#### Need Help?

Get in touch with us  # Interior and Exterior Angles of Triangles

## 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, the 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 of𝒚

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

1. Find m∠1 and m∠2.
1. 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?
1. In ∆ABC, what is m∠C?
1. The measure of ∠F is 110°. The measure of ∠E is 100°. What is the measure of ∠D?

### What we have learned:

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

#### Related topics #### Addition and Multiplication Using Counters & Bar-Diagrams

Introduction: We can find the solution to the word problem by solving it. Here, in this topic, we can use 3 methods to find the solution. 1. Add using counters 2. Use factors to get the product 3. Write equations to find the unknown. Addition Equation: 8+8+8 =? Multiplication equation: 3×8=? Example 1: Andrew has […] #### Dilation: Definitions, Characteristics, and Similarities

Understanding Dilation A dilation is a transformation that produces an image that is of the same shape and different sizes. Dilation that creates a larger image is called enlargement. Describing Dilation Dilation of Scale Factor 2 The following figure undergoes a dilation with a scale factor of 2 giving an image A’ (2, 4), B’ […] #### How to Write and Interpret Numerical Expressions?

Write numerical expressions What is the Meaning of Numerical Expression? A numerical expression is a combination of numbers and integers using basic operations such as addition, subtraction, multiplication, or division. The word PEMDAS stands for: P → Parentheses E → Exponents M → Multiplication D → Division  A → Addition S → Subtraction         Some examples […]   