Triangle Sum Theorem
Bill Zahner
Lori Jordan
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Printed: July 22, 2012
AUTHORS
Bill Zahner
Lori Jordan
www.ck12.org
C ONCEPT
1
1
Triangle Sum Theorem
Here you’ll learn that the sum of the angles in any triangle is the same, due to the Triangle Sum Theorem.
What if you wanted to classify the Bermuda Triangle by its sides and angles? You are probably familiar with the
myth of this triangle; how several ships and planes passed through and mysteriously disappeared.
The measurements of the sides of the triangle are in the image. What type of triangle is this? Using a protractor,
find the measure of each angle in the Bermuda Triangle. What do they add up to? Do you think the three angles
in this image are the same as the three angles in the actual Bermuda triangle? Why or why not? After completing
this Concept, you’ll be able to determine how the three angles in any triangle are related in order to help you answer
these questions.
Watch This
https://vimeo.com/45151775
MEDIA
Click image to the left for more content.
http://www.youtube.com/watch?v=Rt0TbTVbKhA
Now watch this video.
MEDIA
Click image to the left for more content.
http://www.youtube.com/watch?v=fBPTfmU6XaI
Concept 1. Triangle Sum Theorem
2
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Guidance
In polygons, interior angles are the angles inside of a closed figure with straight sides. The vertex is the point
where the sides of a polygon meet.
Triangles have three interior angles, three vertices and three sides. A triangle is labeled by its vertices with a 4. This
triangle can be labeled 4ABC, 4ACB, 4BCA, 4BAC, 4CBA or 4CAB. Order does not matter. The angles in any
polygon are measured in degrees. Each polygon has a different sum of degrees, depending on the number of angles
in the polygon. How many degrees are in a triangle?
Investigation: Triangle Tear-Up
Tools Needed: paper, ruler, pencil, colored pencils
1. Draw a triangle on a piece of paper. Try to make all three angles different sizes. Color the three interior angles
three different colors and label each one, 6 1, 6 2, and 6 3.
2. Tear off the three colored angles, so you have three separate angles.
3. Attempt to line up the angles so their points all match up. What happens? What measure do the three angles
add up to?
This investigation shows us that the sum of the angles in a triangle is 180◦ because the three angles fit together to
form a straight line. Recall that a line is also a straight angle and all straight angles are 180◦ .
The Triangle Sum Theorem states that the interior angles of a triangle add up to 180◦ . The above investigation is
one way to show that the angles in a triangle add up to 180◦ . However, it is not a two-column proof. Here we will
prove the Triangle Sum Theorem.
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3
←
→
Given : 4ABC with AD || BC
Prove : m6 1 + m6 2 + m6 3 = 180◦
TABLE 1.1:
Statement
←
→
1. 4ABC above with AD || BC
2. 6 1 ∼
= 6 4, 6 2 ∼
=6 5
3. m6 1 = m6 4, m6 2 = m6 5
4. m6 4 + m6 CAD = 180◦
5. m6 3 + m6 5 = m6 CAD
6. m6 4 + m6 3 + m6 5 = 180◦
7. m6 1 + m6 3 + m6 2 = 180◦
Reason
Given
Alternate Interior Angles Theorem
∼
= angles have = measures
Linear Pair Postulate
Angle Addition Postulate
Substitution PoE
Substitution PoE
There are two theorems that we can prove as a result of the Triangle Sum Theorem and our knowledge of triangles.
Theorem #1: Each angle in an equiangular triangle measures 60◦ .
Theorem #2: The acute angles in a right triangle are always complementary.
Example A
What is the m6 T ?
From the Triangle Sum Theorem, we know that the three angles add up to 180◦ . Set up an equation to solve for 6 T .
m6 M + m6 A + m6 T = 180◦
82◦ + 27◦ + m6 T = 180◦
109◦ + m6 T = 180◦
m6 T = 71◦
Concept 1. Triangle Sum Theorem
4
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Example B
Show why Theorem #1 is true.
4ABC above is an example of an equiangular triangle, where all three angles are equal. Write an equation.
m6 A + m6 B + m6 C = 180◦
m6 A + m6 A + m6 A = 180◦
3m6 A = 180◦
m6 A = 60◦
If m6 A = 60◦ , then m6 B = 60◦ and m6 C = 60◦ .
Example C
Use the picture below to show why Theorem #2 is true.
m6 O = 41◦ and m6 G = 90◦ because it is a right angle.
m6 D + m6 O + m6 G = 180◦
m6 D + 41◦ + 90◦ = 180◦
m6 D + 41◦ = 90◦
m6 D = 49◦
Notice that m6 D + m6 O = 90◦ because 6 G is a right angle.
Watch this video for help with the Examples above.
https://vimeo.com/45151776
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5
Concept Problem Revisited
The Bermuda Triangle is an acute scalene triangle. The angle measures are in the picture to the right. Your measured
angles should be within a degree or two of these measures. The angles should add up to 180◦ . However, because
your measures are estimates using a protractor, they might not exactly add up.
The angle measures in the picture are the actual measures, based off of the distances given, however, your measured
angles might be off because the drawing is not to scale.
Vocabulary
A triangle is a three sided shape. All triangles have three interior angles, which are the inside angles connecting the
sides of the triangle. The vertex is the point where the sides of a polygon meet. Special types of triangles are listed
below:
Scalene: All three sides are different lengths.
Isosceles: At least two sides are congruent.
Equilateral: All three sides are congruent.
Right: One right angle.
Acute: All three angles are less than 90◦ .
Obtuse: One angle is greater than 90◦ .
Equiangular: All three angles are congruent.
Guided Practice
1. Determine m6 1 in this triangle:
2. Two interior angles of a triangle measure 50◦ and 70◦ . What is the third interior angle of the triangle?
Concept 1. Triangle Sum Theorem
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3. Find the value of x and the measure of each angle.
Answers:
1. 72◦ + 65◦ + m6 1 = 180◦ .
Solve this equation and you find that m6 1 = 43◦ .
2. 50◦ + 70◦ + x = 180◦ .
Solve this equation and you find that the third angle is 60◦ .
3. All the angles add up to 180◦ .
(8x − 1)◦ + (3x + 9)◦ + (3x + 4)◦ = 180◦
(14x + 12)◦ = 180◦
14x = 168
x = 12
Substitute in 12 for x to find each angle.
[3(12) + 9]◦ = 45◦
Practice
Determine m6 1 in each triangle.
1.
2.
3.
[3(12) + 4]◦ = 40◦
[8(12) − 1]◦ = 95◦
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4.
5.
6.
7.
8. Two interior angles of a triangle measure 32◦ and 64◦ . What is the third interior angle of the triangle?
9. Two interior angles of a triangle measure 111◦ and 12◦ . What is the third interior angle of the triangle?
10. Two interior angles of a triangle measure 2◦ and 157◦ . What is the third interior angle of the triangle?
Find the value of x and the measure of each angle.
11.
12.
13.
Concept 1. Triangle Sum Theorem
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14.
15.
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