Triangle Sum Theorem

Bill ZahnerLori Jordan

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Printed: September 15, 2013

AUTHORSBill ZahnerLori Jordan

www.ck12.org Concept 1. Triangle Sum Theorem

CONCEPT 1 Triangle Sum TheoremHere 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 themyth of this triangle; how several ships and planes passed through and mysteriously disappeared.

The measurements of the sides of the triangle from a map are in the image. What type of triangle is this? Usinga protractor, find the measure of each angle in the Bermuda Triangle. What do they add up to? Do you think thethree angles in this image are the same as the three angles in the actual Bermuda triangle? Why or why not? Aftercompleting this Concept, you’ll be able to determine how the three angles in any triangle are related in order to helpyou answer these questions.

Watch This

MEDIAClick image to the left for more content.

CK-12 Foundation: Chapter4TriangleSumTheoremA

MEDIAClick image to the left for more content.

James Sousa:ProvingtheTriangle Sum Theorem

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Guidance

In polygons, interior angles are the angles inside of a closed figure with straight sides. The vertex is the pointwhere the sides of a polygon meet.

Triangles have three interior angles, three vertices and three sides. A triangle is labeled by its vertices with a4. Thistriangle can be labeled 4ABC,4ACB,4BCA,4BAC,4CBA or 4CAB. Order does not matter. The angles in anypolygon are measured in degrees. Each polygon has a different sum of degrees, depending on the number of anglesin 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 anglesthree 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 anglesadd up to?

This investigation shows us that the sum of the angles in a triangle is 180◦ because the three angles fit together toform 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 isone way to show that the angles in a triangle add up to 180◦. However, it is not a two-column proof. Here we willprove the Triangle Sum Theorem.

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www.ck12.org Concept 1. Triangle Sum Theorem

Given: 4ABC with←→AD || BC

Prove: m6 1+m 6 2+m6 3 = 180◦

TABLE 1.1:

Statement Reason1. 4ABC above with

←→AD || BC Given

2. 6 1∼= 6 4, 6 2∼= 6 5 Alternate Interior Angles Theorem3. m6 1 = m6 4,m 6 2 = m6 5 ∼= angles have = measures4. m6 4+m 6 CAD = 180◦ Linear Pair Postulate5. m6 3+m 6 5 = m6 CAD Angle Addition Postulate6. m6 4+m 6 3+m6 5 = 180◦ Substitution PoE7. m6 1+m 6 3+m6 2 = 180◦ 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 m 6 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+m 6 A+m 6 T = 180◦

82◦+27◦+m6 T = 180◦

109◦+m6 T = 180◦

m6 T = 71◦

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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+m 6 B+m 6 C = 180◦

m 6 A+m6 A+m6 A = 180◦

3m6 A = 180◦

m6 A = 60◦

If m6 A = 60◦, then m 6 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.

m 6 D+m 6 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.

MEDIAClick image to the left for more content.

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www.ck12.org Concept 1. Triangle Sum Theorem

CK-12 Foundation: Chapter4TriangleSumTheoremB

Concept Problem Revisited

The Bermuda Triangle is an acute scalene triangle. The angle measures are in the picture below. Your measuredangles should be within a degree or two of these measures. The angles should add up to 180◦. However, becauseyour measures are estimates using a protractor, they might not exactly add up.

The angle measures in the picture are the measures from a map (which is flat). Because the earth is curved, in reallife the measures will be slightly different.

Vocabulary

A triangle is a three sided shape. All triangles have three interior angles, which are the inside angles connecting thesides of the triangle. The vertex is the point where the sides of a polygon meet. Special types of triangles are listedbelow:

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 m 6 1 in this triangle:

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2. Two interior angles of a triangle measure 50◦ and 70◦. What is the third interior angle of the triangle?

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◦ [3(12)+4]◦ = 40◦ [8(12)−1]◦ = 95◦

Practice

Determine m6 1 in each triangle.

1.

2.

3.

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4.

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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.

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