Angles in Triangles

Interior Angles

Interior angles are the angles formed inside a triangle by the intersection of its sides. In any triangle, the sum of the interior angles is always equal to 180 degrees. For example, if a triangle has angles a, b, and c, then a + b + c = 180°.

  • This is known as the Angle Sum Property.

You can prove this by drawing a parallel line to one of the sides of the triangle. In the diagram below, we have drawn a line parallel to the line opposite angle b.

Alternate angles are equal:

  • angle d and angle a are alternate angles, so d = b
  • angle e and angle c are alternate angles, so e = c

Angles d + b + e form a straight line, so they must equal 180°. Therefore, a + b + c must also equal 180°.

Exterior Angles

Exterior angles are the angles formed outside a triangle when one of its sides is extended. To create an exterior angle, you can pick any side of the triangle, extend it beyond the vertex, and measure the angle formed between the extended side and the adjacent side. In a triangle, there are three exterior angles, one for each vertex.

Let’s look at an example in which we have two unidentical triangles facing each other and sharing a vertex. Let’s call these triangles T1 and T2.

Triangle T1 has angle a (19°). Triangle T2 has angle c (29°). The remaining angle b (132°) is the exterior of both angle a and angle c.

Calculating angle c

Since angle a, angle c, and the exterior angle b are all adjacent and together form a straight line, their sum should equal 180°. So, to calculate angle c, we follow this simple formula:

angle c = 180° − angle a − angle b

Plugging in the given values, we get:

angle c = 180° − 19° − 132°

= 29°

So, angle c is indeed 29°.

Calculating angle a

Now let’s reverse the example, where we want to calculate angle a (in triangle T1).

To calculate angle a, we follow a similar formula:

angle a = 180° − angle b − angle c

Plugging in the given values, we get:

angle a = 180° − 132° − 29° = 19°

So, angle a is 19°.

Examples

Example 1:

In a triangle with angles a = 35°, b = 75° and c = 70°, find the exterior angle formed by extending the side with angle c.

Since the exterior angle and angle b together form a straight line, their sum should equal 180°.

To find the exterior angle (let’s call it angle d) formed by extending the side with angle c, we use the following formula:

angle d = 180° − angle c

Plugging in the given values, we get:

angle d = 180° − 70° = 110°

So, the exterior angle formed by extending the side with angle c is 110°.

Example 2:

In a triangle, angle A is 60 degrees, angle B is 50 degrees, and angle C is 70 degrees. Calculate the exterior angles A’, B’, and C’.

Step 1: Verify that the interior angles of the triangle sum up to 180 degrees. A + B + C = 60° + 50° + 70° = 180° (The triangle is valid)

Step 2: Calculate the exterior angles. Recall that the exterior angle of a triangle is supplementary to its corresponding interior angle, which means their sum is 180 degrees.

A’: 180° – A = 180° – 60° = 120°

B’: 180° – B = 180° – 50° = 130°

C’: 180° – C = 180° – 70° = 110°

So, the exterior angles of the triangle are:

A’ = 120°

B’ = 130°

C’ = 110°

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