Lines and Angles

Interior and Exterior Angles of Triangles

Suppose the building that you live in, is polygonal in shape. If you start walking around the building on any side, as you turn each corner, the angle you pivot is the exterior angle of that corner. As you reach your starting point, you are facing the same way as when you started, thus you have made one complete rotation as you walked around the building. That is 360 degrees of rotation. So all your exterior turns are added up to 360 degrees. Let us now understand the topic exterior and interior angles of triangles in detail.

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Exterior and Interior Angles of Triangles

What are the Triangles?

A triangle is a simple closed curve which is created by three line-segments. In geometry, any three points, specifically non-collinear, form a unique triangle and separately, a unique plane.  The basic elements of the triangle are sides, angles, and vertices. Let us now talk about the exterior and interior angles of the triangle.

1) Interior Angles

Interior angles

An interior angle is an angle inside the shape. From the above diagram, we can say that the triangle has three interior angles.  In this triangle ∠ x, ∠y and ∠z are all interior angles. The sum of the interior angles is always 180° implies, ∠ x + ∠y + ∠z = 180°. Let us see the proof of this statement.

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Sum of Interior Angles of a Triangle

Interior angles

Statement: The sum of the interior angles is always 180°

Proof: Let us consider a ΔABC, as shown in the figure above. To prove the above property of triangles, you need to draw a line PQ parallel to the side BC of the given triangle. As we can see that PQ is a straight line, so  it can be concluded that:
∠PAB + ∠BAC + ∠QAC = 180……(1)

Since PQ || BC and AB and AC are transversal, therefore
∠QAC = ∠ACB (a pair of alternate angles)
Also, ∠PAB = ∠CBA (a pair of alternate angles)

Now substitute the value of ∠QAC and ∠PAB in equation (1)
∠ACB + ∠BAC + ∠CBA = 180°
Therefore the sum of the interior angles is always 180°

2) Exterior Angles

Exterior Angles of Triangles

An exterior angle of the triangle is the angle between one side of a triangle and the extension of an adjacent side. In the above triangle, ∠ a ∠b  ∠c are interior angles while ∠ d is an exterior angle. How do we find the value of exterior angles?

We need to find the interior angle and then subtract that value from 360 to get the value of the exterior angle. If you don’t know the value of interior angle you can deduce it by either the shape of it or the number of the other angles.

Solved Examples for You

Question 1: In the figure (not drawn to scale), ADF  and DEF are triangles and EC = ED, find y

Exterior Angles of Triangles

  1. 91°
  2. 92°
  3. 93°
  4. 94°

Answer : The correct option is A. In Δ CED, we have CE = ED as given. Therefore,
∠EDC = ∠ECD (angles opposite to equal sides are equal)
⇒ ∠ECD = 28°

Also, ∠EDC = ∠BCA ( vertically opposite angles)
⇒ ∠BCA = 28°
In Δ BCA, y = 62° + 28°
⇒ y = 90°

Question 2: How to find the interior angle?

Answer: In a regular polygon that is a flat shape whose sides are all equal. Besides, for finding the sum of the measure of interior angles the formula is (n – 2) × 180. However, for finding the measure of one interior angle, we take that formula and divide it by the number of sides n: (n – 2) × 180 ÷ n.

Question 3: What is the interior angle?

Answer: In mathematics, the interior angle refers to the two inner angles that formed where two sides of a polygon come together. In addition, any of the four angles formed in the area between a pair of parallel lines when a third line cuts them.

Question 4: What is the interior angle and exterior angle?

Answer: Interior angles refer to all those angles that are inside a shape. On the other hand, the exterior angle is an angle that is made by the side of the shape and a line drawn out from an adjacent side. Furthermore, the exterior angle is equal to the sum of the non-adjacent interior angle.

Question 5: Do interior angles add up to 180º?

Answer: Two angles that add up to 180º are known as supplementary angles. Moreover, the sum of three interior angles of a triangle is 180º and the sum of interior angles on a line is also 180º.

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