Åk 6–9

4.1 Theorems

Outer Angles
The outer angle to the triangle is marked with an x in this figure.  We will show you how to calculate the outer angle of the triangle by using the triangle’s angle sum and the angle’s sums of supplementary angles.

Supplementary Angles
You know that a straight angle is 180°.  Therefore we can say that:
Λb + Λx = 180°

You also know that the angle sum for a triangle is 180°.
Λa + Λb + Λc = 180°

According to the transitive law (Operator and Calculation Rules 8.5):
Λb + Λx = Λa + Λb + Λc

You subtract the angle b on both sides:
Λx = Λa + Λc

Then we can see that the outer angle of the triangle is equal to the sum of the two opposite angles in the triangle:
Λa + Λc = Λx

Top Triangles

The segment DE is parallel to the triangle’s base BC.

ADE is called the top triangle in the triangle ABC.

DE splits the triangle so that the top triangle is similar to the entire triangle.

Therefore the corresponding angles in the two triangles ABD and ADE are equal and the sides are proportional to each other.


Base Angles in an Isosceles Triangle

In an isosceles triangle the base angles are equal.

Λb = Λc

You will learn to prove these geometric theorems in higher levels of mathematics.