This theorem is Proposition 1.16 in Euclid's Elements, which states that the measure of an exterior angle of a triangle is greater than either of the measures of the remote interior angles. This is a fundamental result in absolute geometry, because its proof does not depend upon the parallel postulate.

In the above diagram,

- m∠1, m∠2, and m∠3 are interior angles.
- m∠4 is an exterior angle.
- m∠1 and m∠2 are remote interior angles to m∠4.

The 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

**Proof : **

There is a special relationship between the measure of an exterior angle and the measures of its remote interior angles.

Let us understand this relationship through the following steps.

**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 : **

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°.

**Problem 2 : **

Find m∠A and m∠B in the triangle given below.

**Solution : **

**Step 1 : **

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

m∠A + m∠B = m∠C

**Step 2 : **

Substitute the given angle measures.

(5y + 3)° + (4y + 8)° = 146°

**Step 3 : **

Solve the equation for y.

(5y + 3)° + (4y + 8)° = 146°

5y + 3 + 4y + 8 = 146

Combine the like terms.

9y + 11 = 146

Subtract 11 from both sides.

9y + 11 - 11 = 146 - 11

Simplify.

9y = 135

Divide both sides by 9.

9y / 9 = 135 / 9

y = 15

**Step 4 : **

Use the value of y to find m∠A and m∠B.

m∠A = 5y + 3

m∠A = 5(15) + 3

m∠A = 75 + 3

m∠A = 78

m∠B = 4y + 8

m∠B = 4(15) + 8

m∠B = 60 + 8

m∠B = 68

So, m∠A = 78° and m∠B = 68°.

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