There are two ways to classify triangles. The first way deals with the size of the largest angle in the triangle.

Obtuse triangle: A triangle is obtuse if the largest angle is greater than 90 degrees in measure. If the sides are given as a, b and c, where c is the longest side, if a^2 + b^2 < c^2, the triangle is obtuse.

Right triangle: If the largest angle is exactly 90 degrees in measure, the triangle is a right triangle. If the sides are given as a, b and c, where c is the longest side, a^2 + b^2 = c^2 means we have a right triangle. This is known as the Pythagorean Theorem.

Acute triangle: If the largest angle is less than 90 degrees, the triangle is called an acute triangle. If the sides are given as a, b and c, where c is the longest side, a^2 + b^2 > c^2 means we have an acute triangle.

The other method of classification has to do with having angles of the same measure or sides of the same length.

Equilateral triangle: All three angles have the same measure, 60 degrees. All three sides are the same length.

Isosceles triangle: At least two sides have the same measure, which also means two angles must have the same measure. Technically, and equilateral triangle is still isosceles, but it is a special case and we usually name it by the more specific name.

Scalene triangle: All three sides are different lengths, and all three angles are different measure.

Practice problems.

1) Can a triangle be both a right triangle and an isosceles triangle?

2) A triangle has side lengths 3, 3 and 4. What are its two classifications?

3) A triangle has two angles, 70 degrees and 15 degrees. What are its two classifications?

4) An isosceles triangle has one angle of measure 40 degrees. What are the two possible sets of angle measures and classify both as acute, right or obtuse.

5) A triangle has side lengths 5, 12 and 13. What are its two classifications?

Answers in the comments.

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1) Can a triangle be both a right triangle and an isosceles triangle?

Answer: Yes. The triangle with angles 45 degrees, 45 degrees and 90 degrees is a right isosceles triangle.2) A triangle has side lengths 3, 3 and 4. What are its two classifications?

Answer: Two side lengths are equal, so it's isosceles. Because 3^2 + 3^2 = 18 and 4^ = 16, it is an acute triangle.3) A triangle has two angles, 70 degrees and 15 degrees. What are its two classifications?

Answer: The third angle is 180-70-15 degrees, which is 95 degrees. Because it is more than 90 degeres the triangle is obtuse. Because all measures are different, it is scalene.4) An isosceles triangle has one angle of measure 40 degrees. What are the two possible sets of angle measures and classify both as acute, right or obtuse.

Answer:

Possibility #1: The angle measures are 40 degrees, 40 degrees and 100 degrees. This triangle is obtuse.

Possibility #2: The other two angles add up to 180-40 = 140 degrees and are of equal measure. This triangle has angles of 70 degrees, 70 degrees and 40 degrees, so it is acute.

5) A triangle has side lengths 5, 12 and 13. What are its two classifications?

Answer: 5^2 + 12^2 = 24 + 144 = 169 = 13^2. This is a right triangle.

All sides are different lengths, so it is a scalene triangle as well.

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