Types of Triangles
A triangle can be classified by the size of its angles and sides:
Equilateral Triangle
- 3 congruent angles, each 60°
- 3 congruent sides
Hook: The word equilateral comes from equi, meaning equal, and lat, meaning side. Thus, all equal sides.
Isosceles Triangle
- 2 congruent angles, called base angles; the third angle is the vertex angle.
- Sides opposite the base angles are congruent.
- An equilateral triangle is also isosceles.
Hook: Think of the "I" sound in isosceles as two equal eyes, which almost rhymes with 2 equal sides.
Scalene Triangle
- No congruent sides
- No congruent angles
Hook: Scalene, like scaly, could describe a hideous monster or something else as uneven as a triangle where every side and every angle is different.
Right Triangle
- 1 right (90°) angle, the largest angle in the triangle
- Side opposite the right angle is the hypotenuse, the longest side of the triangle. (Hook: The word hypotenuse reminds us of hippopotamus, a very large animal.)
- The other two sides are called legs.
- A right triangle may be isosceles or scalene.
Area of a Triangle
To find the area of a triangle, use this formula:
Although any side of a triangle may be called its base, it's often easiest to use the side on the bottom. To use another side, rotate the page and view the triangle from another perspective.
A triangle's height (or altitude) is represented by a perpendicular line drawn from the angle opposite the base to the base. Depending on the triangle, the height may be inside, outside, or on the legs of the triangle. Notice the height of the second triangle: We extended the base to draw the height perpendicular to the base. The third triangle is a right triangle: One leg may be its base and the other its height.
Hook: Think of a triangle as being half a rectangle. The area of that triangle (as well as the area of the largest triangle that fits inside a rectangle) is half the area of the rectangle.
Example: Find the area of a triangle with a 2-inch base and a 3-inch height.
- Draw the triangle as close to scale as you can.
- Label the size of the base and height.
- Write the area formula; then substitute the base and height numbers into it:
- The area of the triangle is 3 square inches.
Triangle Rules
The following rules tend to appear more frequently on tests than other rules. A typical test question follows each rule.
Example: One base angle of an isosceles triangle is 30°. Find the vertex angle.
- Draw a picture of an isosceles triangle. Drawing it to scale helps: Since it is an isosceles triangle, draw both base angles the same size (as close to 30° as you can), and make sure the sides opposite them are the same length. Label one base angle as 30°.
- Since the base angles are congruent, label the other base angle as 30°.
- There are two steps needed to find the vertex angle:
- Add the two base angles together: 30°+ 30°= 60°
- The sum of all three angles is 180°. To find the vertex angle, subtract the sum of the two base angles (60°) from 180: 180°–60° = 120°
- Thus, the vertex angle is 120°.
Check: Add all 3 angles together to make sure their sum is 180°: 30°+ 30°+ 120°=180°
TipThe longest side of a triangle is always opposite the largest angle, and the smallest side is opposite the smallest angle. That is why in an isosceles triangle, the two base angles are equal to and opposite of the two equal sides. This rule implies that the second longest side is opposite the second largest angle, and the shortest side is opposite the smallest angle. Hook: Visualize a door and its hinge. The more the hinge is open (largest angle), the fatter the person who can get through (longest side is opposite); similarly, for a door that's hardly open at all (smallest angle), only a very skinny person can get through (shortest side is opposite). |
Example: In the triangle shown at the right, which side is the shortest?
- Determine the size of A, the missing angle, by adding the two known angles, and then subtracting their sum from 180°: Thus, A is 44°.
- Since A is the smallest angle, side a (opposite A) is the shortest side.
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