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Describing Triangles

Concept

Types of Triangles

A triangle is a polygon with three sides and three interior angles. Triangles can be classified using their side lengths and angle measures.
Concept

Classification by Sides

Concept

Classification by Angles

Rule

Isosceles Triangle Theorem

If two sides of a triangle are congruent, then the angles opposite them are congruent.

An isosceles triangle with two congruent sides and base angles

Based on the diagram above, the following relation holds true.

The Isosceles Triangle Theorem is also known as the Base Angles Theorem.

Proof

This theorem will be proven using congruent triangles. Consider a triangle with two congruent sides.

An isosceles triangles with two congruent sides

In this triangle, let be the point of intersection of the angle bisector of and

An isosceles triangle with two congruent sides and an angle bisector

From the diagram, the following features of and can be observed.

Feature Reasoning
Definition of an angle bisector.
Given.
Reflexive Property of Congruence.

Therefore, and have two pairs of corresponding congruent sides and one pair of included congruent angles. By the Side-Angle-Side Congruence Theorem, and are congruent triangles. Corresponding parts of congruent figures are congruent. Therefore, and are congruent.

It has been proven that if two sides in a triangle are congruent, the angles opposite them are congruent.

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Exercise

Classify the triangle by its sides and its angles.

Show Solution
Solution
We can begin by classifying the triangle by its side lengths. First, we must use the distance formula to calculate the length of the sides. We'll start with
Thus, the side is units long. The lengths of the other sides can be calculated in the same way.
Side Distance formula Length

We have found that two of the sides, and have the same length. Thus, it is an isosceles triangle. Consequently, the base angles, and are congruent and acute. Since the third angle, is also acute we can conclude that it is an acute triangle. Thus, the triangle can be classified as isosceles and acute.

Rule

Angles of a Triangle

A triangle contains three interior angles and creates three exterior angles.
Theory

Interior Angles of a Triangle

The interior angles are the angles on the inside of the triangle.

Concept

Exterior Angles of a Triangle

Suppose one side of a triangle is extended. The angle created is called an exterior angle.

Because a triangle has three sides, three exterior angles exist. Each exterior angle and its corresponding interior angle are supplementary.

At each vertex, the exterior angle can be defined in two ways.

Rule

Interior Angles Theorem

Consider

The sum of the interior angles of is

Proof

Interior Angles Theorem


To begin, draw a line, that passes through and is parallel to and create three angles and

Together, the three angles make a straight angle. Thus, the sum of their measures is Because, and is a transversal, and are alternate interior angles. Thus, according to the Alternate Interior Angles Theorem, By the same reasoning,

Two congruent angles have the same measure. This can be used to rewrite the sum of the three angles. Therefore, the sum of the interior angles of a triangle is This can be summarized in a flowchart proof.


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Exercise

Find the measures of the angles that are marked in the figure.

Show Solution
Solution
To begin, we'll determine the unknown interior angle measures. The Interior Angles Theorem states that the sum of the measures of the interior angles is We can use this to find and in turn to find and
Since We can substitute into the given expression for Lastly, we can find the measure of the exterior angle, Since this angle is supplementary to the sum of their measures is
The measures of the angles marked in the figure are shown below.
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