Download Chapter 3.7 Angle

Chapter 3.7
Angle-Side Theorems.
Erin Sanderson Mod 9.
Objective.
• This section will teach you how to apply
theorems relating to the angle measure
and side lengths of triangles.
Triangle =D
Theorem 20.
• If two sides of a triangle are congruent, the
angles opposite the sides are congruent.
• (If  , then  .)
But Why?
A
Statement.
Reason.
1. AB  AC
1. Given
2. A
A 2. Reflexive
Property
3. ABC  ACB
3. SAS
4. B  C
(1,2,1)
4. CPCTC
B
C
Given: AB  AC
Prove:B  C
Theorem 21; the Reverse.
• If two angles of a triangle are congruent,
the sides opposite the angles are
congruent.
• (if  , then  .)
How Come?
Statement.
1. M  E
2.
ME  ME
G
Reason.
1.
2.
3. MEG  EMG
4. MG  EG 3.
4.
Given
Reflexive
Property
ASA (1,2,1)
CPCTC
E
M
Given: M  E
Prove:
DF  EF
How Do I know if a
Isosceles?

is
1. If at least two sides of a triangle are
congruent, the triangle is isosceles.
2. If at least two angles of a triangle are
congruent, the triangle is isosceles.
The Inverses Also Work...
• 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.
• 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.
Basically;
• This means that the
longest side is across
from the largest angle
and the shortest side
is across from the
smallest angle.
It Would Kind of Look Like...
LARGER
S
M
A
L
L
E
R
S
H
O
R
T
E
R
LONGER
That.
This means...
• Equilateral triangles are also equiangular
because all of the sides are congruent,
thus all of the angles are congruent.
Sample Problems.
Statement.
1.
2.
ACDE is a
square.
B bisects
Reason.
1.
2.
3.
AC
3. AE  CD
4. AB  BC
5. A  C
6. AEB  CDB
7. BE  BD
8. BED  BDE
4.
5.
6.
7.
8.
Given.
Given
All sides of a
square are
cong.
If a line is
bisected, it is
divided into 2
cong. lines
All angles of a
square are
cong.
SAS (3,4,5)
CPCTC
If sides, then
angles
A
B
E
C
D
Given: ACDE is a square.
B bisects AC .
Prove: BED  BDE
#2
B
C
x+40
9x-72
Given: Angle measures
as shown; ABC is
isosceles.
Since you know that
B
C, you can say that
x+40=9x-72
8x=112
x=14
Find: The measure of
angle A.
Then, you can substitute
14 in for the x in A.
6(14)-12
The answer is 72.
A
6x-12
Now, do some on your own.
U
Q
4
3
R
2
1
S
Given: QR
Prove:
T
ST; UR
QUS
US
TUR
E
G
D
F
Given:
F; GE
Prove: EF bisects
GFD
ED
Answers.
Statement.
1.
QR ST;
UR US
2.
QS RT
3.
3
2
4.
QUS
TUR
Reason.
1.
Given
2.
Addition
3.
If sides, then
angles
4.
SAS (1,2,3)
And another…
Statement.
1.
2.
3.
4.
5.
6.
F; GE
ED
GF FD
EGF
EDF
EGF
EDF
GFE
DFE
EF bisects
GFD
Reason
1.
2.
3.
4.
5.
6.
Given
Radii of a
circle are
congruent.
If sides, then
angles.
SAS (1,2,3)
CPCTC
If a ray
divides an
angle into 2
congruent
angles, the
ray bisects
the angle
Works Cited
• Geometry for Enjoyment and Challenge. New
Edition. Evanston, Illinois: McDougal Littell,
1991.
• “Isosceles Triangle Proofs.” Math Warehouse.
29 May 2008.
<http://www.mathwarehouse.com/geometry/con
gruent_triangles/isosceles-triangle-theoremsproofs.php>.