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Six Triangle Facts To Know

Triangle problems show up frequently on the ACT Math test.  Here are six triangles facts that you should know and that will help you answer triangle problems correctly and get them done quickly.

1.  Triangle Height

Knowing how to get the area of a triangle is important. As you probably know, the area of a triangle is one-half of the base times the height.  But make sure that you are getting the height correct.  If you orient the triangle so that one side is horizontal, then that side is the base.  If you draw a perpendicular line from the base to the vertex opposite the base, that is the height.  You may or may not know that length of the height with one side as the base, but you might know it with another side as the base.  Use the orientation that is easiest for you to work the problem.

Triangle Height

2.  Equal Angles

Angles that are opposite sides that are equal (as in an isosceles triangle) are also equal.  That’s pretty simple.

Equal Angles

3.  Largest and Smallest Angles

The angle that is opposite the longest side is the largest angle.  And the side that is opposite the largest angle is also the largest side.  This works for the smallest angle and side as well.

4.  Sum of Two Sides

The sum of the length of two sides of a triangle is always larger than the length of the third side.  So in this triangle, if you add the length of sides 1 and 2 together, it will always be greater than the length of side 3.  If you add the length of sides 2 and 3 together, it will always be greater than the length of side 1.  And so on.

Triangle Side Lengths

This may seem obvious and trivial, but it is one of the properties of triangles and is occasionally useful on the ACT Test.

5.  Exterior Angles

If one side of a triangle is extended, then the angle formed by that line on the outside of the triangle (the exterior angle) is equal in size to the sum of the opposite angles.

Exterior Angles

Angle 1 represents one of the exterior angles in this triangle–the line representing one of the sides of the triangle is extended.  Angle 1 is equal to the sum of angles 3 and 4.  With just a little thought, you can see why this would be so. Remember that angle 1 + angle 2 equals 180 (straight angle).  Also remember that angle 2 + angle 3 + angle 4 = 180 (sum of interior angles of a triangle – 180).  Since these both equal 180, you can set them two equal to one another:

Angle 1 + Angle 2 = Angle 2 + Angle 3 + Angle 4.

Subtract Angle 2 from both sides and you are left with Angle 1 = Angle 3 + Angle 4.

So the exterior angle is equal to the sum of the opposite angles.

6.  The Pythagorean Theorem

The last triangle fact that we are going to deal with right now is the Pythagorean Theorem.  This is something that you should know, because I can just about guarantee that there will be problems where you will need use it.

This theorem defines the relationships between the length of the sides of all right triangles.  It’s pretty simple:  The length of the hypotenuse squared is equal to the sum of the squares of the lengths of the other two sides.

Remember that the hypotenuse is the side opposite the right angle.  Take the length of one of the other sides, square it, add that to the square of the length of the other side, and that will be the same as the length of the hypotenuse squared.

Pythagorean Theorum

What this means is that for all right triangles, if you know the length of two of the sides, you can find the length of the third side.

I think we are just about done with triangles.  If you have any questions, or problems that you don’t know how to do, please let me know in the comments and I’ll get them addressed.

Good Luck!

Scott

P.S. ACT Exam Secrets has some great tips for doing well on the ACT.  Make sure that you check it out.

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