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Interior and Exterior Angles of Triangles

Grade 8
Sep 9, 2022
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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) 
Interior and Exterior Angles of Triangles
  • 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 

Interior and Exterior Angles of Triangles 

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. 

parallel
Relate interior angle measures in a triangles 

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

 A straight line has an angle of 180°

∴∠A+∠B+∠C=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

parallel

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

Exterior angle. 

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

 the interior opposite angles. 

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

 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

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.
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?
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?
 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 is the measure of ∠D?

Concept Map: 

Concept Map: 

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.

Comments:

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