Introduction to Classifying Triangles
The only angle from which to approach a problem is the TRY-Angle.
—Author Unknown
The triangle is the fundamental figure in geometry. This lesson will expose the different types of triangles and how to use the Pythagorean theorem.
A Triangle is a figure with three equal sides. The sum of the measure of the angles in a triangle is equal to 180°.
A triangle can be classified by its angles: acute, right, or obtuse.
Acute triangles have all angles less than 90°.
Equilateral triangles have all angles equal to 60°. All sides of equilateral triangles are congruent (equal).
Obtuse triangles have one angle that is greater than 90°.
Right triangles have one right (90°) angle.
Tip:Be careful when you classify triangles by angle measure. Notice that even though right triangles and obtuse triangles each have two acute angles, their classification is not affected by these angles. Acute triangles have all three acute angles. |
A triangle can also be classified by its sides: scalene, isosceles, or equilateral. Isosceles triangles have two congruent sides (and the angles opposite these equal sides are equal as well).
Scalene triangles have no sides that are congruent.
Congruent Triangles
Triangles with the same size and shape are congruent triangles. The matching parts of congruent triangles are called congruent parts.
You can determine that two triangles are congruent if the following corresponding parts are congruent:
3 sides or side-side-side (SSS)
2 angles and the included side or angle-side-angle (ASA)
2 sides and the included angle or side-angle-side (SAS)
Similar Triangles
Triangles with the same shape, but different sizes, are similar triangles. The angles are equal, but the sides vary in length. Similarity is indicated by the ~ symbol.
Right Triangles and The Pythagorean Theorem
Right triangles have one right angle. They are special because you can use the Pythagorean theorem:
a^{2} + b^{2} = c^{2}
Here, a and b are the legs of the triangle and c is the hypotenuse. The hypotenuse is the side opposite the right angle.
The hypotenuse is equal to the sum of the squares of the lengths of the legs. Let's look at an example. What is the hypotenuse for the following triangle?
Using the Pythagorean theorem, substitute the values that you know:
a^{2} + b^{2} = c^{2}
6^{2} + 8^{2} = c^{2}
36 + 64 = c^{2}
100 = c^{2}
√100 = c^{2}
10 = c
So, the hypotenuse length is 10. This also means that the lengths 6, 8, and 10 work in a right triangle. Three numbers that prove the Pythagorean theorem are called Pythagorean triples. Three other Pythagorean triples include 3-4-5, 5-12-13, and 8-15-17.
Find practice problems and solutions for these concepts at Classifying Triangles Practice Questions.
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