How to Determine if a Triangle is Acute: Angle Measurement, Triangle Inequality Theorem, and Pythagorean Theorem

acute triangle

An acute triangle is a triangle in which all three angles are less than 90 degrees

An acute triangle is a triangle in which all three angles are less than 90 degrees. In other words, the angles of an acute triangle are all acute angles.

To determine if a triangle is acute, we need to examine the measures of its angles. If all three angles are less than 90 degrees, then the triangle is acute.

Here are some methods to determine if a triangle is acute:

1. Angle measurement: Measure all three angles of the triangle using a protractor. If each angle measures less than 90 degrees, then the triangle is acute.

2. Triangle inequality theorem: According to the triangle inequality theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. If this condition is satisfied for the three sides of a triangle, then the triangle is acute.

This theorem can be used to determine if a triangle is acute by comparing the lengths of its sides. If the sum of the squares of the two shorter sides is greater than the square of the longest side, then the triangle is acute.

For example, consider a triangle with side lengths of 5, 6, and 8 units. The sum of the squares of the two shorter sides is 5^2 + 6^2 = 61, which is greater than the square of the longest side (8^2 = 64). So, this triangle is acute.

3. Pythagorean theorem: If a triangle is a right triangle, it cannot be acute because it has a right angle measuring 90 degrees. Therefore, if all angles of a triangle are acute, it cannot be a right triangle.

By using the Pythagorean theorem, we can determine if a triangle is acute or right. If the sum of the squares of the two shorter sides is less than the square of the longest side, then the triangle is acute. If it is equal to the square of the longest side, then the triangle is right.

For example, consider a triangle with side lengths of 3, 4, and 5 units. The sum of the squares of the two shorter sides is 3^2 + 4^2 = 25, which is equal to the square of the longest side (5^2 = 25). So, this triangle is right and not acute.

Overall, an acute triangle is a triangle with all three angles measuring less than 90 degrees. You can determine if a triangle is acute by measuring its angles, using the triangle inequality theorem, or applying the Pythagorean theorem.

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