What are the remote interior angles to the exterior angle 4?

What are the remote and interior angles?

It's all about extending a side of the triangle

An exterior angle of a triangle, or any polygon, is formed by extending one of the sides.

In a triangle, each exterior angle has two remote interior angles . The remote interior angles are just the two angles that are inside the triangle and opposite from the exterior angle.

The Formula

As the picture above shows, the formula for remote and interior angles states that the measure of a an exterior angle $$ \angle A $$ equals the sum of the remote interior angles.

To rephrase it, the angle 'outside the triangle' (exterior angle A) equals D + C (the sum of the remote interior angles).

If one side of a triangle is extended beyond the vertex, an exterior angle is formed. This exterior angle is supplementary with its adjacent, linear angle. Since the angle sum in a triangle is also 180 degrees, the exterior angle must have a measure equal to the sum of the remaining angles, called the remote interior angles.

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.

Exterior Angle Theorem – Explanation & Examples

So, we all know that a triangle is a 3-sided figure with three interior angles. But there exist other angles outside the triangle, which we call exterior angles.

We know that the sum of all three interior angles is always equal to 180 degrees in a triangle.

Similarly, this property holds for exterior angles as well. Also, each interior angle of a triangle is more than zero degrees but less than 180 degrees. The same goes for exterior angles.

In this article, we will learn about:

• Triangle exterior angle theorem,
• exterior angles of a triangle, and,
• how to find the unknown exterior angle of a triangle.

What is the Exterior Angle of a Triangle?

The exterior angle of a triangle is the angle formed between one side of a triangle and the extension of its adjacent side.

In the illustration above, the triangle ABC’s interior angles are a, b, c, and the exterior angles are d, e, and f. Adjacent interior and exterior angles are supplementary angles.

In other words, the sum of each interior angle and its adjacent exterior angle is equal to 180 degrees (straight line).

Triangle Exterior Angle Theorem

The exterior angle theorem states that the measure of each exterior angle of a triangle is equal to the sum of the opposite and non-adjacent interior angles.

Remember that the two non-adjacent interior angles opposite the exterior angle are sometimes referred to as remote interior angles.

For example, in triangle ABC above;

⇒ d = b + a

⇒ e = a + c

⇒ f = b + c

Properties of exterior angles

• An exterior angle of a triangle is equal to the sum of the two opposite interior angles.
• The sum of exterior angle and interior angle is equal to 180 degrees.

⇒ c + d = 180°

⇒ a + f = 180°

⇒ b + e = 180°

• All exterior angles of a triangle add up to 360°.

Proof: 

⇒ d + e + f = b + a + a + c + b + c

⇒ d +e + f = 2a + 2b + 2c

= 2(a + b + c)

But, according to triangle angle sum theorem,

a + b + c = 180 degrees

Therefore, ⇒ d +e + f = 2(180°)

= 360°

How to Find the Exterior Angles of a Triangle?

Rules to find the exterior angles of a triangle are pretty similar to the rules to find the interior angles. It is because wherever there is an exterior angle, there is an interior angle with it, and both add up to 180 degrees.

Let’s take a look at a few example problems.

Example 1

Given that for a triangle, the two interior angles 25° and (x + 15) ° are non-adjacent to an exterior angle (3x – 10) °, find the value of x.

Solution

Apply the triangle exterior angle theorem:

⇒ (3x − 10) = (25) + (x + 15)

⇒ (3x − 10) = (25) + (x +15)

⇒ 3x −10 = x + 40

⇒ 3x – 10 = x + 40

⇒ 3x = x + 50

⇒ 3x = x + 50

⇒ 2x = 50

x =25

Hence, x = 25°

Substitute the value of x into the three equations.

⇒ (3x − 10) = 3(25°) – 10°

= (75 – 10) ° = 65°

⇒ (x+15) = (25 + 15) ° = 40°

Therefore, the angles are 25°, 40° and 65°.

Example 2

Calculate values of x and y in the following triangle.

Solution

It is clear from the figure that y is an interior angle and x is an exterior angle.

By Triangle exterior angle theorem.

⇒ x = 60° + 80°

x = 140°

The sum of exterior angle and interior angle is equal to 180 degrees (property of exterior angles). So, we have;

⇒ y + x = 180°

⇒ 140° + y = 180°

subtract 140° from both sides.

⇒ y = 180° – 140°

y = 40°

Therefore, the values of x and y are 140° and 40°, respectively.

Example 3

The exterior angle of a triangle is 120°. Find the value of x if the opposite non-adjacent interior angles are (4x + 40) ° and 60°.

Solution

Exterior angle = sum of two opposite non-adjacent interior angles.

⇒120° =4x + 40 + 60

Simplify.

⇒ 120° = 4x + 100°

Subtract 120° from both sides.

⇒ 120° – 100° = 4x + 100° – 100°

⇒ 20° = 4x

Divide both sides by to get,

x = 5°

Therefore, the value of x is 5 degrees.

Verify the answer by substitution.

120°= 4x + 40 + 60

120° = 4° (5) + 40° + 60°

120° = 120° (RHS = LHS)

Example 4

Determine the value of x and y in the figure below.

Solution

Sum of interior angles = 180 degrees

y + 41° + 92° = 180°

Simplify.

y + 133° = 180°

subtract 133° from both sides.

y = 180° – 133°

y = 47°

Apply the triangle exterior angle theorem.

x = 41° + 47°

x = 88°

Hence, the value of x and y is 88° and 47°, respectively.

Which are the remote interior angles of ∠ 4?

What is are the exterior angles remote interior angles?

What is the interior angle of 4 sides?