Download T1. Triangle Sum Theorem

December 13, 2012
December 12, 2012
Objective: SWBAT identify congruent triangles and their
corresponding sides.
Pre-Class:
Tell whether it is possible to draw the following figures:
1) acute, scalene triangle.
2) obtuse, equilateral triangle.
3) right, isosceles triangle.
4) equiangular, scalene triangle.
5) right, scalene triangle.
T1. Triangle Sum Theorem
The measures of the interior angles of a triangle sum to 180°
C
A
If you have a triangle, then you know the sum of
its three interior angles must sum to 180°
B
December 13, 2012
Example: Triangle Sum Theorem
Find the measure of the missing angle
J
32°
20°
K
L
Theorem T1. The Triangle Sum Theorem says that the interior
angles of
must sum to 180°.
So,
32° +
+ 20° = 180°
and substituting the
information from the diagram
+ 52° = 180°
= 128°
1
Check: 128+32+20=180
What is the measurement of the missing angle?
C
52°
A
53°
B
December 13, 2012
2
What is the measurement of the missing angle?
M
57°
N
3
L
What is the measure of the missing angle?
34°
47°
x°
x=
December 13, 2012
4
In
, if
what is the
is 84° and
?
is 36°,
(draw a diagram)
triangle ABC = 180°
So, 84 + 36 +
= 180°
Answer
=180° - 120°
= 60°
5
A triangle can have more than one obtuse angle.
True
False
December 13, 2012
6
A triangle can have more than one right angle.
True
False
7
Each angle in an equiangular triangle measures 60°
True
False
December 13, 2012
8
An equilateral triangle is also an isosceles triangle
True
False
Example
We can solve more "complicated" problems using the
Triangle Sum Theorem.
Q
Solve for x
(12x+8)°
P
55°
From the Triangle Sum Theorem
(8x-3)°
R
55 + (12x+8) + (8x-3) = 180 Substituting from the diagram
20x + 60 = 180 Combining like terms
20x = 120 Isolating x using inverse operations
x=6
December 13, 2012
9
Solve for x in the diagram.
R
8x°
2x°
Q
5x°
S
Answer
2x+5x+8x = 180 What is
15x = 180
x = 12
10
What is the measure of angle B?
B
A
C
Hint
Solve for x
3x-17 +x+40 +2x-5 = 180°
December 13, 2012
Corollary to Triangle Sum Theorem
The acute angle of a right triangle are complementary.
C
A
B
Since T1. the Triangle Sum Theorem says the interior
angles of a triangle must sum to 180°. So, 180° - 90° (the
right angle) = 90° left between
and
.
Recall: two angles that sum up to 90° are called
complementary
Example
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.
5x°
x°
Since this is a right triangle, we can use the Corollary to the
Triangle Sum Theorem which says the two acute angles are
complementary. So,
x + 5x = 90
(using the Triangle Sum
6x = 90
Theorem is a little more work)
x = 15
One acute angle is 15° and the other is 75°
December 13, 2012
11
In a right triangle, the two acute angles sum to 90°
True
False
12
Solve for x
What are the
Challenge
three angles?
December 13, 2012
13
Solve for x
What are the
Challenge
angle measures?
14
In the right triangle given, what is the measurement of each
acute angle?
2x°
x°
December 13, 2012
What is the measurement of the missing angle?
15
M
57°
N
L
Note: we solved this problem earlier using the Triangle Sum
Theorem. Use the Corollary to the Triangle Sum this time.
Answer
x + 57 = 90
x = 33
16
1
2
3
December 13, 2012
17
1
2
3
Answer
December 13, 2012
18
Find the value of x in the diagram
X°
Mark yourHint
vertical angles!
20°
Example
Find the missing angles in the diagram.
December 13, 2012
T2. Exterior Angle Theorem
The measure of an exterior angle of a triangle is equal to the
sum of the two nonadjacent interior angles measures
C
1
A
B
Recall that adjacent angles share a side or ray
The adjacent angle to
would be
The two nonadjacent sides are then
and
Example
Solve for x using the Exterior Angle Theorem
21°
x°
y°
34°
The Exterior Angle Theorem says that the exterior angle,
marked x°, is equal to the two nonadjacent interior angles.
x = 21 + 34
So, the exterior angle
x = 55°
We also know what
is - how?
What does x° + y° have to equal?
December 13, 2012
Find the value of the missing angle
Y°
Q
1
Q
2
Q
3
Q
4
25°
65°
90°
40°
110°
30°
70°
100°
95°
115°
3
Y°
1
19
2
Solve for the exterior angle, x.
60°
x°
Y°
55°
What is Y°?
December 13, 2012
20
Use the Exterior Angle Theorem to solve for x.
Y°
94 = 60 +Answer
2x
34 = 2x
What is Y°?
17 = x
21
Using the Exterior Angle Theorem, solve for x.
Y°
100 = 2x +3 +51
Answer
100 = 2x
+54
46 = 2x
What is Y°?
23 = x
December 13, 2012
22
Solve for x
(3x - 5)°
(x + 2)°
Y°
33°
3x - 5 = (x + 2) + 33
3x - 5 =Answer
x + 35
2x = 40 What is Y°?
x = 20