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Watch Video on Area of a Triangle - Pre Algebra Help
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Triangles
The sum of the measures of the angles in any triangle is 180°.
Equilateral Triangles
The three sides of an equilateral triangle (a, b, c) are equal in length.
The three angles are also equal and they each measure 60° (x = y = z = 60).
Isosceles Triangles
An isosceles triangle is a triangle with two sides of equal length ( m = n ).
The angles opposite the equal sides are also equal ( x = y ).
Right Triangles and the Pythagorean Theorem
A right triangle is a triangle with a right angle. ( Note that the other two angles in a right triangle are complementary angles. ) You can get a lot of information from figures that contain right triangles. This information frequently involves the Pythagorean theorem: The square of the length of the hypotenuse of a right triangle is equal to the sum of the squares of the lengths of the other two sides.
The hypotenuse is the longest side of the triangle and is opposite the right angle.
The other two sides are usually referred to as legs. In the figure above :
* A-B is the hypotenuse with length c.
* B-C and A-C are the two legs, with lengths a and b, respectively.
* The Pythagorean theorem leads to the equation:
a² + b² = c²
* If you know the lengths of any two sides, you can use the Pythagorean Theorem to find the length of the third side.
30˚ - 60˚ - 90˚ Triangles
The lengths of the sides of a 30 60 90 triangle are in the ratio of 1: Root 3 : 2, as shown in the figure:
Short leg = x
Long leg = x Root 3
Hypotenuse = 2x
If you know the lengths of any one side, you can find the lengths of the other two sides.
For instance, if you know the length of the short leg is 1, then the length of the hypotenuse is 2, and the Pythagorean Theorem gives you the length of the longer leg:
c² = a² + b²
c = 2 , b = 1
2² = a² + 1
4 = a² + 1
3 = a ²
Root 3 = a
45ˇ - 45˚ - 90˚ Triangles
The lengths of the sides of a 45 - 45 - 90 triangle are in the ratio of 1 : 1 : Root 2 , as shown in the
figure below. To verify this ratio when the equal sides are of length 1, apply the Pythagorean Theorem to find the length of the hypotenuse:
c² = a² + b²
a = 1 , b = 1
c² = 1² + 1²
c² = 1 + 1
c² = 2
c = Root 2
Congruent Triangles
Congruent triangles are triangles that have the same size and shape.
In the figure, each side of [ ] ABC has the same length as the corresponding side of [ ] DEF.
AB = DE = r
BC = EF = s
CA = FD = t
Each angle of [ ] ABC is also equal to its corresponding angle is [ ] DEF.
Two triangles are congruent if any of the following is true:
* Each pair of corresponding sides has the same length.
* Two pairs of corresponding sides each have the same length, and the angles formed by these sides
have the same measure.
* One pair of corresponding sides have the same length, and two pairs of corresponding angles each
have the same measure.