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.

**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,

m∠A + m∠B + m∠C = 180°

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°

**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°

Because 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°

So, the three angles of a triangle are 55°, 60° and 65°.

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