Proving Congruent Triangles
Four Postulates that will work for any
type of triangle:
SSS, SAS, ASA, AAS
*Just remember: Any Combination
works as longs as it does not spell a
bad word forward or back.
Example 1
Find x.
Use the Angle Sum Theorem!
3x
3x + x +2x = 180
2x
x
6x = 180
x = 30
Example 2
Find x.
Use the Exterior Angle Theorem!
2x
(103-x)
(103 – x) +2x = 6x - 7
103 + x = 6x - 7
103 = 5x - 7
110 = 5x
(6x-7)
x = 22
Example 3
Find x.
Use the Angle Sum Theorem!
y + 53 + 80 = 180
x = 65
y + 133 = 180
y = 47
50
53
n + 50 + 62 = 180
n + 112 = 180
x
n = 68
Vertical angles are congruent, so…
62
47
y 68
n
80
x + 47 + 68 = 180
x + 115 = 180
Example 4
Are the two triangles congruent? If so,
what postulate identifies the two triangles
as congruent?
Yes, SAS
Example 5
Are the two triangles congruent? If so,
what Postulate identifies the two triangles
as congruent?
Not
Congruent!
Example 6
R
Given: RQ  RT ; S is the midpoint of
Prove: QRS  TRS
QT
Q
S
Statements
Reasons
1. RQ  RT ; S is the midpoint of QT
1. Given
2.
QS  ST
2. Midpoint Theorem
3.
RS 
3. Reflexive Property
RS
4. QRS  TRS
4. SSS
T
Example 7
Given: AD bisects BAC and BDC.
Prove:
BAD  CAD
B
A
Proof:
D
C
Since AD bisects BAC and BDC, BAD  CAD and BDA  CDA.
AD  AD by the Reflexive Property.
By ASA, BAD  CAD.