## Different Angles of Triangles

### Key Concepts

- Relate interior angle measures in a triangle
- Find the exterior angle measures
- Find the unknown angle measures using algebra

## Interior and Exterior Angles of Triangles

- A triangle is a three-sided polygon that consists of three edges and three vertices.

- If xx and yy are two parallel lines, a line that intersects two or more lines at different points is called a transversal. (Say tt)

- We know the corresponding angles are congruent.

So, ∠1=∠5, ∠2=∠6, ∠3=∠7 and ∠4=∠8

- The alternate interior angles are congruent.

So, ∠4=∠6and ∠3=∠5

- The same-side interior angles are supplementary.

So, ∠3+∠6=180° and ∠4+∠5=180°

**Interior and Exterior Angles of Triangles**

Consider a triangle △ABC as shown

**Relate interior angle measures in a triangles**

Let us rotate the copies of △ABC and place them in order to bring all the angles together.

∠A ∠B and ∠C appear to form a straight line.

A straight line has an angle of 180°

∴∠A+∠B+∠C=180°

**Hence, the sum of the measures of interior angles of a triangle is** **180°**

**Find exterior angle measures**

If we extend any side of a triangle, the angle is called an **Exterior angle**.

In △PQR if we extend QR towards R, ∠PRS is the exterior angle.

For ∠PRS, ∠QRP is the interior adjacent angle and ∠PQR and ∠RPQ are the interior opposite angles.

Let us add the measures of ∠P and ∠Q and compare it to the measure of the exterior angle.

∴∠PRS = ∠PQR+∠RPQ

**Hence, t****he measure of an exterior angle of a triangle is equal to the sum of the measures of ****its interior opposite angles.**

### 1.6.3: Use algebra to find unknown angle measures

**Example 1: **Find the measure of x and y

**Solution: **

**Step 1: Find the measure of** 𝒙

In the given triangle, x and 120° form a linear pair.

A linear pair of angles must add up to 180°

⇒120°+x=180°

⇒x=180°−120°

⇒x=60°

∴ **The measure of** 𝒙** is** 𝟔𝟎°

**Step 2: Find the measure of**𝒚

We know, x, y and 70° are the angles of the triangle.

The sum of the interior angles of a triangle is 180°

⇒x+y+70°=180°

⇒60°+y+70°=180°

⇒ y=180°−70°−60°

⇒ y=50°

∴**The measure of** 𝒚** is** 𝟓𝟎°

**Exercise**

- Find m∠1 and m∠2.

- In the figure, m ∠1=(8x+7)°, m∠2=(4x+14)°, and m∠4=(13x+12)°. Your friend incorrectly says that m∠4=51°. What is m∠4? What mistake might your friend have made?

- In ∆ABC, what is m∠C?

- The measure of ∠F is 110°. The measure of ∠E is 100°. What is the measure of ∠D?

**Concept Map:**

### What we have learned:

- Sum of the measures of interior angles of a triangle is 180°.
- The measure of an exterior angle of a triangle is equal to the sum of the measures of its interior opposite angles.

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