Exterior angle inequality theorem worksheet with answers

In this lesson, I will talk about triangle inequalities, exterior angle, and hinge theorem. These theorems can help us arrange the sides and angles of a triangle from smallest to largest.

Table of Contents Show

What is Exterior Angle Theorem?
Proof of Exterior Angle Theorem
Exterior Angle Inequality Theorem
FAQs on Exterior Angle Theorem
What is the Exterior Angle Theorem?
How do you use the Exterior Angle Theorem?
What are Exterior Angles?
What is the Exterior Angle Inequality Theorem?
What is the Exterior Angle Property?
What is the Exterior Angle Theorem Formula?
Where Should We Use Exterior Angle Theorem?
Do All Polygons Exterior Angles Add up to 360?
What is the example of exterior angle inequality theorem?
Which statement defines exterior angle inequality theorem?

What is the triangle inequality theorem?

This states that the sum of the lengths of any two sides of a triangle is greater than the length of the third side.

If one side of a triangle is longer than a second side, then the angle opposite the first side is larger than the angle opposite the second side.

If one angle of a triangle is larger than a second angle, then the side opposite the first angle is longer than the side opposite the second angle.

What is the Exterior Angle Theorem?

The measure of an exterior angle is equal to the sum of two remote interior angle.

What is the Hinge Theorem

If two triangles have two sets of congruent sides, the larger included angle will have the longer third side.

Useful Resources

Practice what you’ve learned in these problems.

Exercises 1

Exercises 2

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The exterior angle theorem states that when a triangle's side is extended, the resultant exterior angle formed is equal to the sum of the measures of the two opposite interior angles of the triangle. The theorem can be used to find the measure of an unknown angle in a triangle. To apply the theorem, we first need to identify the exterior angle and then the associated two remote interior angles of the triangle.

What is Exterior Angle Theorem?

The exterior angle theorem states that the measure of an exterior angle is equal to the sum of the measures of the two opposite(remote) interior angles of the triangle. Let us recall a few common properties about the angles of a triangle: A triangle has 3 internal angles which always sum up to 180 degrees. It has 6 exterior angles and this theorem gets applied to each of the exterior angles. Note that an exterior angle is supplementary to its adjacent interior angle as they form a linear pair of angles. Exterior angles are defined as the angles formed between the side of the polygon and the extended adjacent side of the polygon.

We can verify the exterior angle theorem with the known properties of a triangle. Consider a Δ ABC.

The three angles a + b + c = 180 (angle sum property of a triangle) ----- Equation 1

c= 180 - (a+b) ----- Equation 2 (rewriting equation 1)

e = 180 - c----- Equation 3 (linear pair of angles)

Substituting the value of c in equation 3, we get

e = 180 - [180 - (a + b)]

e = 180 - 180 + (a + b)

e = a + b

Hence verified.

Proof of Exterior Angle Theorem

Consider a ΔABC. a, b and c are the angles formed. Extend the side BC to D. Now an exterior angle ∠ACD is formed. Draw a line CE parallel to AB. Now x and y are the angles formed, where, ∠ACD = ∠x + ∠y

Statement	Reason
∠a = ∠x	Pair of alternate angles. (Since BA is parallel to CE and AC is the transversal).
∠b = ∠y	Pair of corresponding angles. (Since BA is parallel to CE and BD is the transversal).
∠a + ∠b = ∠x + ∠y	From the above statements
∠ACD = ∠x + ∠y	From the construction of CE
∠a + ∠b = ∠ACD	From the above statements

Hence proved that the exterior angle of a triangle is equal to the sum of the two opposite interior angles.

Exterior Angle Inequality Theorem

The exterior angle inequality theorem states that the measure of any exterior angle of a triangle is greater than either of the opposite interior angles. This condition is satisfied by all the six external angles of a triangle.

Related Articles

Check out a few interesting articles related to Exterior Angle Theorem.

Exterior Angle Formula
Exterior Angle Theorem Worksheets
Alternate Exterior Angles
How to find the measure of each exterior angle of a regular pentagon?
Properties of Triangle
Interior and Exterior Angles Worksheets
Sum of Exterior Angles Formula

Important notes

The exterior angle theorem states that the measure of an exterior angle is equal to the sum of the measures of the two remote interior angles of the triangle.
The exterior angle inequality theorem states that the measure of any exterior angle of a triangle is greater than either of the opposite interior angles.
The exterior angle and the adjacent interior angle are supplementary. All the exterior angles of a triangle sum up to 360º.

FAQs on Exterior Angle Theorem

What is the Exterior Angle Theorem?

The exterior angle theorem states that the measure of an exterior angle is equal to the sum of the measures of the two remote interior angles of the triangle. The remote interior angles are also called opposite interior angles.

How do you use the Exterior Angle Theorem?

To use the exterior angle theorem in a triangle we first need to identify the exterior angle and then the associated two remote interior angles of the triangle. A common mistake of considering the adjacent interior angle should be avoided. After identifying the exterior angles and the related interior angles, we can apply the formula to find the missing angles or to establish a relationship between sides and angles in a triangle.

What are Exterior Angles?

An exterior angle of a triangle is formed when any side of a triangle is extended. There are 6 exterior angles of a triangle as each of the 3 sides can be extended on both sides and 6 such exterior angles are formed.

What is the Exterior Angle Inequality Theorem?

The measure of an exterior angle of a triangle is always greater than the measure of either of the opposite interior angles of the triangle.

What is the Exterior Angle Property?

An exterior angle of a triangle is equal to the sum of its two opposite non-adjacent interior angles. The sum of the exterior angle and the adjacent interior angle that is not opposite is equal to 180º.

What is the Exterior Angle Theorem Formula?

The sum of the exterior angle = the sum of two non-adjacent interior opposite angles. An exterior angle of a triangle is equal to the sum of its two opposite non-adjacent interior angles.