Right Triangles Made Simple

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Right Triangles Made Simple

Marilyn L. Brown

By Marilyn L. Brown

Right triangles are special kind of triangle because by definition one angle measures 90 degrees.

Recognize these patterns of steps as they are common. Depending upon what you are given, you may need to begin with an intermediate step and follow it to what you are trying to prove.

Proving Right Triangles

If a shape exists such that it has three sides and three angles it is a triangle. So begin by using the definition of triangle to prove the shape is a triangle.

Next use what you are given to prove that it is a right angle. If you have a straight line that is bisected the result is a right angle. If you are given perpendicular lines the angle is a right angle.

Since you have a triangle with a right angle, you can use the definition of right triangle to prove that it is a right triangle.

Proving They Are Congruent

Once you know they are right triangles, you can prove they are congruent. If you have this kind of triangle use one of the special right triangle rules. To use these you must have a pair of right triangles, meaning they have a pair of congruent right angles.

HA
If in a pair of right triangles the hypotenuse (long side) and an acute (small) angle are congruent, then the triangles are congruent. This is called the Hypotenuse-Angle theorem.

HL
If in a pair of right triangles the hypotenuse (long side) and a leg (side) are congruent, then the triangles are congruent. This is called the Hypotenuse-Leg theorem.

LL
If in a pair of right triangles legs (side) are congruent, then the triangles are congruent. This is called the Leg-Leg theorem.

LA
If in a pair of right triangles a leg (side) and an acute (small) angle are congruent, then the triangles are congruent. This is called the Leg-Angle theorem.

Using These Triangles

Once the triangles are proven congruent you can use the corresponding parts of congruent triangles are congruent (CPCTC). This may be a necessary step to your final goal.

In a right triangle there is a special mathematical relationship called the Pythagorean Theorem. To use this theorem you must prove that the segments are to certain lengths. Then the each leg, called a and b, squared and added equal the hypotenuse, called c, squared. In mathematical symbols, it is a2+b2=c2.

Using algebra skills if you are given any two of a,b,c; then you can find the length of the last one.

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