Download Triangles & Angle Theorems

Review:
Begin at the word
“Tomorrow”. Every Time
you move, write down the
word(s) upon which you land.
is
show spirit
1. Move to the consecutive interior angle.
homecoming!
2. Move to the alternate interior angle.
Tomorrow
3. Move to the corresponding angle.
4. Move to the alternate exterior.
5. Move to the exterior linear pair.
because
your
6. Move to the alternate exterior angle.
it
7. Move to the vertical angle.
Triangles & Angle Theorems
Days 107 & 108
Learning Target:
Students can prove geometric theorems
involving lines, angles, triangles and
parallelograms.
4.1 Classifying Triangles
Triangle – A figure formed when three
noncollinear points are connected by
segments.
The sides are DE, EF, and DF.
E
Angle
Side
The vertices are D, E, and F.
The angles are D,  E,  F.
Vertex
D
F
Triangles Classified by Angles
Acute
Obtuse
Right
17º
50º
60º
120º
30°
70º
43º
60º
All acute angles
One obtuse angle
One right angle
Triangles Classified by Sides
Scalene
no sides
congruent
Isosceles
at least two
sides congruent
Equilateral
all sides
congruent
Classify each triangle by its angles and by its sides.
C
E
60°
45°
45°
F
60°
G
EFG is a right
isosceles triangle.
A
60°
B
ABC is an acute
equilateral triangle
Fill in the table
Acute
Scalene
Isosceles
Equilateral
Obtuse
Right
Adjacent Sides- share a vertex
ex. The sides DE & EF are adjacent to E.
Opposite Side- opposite the vertex
ex. DF is opposite E.
E
D
F
Parts of Isosceles Triangles
The angle formed by the congruent
sides is called the vertex angle.
The two angles formed
by the base and one of
the congruent sides are
called base angles.
leg
leg
The congruent
sides are called
legs.
base angle
base angle
The side opposite the vertex
is the base.
Base Angles Theorem (4.9)
If two sides of a triangle are congruent, then the
angles opposite them are congruent.
If AB  AC , then B  C
Converse of Base Angles
Theorem (4.10)
If two angles of a triangle are congruent, then
the sides opposite them are congruent.
If B  C , then AB  AC
EXAMPLE 1
Apply the Base Angles Theorem
Find the measures of the angles.
SOLUTION
Q
P
Since a triangle has 180°, 180 – 30 = 150°
for the other two angles.
Since the opposite sides are congruent,
angles Q and P must be congruent.
150/2 = 75° each.
(30)°
R
EXAMPLE 2
Apply the Base Angles Theorem
Find the measures of the angles.
Q
P
(48)°
R
EXAMPLE 4
Apply the Base Angles Theorem
Find the value of x. Then find the measure of each angle.
P
Q
(20x-4)°
SOLUTION
R
Plugging back in,
Since there are two congruent
sides, the angles opposite them
must be congruent also. Therefore,
12x + 20 = 20x – 4
20 = 8x – 4
24 = 8x
3=x
mP  12(3)  20  56
mR  20(3)  4  56
And since there must be 180 degrees in the triangle,
mQ  180  56  56  68
EXAMPLE 5
Apply the Base Angles Theorem
Find the value of x. Then find the measure of each angle.
Q
(11x+8)°
(5x+50)°
R
P
EXAMPLE 6
Apply the Base Angles Theorem
Find the value of x. Then find the length of the labeled sides.
Q
(80)°
(80)°
P
SOLUTION
Since there are two congruent
sides, the angles opposite them
must be congruent also. Therefore,
7x = 3x + 40
3x+40
7x
4x = 40
x = 10
Plugging back in,
QR = 7(10)= 70
PR = 3(10) + 40 = 70
R
EXAMPLE 7
Apply the Base Angles Theorem
Find the value of x. Then find the length of the labeled sides.
P
(50)°
5x+3
(50)°
Q
R
10x – 2
HYPOTENUSE
LEG
LEG
Interior Angles
Exterior Angles
Triangle Sum Theorem (4.1)
The measures of the three interior angles
in a triangle add up to be 180º.
x°
y°
x + y + z = 180°
z°
Find mT in RST.
R
m R + m S + m T = 180º
54º + 67º + m T = 180º
121º + m T = 180º
54°
S
67°
T
m T = 59º
Find the value of each variable in DCE
E
B
y°
C
x°
85°
55°
A
m  D + m DCE + m E = 180º
55º + 85º + y = 180º
140º + y = 180º
y = 40º
D
Find the value of each variable.
x°
43°
x°
x = 50º
57°
Find the value of each variable.
55°43°
28°
x = 22º
y = 57º
Find the value of each variable.
50°
53°
x°
62°
x = 65º
50°
Exterior Angle Theorem (4.3)
The measure of
the exterior
angle is equal to
the sum of two
nonadjacent
interior angles
1
m1+m2 =m3
2
3
Ex. 1: Find x.
B.
A.
76
81
72
43
x
38
x
148
Corollary to the Triangle Sum Theorem
The acute angles of a right triangle are
complementary.
x + y = 90º
x°
y°
Find mA and mB in right triangle
ABC.
mA + m B = 90
2x
+ 3x = 90
A
2x°
3x°
C
mA = 2x
= 2(18)
= 36
5x = 90
x = 18
B
mB = 3x
= 3(18)
= 54