Triangle Outer Angle Theorem: Statement and Problems

In this publication, we will consider one of the main theorems in class 7 geometry – about the external angle of a triangle. We will also analyze examples of solving problems in order to consolidate the presented material.

Definition of an outside corner

First, let’s remember what an external corner is. Let’s say we have a triangle:

Triangle Outer Angle Theorem: Statement and Problems

Adjacent to an internal corner (λ) triangle angle at the same vertex is external. In our figure, it is indicated by the letter γ.

Wherein:

  • the sum of these angles is 180 degrees, i.e. c + λ = 180° (property of the outer corner);
  • 0 и 0.

Statement of the theorem

The exterior angle of a triangle is equal to the sum of the two angles of the triangle that are not adjacent to it.

c = a + b

Triangle Outer Angle Theorem: Statement and Problems

From this theorem it follows that the external angle of a triangle is greater than any of the internal angles that are not adjacent to it.

Examples of tasks

Task 1

A triangle is given in which the values ​​of two angles are known – 45 ° and 58 °. Find the exterior angle adjacent to the unknown angle of the triangle.

Solution

Using the formula of the theorem, we get: 45° + 58° = 103°.

Task 1

The external angle of a triangle is 115°, and one of the non-adjacent internal angles is 28°. Calculate the values ​​of the remaining angles of the triangle.

Solution

For convenience, we will use the notation shown in the figures above. The known internal angle is taken as α.

Based on the theorem: β = γ – α = 115° – 28° = 87°.

Angle λ is adjacent to the outer, and therefore is calculated by the following formula (follows from the property of the outer corner): λ = 180° – γ = 180° – 115° = 65°.

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