An exterior angle of a triangle is equal to the sum of the two remote interior angles. true false

When we talk about exterior angles and the remote interior angles it helps that we think about what in English what do they mean do they mean? Exterior means outside, so this is angle 1 outside of our triangle. The angles that are inside the triangles are the interior but there's only two that are remote. Remote means far away which is why when you're trying to use your TV you use a remote because your far away, so the remote interior angles are 3 and 4 so again 1 is your exterior angle because it's outside and the two angles that are not adjacent to angle 1 are your remote interior angles.There's a special relationship that exists here and that is angle 1 is equal to angle 3 plus angle 4 but you're not just going to take my word for it you're going say "Mr. McCall you need to prove that," so what I'm going to do is I'm going to say angle 1 and angle 2 must sum to 180 degrees because if I add those two angles up we get a straight line. The second thing I'm going to say is that these 3 angles 2, 3, 3. 2, 3 and 4 must sum to 180 degrees because they make a triangle.If I solve this equation for 2, I'm sorry that 2 is a little messy, then I can substitute in to my first equation, so I'm going to subtract angle 3 and I'm going to subtract angle 4 so what I'm doing is just moving everything to the other side of that equation so subtract angle 3 subtract angle 4 and I find that angle 2 must equal 180 minus those two angles, so 180 degrees minus angle 3 minus angle 4 so I know angle 2 in terms of angle 3 and 4 and I'm going to substitute that in right over there, so we're going to shift and I'm going to say angle 1 plus angle 2 which we said was 180 minus angle 3 minus angle 4 if we go back to our original equation here that has to equal 180 degrees. I see I have 180 degrees on both sides so I'm just going to minus 180 and then that will make them disappear and if I move negatives angle 3 and negative angle 4 to the other side by adding angle 3 and angle 4 then all I have left is angle 1 is equal to 180 and negative 180 is 0 so we have angle 3 plus angle 4 which has proven that the remote exterior angle excuse me the exterior angle is equal to the sum of the remote interior angles.

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

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

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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.

1. Example 1: Find the values of x and y by using the exterior angle theorem of a triangle.

   

   Solution:

   ∠x is the exterior angle.

   ∠x + 92 = 180º (linear pair of angles)

   ∠x = 180 - 92 = 88º

   Applying the exterior angle theorem, we get, ∠y + 41 = 88

   ∠y = 88 - 41 = 47º

   Therefore, the values of x and y are 88º and 47º respectively.

2. Example 2: Find ∠BAC and ∠ABC.

   

   Solution:

   160º is an exterior angle of the Δ ABC. So, by using the exterior angle theorem, we have, ∠BAC + ∠ABC = 160º

   x + 3x = 160º

   4x = 160º

   x = 40º

   Therefore, ∠BAC = x = 40º and ∠ABC = 3xº = 120º

3. Example 3: Find ∠ BAC, if ∠CAD = ∠ADC

   

   Solution:

   Solving the linear pair at vertex D, we get ∠ADC + ∠ADE = 180º

   ∠ADC = 180º - 150º = 30º

   Using the angle sum property, for Δ ACD,

   ∠ADC + ∠ACD + ∠CAD = 180º

   ∠ACD = 180 - ∠CAD -∠ADC

   180º - ∠ADC -∠ADC (given ∠CAD= ∠ADC)

   180º - 2∠ADC

   180º - 2 × 30º

   ∠ACD = 180º - 60º = 120º

   ∠ACD is the exterior angle of ∠ABC

   Using the exterior angle theorem, for Δ ABC, ∠ACD = ∠ABC + ∠BAC

   120º = 60º + ∠BAC

   Therefore, ∠BAC = 120º - 60º = 60º.

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FAQs on 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.

Where Should We Use Exterior Angle Theorem?

Exterior angle theorem could be used to determine the measures of the unknown interior and exterior angles of a triangle.

Do All Polygons Exterior Angles Add up to 360?

The exterior angles of a polygon are formed when a side of a polygon is extended. All the exterior angles in all the polygons sum up to 360º.