**Angle theorems for triangles :**

In this section, we are going to see the following two important angle theorems in triangles.

1. Triangle sum theorem

2. Exterior angle theorem

**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°

**Step 1 :**

**Sketch a triangle and label the angles as **m∠1, m∠2 and m∠3.

**Step 2 :**

According to Triangle Sum Theorem, we have

m∠1 + m∠2 + m∠3 = 180° ------ (1)

**Step 3 :**

Extend the base of the triangle and label the exterior angle as m∠4.

**Step 4 :**

m∠3 and m∠4 are the angles on a straight line.

So, we have

m∠3 + m∠4 = 180° ------ (2)

**Step 5 :**

Use the equations (1) and (2) to complete the following equation,

m∠1 + m∠2 + m∠3 = m∠3 + m∠4 ------ (3)

**Step 6 :**

Use properties of equality to simplify the equation (3).

m∠1 + m∠2 + m∠3 = m∠3 + m∠4

Subtract m∠3 from both sides.

aaaaaaaaaaa m∠1 + m∠2 + m∠3 = m∠3 + m∠4 aaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaa - m∠3 - m∠3 aaaaaaaaaaaaaaaaa aaaaaaaaaaa ------------------------------------ aaaaaaaaaaa aaaaaaaaaaa m∠1 + m∠2 = m∠4 aaaaaaaaaaa aaaaaaaaaaa ------------------------------------ aaaaaaaaaaa

Hence, the Exterior Angle Theorem states that the measure of an** exterior** angle is equal to the sum of its **remote interior** angles.

That is,

m∠1 + m∠2 = m∠4

**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°.

**Problem 4 : **

Find m∠W and m∠X in the triangle given below.

**Solution : **

**Step 1 : **

Write the Exterior Angle Theorem as it applies to this triangle.

m∠W + m∠X = m∠WYZ

**Step 2 : **

Substitute the given angle measures.

(4y - 4)° + 3y° = 52°

**Step 3 : **

Solve the equation for y.

(4y - 4)° + 3y° = 52°

4y - 4 + 3y = 52

Combine the like terms.

7y - 4 = 52

Add 4 to both sides.

7y - 4 + 4 = 52 + 4

Simplify.

7y = 56

Divide both sides by 7.

7y / 7 = 56 / 7

y = 8

**Step 4 : **

Use the value of y to find m∠W and m∠X.

m∠W = 4y - 4

m∠W = 4(8) - 4

m∠W = 28

m∠X = 3y

m∠X = 3(8)

m∠X = 24

So, m∠W = 28° and m∠X = 24°.

After having gone through the stuff given above, we hope that the students would have understood "Angle theorems for triangles".

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