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Geometry

# GED Mathematical Reasoning: Triangles

A triangle can be classified based on the length of its sides and on the measure of its angles. Based on side lengths, triangles can be classified as either equilateral, isosceles or scalene.

Equilateral triangle

First, let’s talk about the equilateral triangle. You might think of the word “equal” when I say “equilateral,” which is appropriate because in an equilateral triangle all three sides are equal length. Notice in the example shown that all three sides have the same length of 5 inches.

Also, all three angles have the same measure of 60 degrees. And this is always the case: an equilateral triangle has three equal sides and three equal angles.

• If all three sides of a triangle are equal, then it can be assumed that all three angles are equal, as well.
• If three angles are known to be equal, it follows that the three sides are equal.
• Equal sides are always opposite equal angles and vice versa.

Isosceles triangle

In an isosceles triangle, two sides are of equal length. Notice in the example shown that two sides have the same length of 8 centimeters.

Also, two angles have the same measure of 40 degrees. And this is always the case: an isosceles triangle has two equal sides and two equal angles.

Similar to equilateral triangles, if it is known that two sides of a triangle are equal, then it can be assumed that two angles are equal, as well. Conversely, if two angles are known to be equal, it follows that the two sides are equal. Also, equal sides are always opposite equal angles and vice versa.

• If it is known that two sides of a triangle are equal, then it can be assumed that two angles are equal.
• If two angles are known to be equal, it follows that the two sides are equal.
• Equal sides are always opposite equal angles and vice versa.

Scalene triangle

A scalene triangle has no sides that are equal. And it follows that no angles are equal. In other words, there are no sides that have the same length and no angles that have the same measure.

Now that we’ve discussed the three triangle classifications that are based on side length, let’s talk about the classifications that are based on angle measure.

Based on the measure of its angles, a triangle may be called right, acute or obtuse.

Right triangle

A right triangle contains a right angle which is an angle that measures 90 degrees.

Acute triangle

In an acute triangle, all angles measure less than 90 degrees.

The equilateral triangle I showed at the beginning of the video can also be considered “acute” since all its angles measure less than 90 degrees.

Obtuse triangle

An obtuse triangle contains one angle that is larger than 90 degrees.

Properties of triangles

• When it comes to equilateral and isosceles triangles, equal sides are opposite equal angles and vice versa.
• The longest side of any triangle is opposite the largest angle.

• The sum of all three angles in ANY triangle is 180 degrees.

[latex]\angle a + \angle b+ \angle c = 180^{\circ}[/latex]

Example 1

Classify the triangle.

Well, we’re given that two sides have the same measure of 1.5 meters so we can assume that two angles have the same measure. A triangle that contains two equal sides and two equal angles is the isosceles triangle.

It might be tempting to state that this is this also a right triangle, because it looks as though the angle made up of the two known sides is 90 degrees.

However, since the angle measures are not provided we cannot know for sure.

This is an important thing for us to discuss. When classifying figures, we need to be sure to base our decision off of what is known for sure – not what appears to be. Looks can be deceiving and not all figures are drawn to scale.

Example 2

Find the measure of the missing angle.

Here, we are given two angle measures and we are being asked to find the third angle measure.

Suppose the two angle measures we are given are angle a, which is 42 degrees and angle b, which is 77 degrees. This means the variable c represents the measure of the missing angle.

We know that all three angles will add to be 180 degrees, meaning that angle a plus angle b plus angle c is equal to 180.

In our equation, we can let a equal 42 and b equal 77 and then use our equation solving skills to find angle c.

In closing, what type of triangle is this? If you’re thinking it’s a scalene triangle – you’re right! Since none of the angle measures are equal, we can assume none of the side lengths are equal – making this a scalene triangle.

And because all the angles are less than 90 degrees it is also considered an acute triangle.

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