How can I tell if a triangle is a 30, 60 90 triangle or a 45 45 90? What are the characteristics?

A 45-45-90 triangle has two equal angles (and sides). It is an isoceles right triangle, in other words. It is easy to tell by looking that a 30-60-90 triangle does not have two equal angles.

To determine if a triangle is a 30-60-90 or a 45-45-90 triangle, you can examine the angles or the side lengths of the triangle. Here are the characteristics of each type:

1. 30-60-90 triangle:
- Angles: It has one 90-degree angle, one 30-degree angle, and one 60-degree angle.
- Side lengths: The side opposite the 30-degree angle is usually the shortest side and is represented by the length "x". The side opposite the 60-degree angle is typically longer and is represented by the length "x√3". The side opposite the 90-degree angle, which is the longest side, is twice the length of the side opposite the 30-degree angle and is represented by "2x".

2. 45-45-90 triangle:
- Angles: It has two equal 45-degree angles and one 90-degree angle.
- Side lengths: Each side opposite the 45-degree angles has the same length and is represented by "x". The side opposite the 90-degree angle, which is the hypotenuse, is equal to the length "x√2".

To identify which type of triangle you have, you can measure the angles using a protractor or use a carpenter's square to compare the lengths of the sides. If the angles are not easily measurable or you don't have the necessary tools, you can look at the side lengths relationship. If the ratios of the side lengths match the characteristics mentioned above, you can determine whether it is a 30-60-90 or a 45-45-90 triangle.