Session 5Warm-up

1
Session 5 Warm-up
Begin at the word Tomorrow. Every Time you
move, write down the word(s) upon which you land.
is
spirit
show
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.
2
CCGPS Analytic Geometry
UNIT QUESTION How do I prove geometric theorems
involving lines, angles, triangles and
parallelograms? Standards MCC9-12.G.SRT.1-5,
MCC9-12.A.CO.6-13 Todays Question If the legs
of an isosceles triangle are congruent, what do
we know about the angles opposite them? Standard
MCC9-12.G.CO.10
3
4.1 Triangles Angles
4
4.1 Classifying Triangles
Triangle A figure formed when three
noncollinear points are connected by segments.
The sides are DE, EF, and DF. The vertices are D,
E, and F. The angles are ?D, ? E, ? F.
Angle
E
Side
Vertex
F
D
5
Triangles Classified by Angles
Acute
Obtuse
Right
17º
50º
120º
60º
30
70º
43º
60º
All acute angles
One obtuse angle
One right angle
6
Triangles Classified by Sides
Isosceles
Equilateral
Scalene
no sides congruent
all sides congruent
at least two sides congruent
7
Classify each triangle by its angles and by its
sides.
8
Fill in the table
Acute Obtuse Right
Scalene
Isosceles
Equilateral
9
Try These

1. ? ABC has angles that measure 110, 50, and 20.
   Classify the triangle by its angles.
2. ? RST has sides that measure 3 feet, 4 feet, and
   5 feet. Classify the triangle by its sides.

10
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
F
D
11
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.
The congruent sides are called legs.
leg
leg
base angle
base angle
The side opposite the vertex is the base.
12
Base Angles Theorem

• If two sides of a triangle are congruent, then
  the angles opposite them are congruent.
• If , then

13
Converse of Base Angles Theorem

• If two angles of a triangle are congruent, then
  the sides opposite them are congruent.
• If , then

14
EXAMPLE 1
Apply the Base Angles Theorem
Find the measures of the angles.
SOLUTION
P
Q
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
15
EXAMPLE 2
Apply the Base Angles Theorem
Find the measures of the angles.
P
Q
(48)
R
16
EXAMPLE 3
Apply the Base Angles Theorem
Find the measures of the angles.
P
Q
(62)
R
17
EXAMPLE 4
Apply the Base Angles Theorem
Find the value of x. Then find the measure of
each angle.
P
SOLUTION
(12x20)
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
(20x-4)
R
Q
Plugging back in, And since there must be 180
degrees in the triangle,
18
EXAMPLE 5
Apply the Base Angles Theorem
Find the value of x. Then find the measure of
each angle.
P
Q
(11x8)
(5x50)
R
19
EXAMPLE 6
Apply the Base Angles Theorem
Find the value of x. Then find the length of the
labeled sides.
SOLUTION
P
Q
(80)
(80)
Since there are two congruent sides, the angles
opposite them must be congruent also. Therefore,
7x 3x 40 4x 40 x 10
3x40
7x
Plugging back in, QR 7(10) 70 PR 3(10) 40
70
R
20
EXAMPLE 7
Apply the Base Angles Theorem
Find the value of x. Then find the length of the
labeled sides.
P
(50)
5x3
(50)
R
Q
10x 2
21
Right Triangles
HYPOTENUSE
LEG
LEG
22
Interior Angles
Exterior Angles
23
Triangle Sum Theorem
The measures of the three interior angles in a
triangle add up to be 180º.
x y z 180
x
y
z
24
R
m ?R m ?S m ?T 180º
54
54º 67º m ?T 180º
121º m ?T 180º
67
S
T
m ?T 59º
25
m ? D m ?DCE m ?E 180º
E
55º 85º y 180º
B
y
140º y 180º
C
x
85
y 40º
55
D
A
26
Find the value of each variable.
x
43
x
57
x 50º
27
Find the value of each variable.
55
43
(6x 7)
(40 y)
28
x 22º
y 57º
28
Find the value of each variable.
50
53
x
50
62
x 35º
29
Exterior Angle Theorem
The measure of the exterior angle is equal to the
sum of two nonadjacent interior angles
1
m?1m?2 m?3
2
3
30
Ex. 1 Find x.
B.
A.
72
43
148
x
76
x
38
81
31
Corollary to the Triangle Sum Theorem
The acute angles of a right triangle
are complementary.
x y 90º
x
y
32
Find m?A and m?B in right triangle ABC.
m?A m? B 90
A
2x 3x 90
2x
5x 90
x 18
3x
C
B
m?A 2x
m?B 3x
2(18)
3(18)
54
36