# SUM OF THE ANGLE MEASURES IN A TRIANGLE

Sum of the Angle Measures in a Triangle :

There is a special relationship between the measures of the interior angles of a triangle.

That is,

Sum of the three angles in any triangle  =  180°

In the next part, we are going to justify this relationship.

## Sum of the angle measures in a triangle is 180° - Justify

Step 1 :

Draw a triangle and cut it out. Label the angles A, B, and C. Step 2 :

Tear off each “corner” of the triangle. Each corner includes the vertex of one angle of the triangle.

Step 3 :

Arrange the vertices of the triangle around a point so that none of your corners overlap and there are no gaps between them. Step 4 :

What do you notice about how the angles fit together around a point ?

The angles form a straight angle.

Step 5 :

What do you notice about how the angles fit together around a point ?

180°

Step 6 :

Describe the relationship among the measures of the angles of triangle ABC ?

The sum of the angle measures is 180°.

Step 7 :

What does the triangle sum theorem state ?

The triangle sum theorem states that for triangle ABC,

mA + mB + mC =  180°

## Reflect

1.  Can a triangle have two right angles ? Explain.

No

The sum of the measures of two right angles is 180°. That means the measure of the third angle would be

180° - 180°  =  0°

which is impossible.

2. Describe the relationship between the two acute angles in a right triangle. Explain your reasoning.

No

They are complementary.

The sum of their measures must be

180° - (measure of the right angle)  =  180° - 90°  =  90°

## Sum of the Angle Measures in a Triangle - Practice Problems

Problem 1 :

Can 30°, 60° and 90° be the angles of a triangle ?

Solution :

Let us add all the three given angles and check whether the sum is equal to 180°.

30° +  60° + 90°  =  180°

Since the sum of the angles is equal 180°, the given three angles can be the angles of a triangle.

Problem 2 :

Can 35°, 55° and 95° be the angles of a triangle ?

Solution :

Let us add all the three given angles and check whether the sum is equal to 180°.

35° +  55° + 95°  =  185°

Since the sum of the angles is not equal 180°, the given three angles can not be the angles of a triangle.

Problem 3 :

In a triangle, if the second angle is 5° greater than the first angle and the third angle is 5° greater than second angle, find the three angles of the triangle.

Solution :

Let "x" be the first angle.

The second angle  =  x + 5

The third angle  =  x + 5 + 5  =  x + 10

We know that,

the sum of the three angles of a triangle  =  180°

x + (x+5) + (x+10)  =  180°

3x + 15  =  180

3x  =  165

x  =  55

The first angle  =  55°

The second angle  =  55 + 5  =  60°

The third angle  =  60 + 5  =  65°

Hence, the three angles of a triangle are 55°, 60° and 65°. After having gone through the stuff given above, we hope that the students would have understood the sum of the angle measures in a triangle

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