section5

VARIOUS TYPES OF TRIANGLES: ISOSCELES AND EQUILATERAL TRIANGLES.

In this section we are going to continue our discussion of triangles. We start with the considering of various types of triangles.
A triangle is called an isosceles triangle if it has two sides with equal lengths. Consider an isosceles triangle ABC with AC=CB. Then the side AB (i.e. the side that is not equal to the other sides) is called the base side. The angle opposite the base side is called the vertex angle and the rest two angles of an isosceles triangle are called base angles.

Theorem 5.1 Suppose that ABC is an isosceles triangle then its base angles are equal.

Proof We are given the fact that ABC is an isosceles triangle e.g. we have that AC=CB. Then we need to prove that

But the triangle CAB is equal to the triangle CBA by SAS (essentially we are dealing with the same triangle):

Therefore

The next theorem that we are going to discuss is the theorem 5.1* which is converse of the theorem 5.1. We will say that a theorem B is the converse of the theorem A if the conclusion of the theorem A is the assumption in the theorem B and the assumption of the theorem A is the conclusion of the theorem B. Please note that not every theorem has a converse one e.g. the theorem that says, "If two angles are vertical angles then they are equal.angles." Indeed, the converse theorem will be "If two angles are equal then they are vertical angles", which is not true in general, we may have two equal angles that are not vertical.

Theorem 5.1* If a triangle ABC has two equal angles then this triangle is an isosceles one.
Proof The proof of this theorem follows the same pattern as the proof for the theorem 5.1. Now it is based on ASA.

Since we have introduced a special definition of a triangle with two equal sides, we are going to make a step forward in this direction. A triangle with three sides of the same length is called an equilateral triangle.

Theorem 5.2 In an equilateral triangle all angles are equal.

Proof Suppose that triangle ABC is an equilateral one. Then
Since AB = BC the triangle ABC is an isosceles triangle with the base side AC. By theorem 5.1 it follows that
Similarly since BC = CA the triangle ABC is an isosceles triangle with the base side AB and by theorem 5.1 it follows that
Therefore

Now we are going to leave for a while special types of triangles and discuss a few notions connected with all types of triangles.
Consider a triangle ABC. A segment perpendicular to the line containing AC and passing through the vertex B is called the altitude of the triangle ABC falling from the vertex B. We can similarly define altitudes falling from vertices A and C.

Consider a triangle ABC. A segment passing through the vertex B between sides AB and BC and dividing the angle B into two equal angles is called the bisector of the angle B. We can similarly define bisectors of the angles A and C.

Finally, in a triangle ABC a segment that connects the vertex B and the middle point of the side opposite to the vertex B is called the median falling from the vertex B. Medians falling from the vertices A and C are defined in the same way.

After introducing the three definitions above we are ready to discuss one more property of isosceles triangles.

Theorem 5.3 In an isosceles triangle a median falling from the vertex angle is also the bisector of the vertex angle and the altitude falling from the vertex angle.
Proof Suppose that a triangle ABC is an isosceles triangle with AC = BC and median CD falling from the vertex angle. We need to prove that CD is also the bisector of the angle C and the altitude falling from C.

Triangles ACD and BCD are equal by SAS: AD = DB since D is the middle point of AB, AC = CB since the triangle ABC is an isosceles triangle and angles A and B are equal as base angles of an isosceles triangle ABC. Therefore angles ACD and BCD are equal and CD is the angle bisector. Angles ADC and BDC are equal as well. Let's denote the degree measure of the angle ADC by x. Then we have the following equation:

since the angles ADC and BDC are complimentary ones. Therefore CD is the altitude as well.

EXERCISE SET #5

Problem 1. Formulate and prove a theorem that will be the converse of the theorem 5.2.

Problem 2. The sum of the lengths of the sides of a triangle is called the triangle's perimeter.

If the perimeter of an isosceles triangle is equal to 20 inches and the base side is equal to 10 inches, find the length of one of the two equal sides of this triangle.