**Corresponding Angles : **

Angles lie on the same side of the transversal t, on the same side of lines a and b.

**Example : ∠ 1 and ∠ 5**

**Alternate Interior Angles :**

Angles are nonadjacent angles that lie on opposite sides of the transversal t, between lines a and b.

**Example : ∠ 3 and ∠ 6**

**Alternate Exterior Angles :**

Angles lie on opposite sides of the transversal t, outside lines a and b.

**Example : ∠ 1 and ∠ 8**

**Same-Side Interior Angles :**

Angles lie on the same side of the transversal t, between lines a and b.

**Example : ∠ 3 and ∠ 5**

In this section, we are going to see the angle relationships in triangles through the following steps.

**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°

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

**Problem 1 :**

In the figure given below, let the lines l₁ and l₂ be parallel and m is transversal. If ∠F = 65°, find the measure of each of the remaining angles.

**Solution : **

From the given figure,

∠F and ∠H are vertically opposite angles and they are equal.

Then,

∠H = ∠F -------> ∠H = 65°

∠H and ∠D are corresponding angles and they are equal.

Then,

∠D = ∠H -------> ∠D = 65°

∠D and ∠B are vertically opposite angles and they are equal.

Then,

∠B = ∠D -------> ∠B = 65°

∠F and ∠E are together form a straight angle.

Then, we have

∠F + ∠E = 180°

Plug ∠F = 65°

∠F + ∠E = 180°

65° + ∠E = 180°

∠E = 115°

∠E and ∠G are vertically opposite angles and they are equal.

Then,

∠G = ∠E -------> ∠G = 115°

∠G and ∠C are corresponding angles and they are equal.

Then,

∠C = ∠G -------> ∠C = 115°

∠C and ∠A are vertically opposite angles and they are equal.

Then,

∠A = ∠C -------> ∠A = 115°

Therefore,

∠A = ∠C = ∠E = ∠G = 115°

∠B = ∠D = ∠F = ∠H = 65°

**Problem 2 :**

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°.

**Problem 3 :**

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°.

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