Triangles and Pythagorean Theorem

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Triangles are three-sided shapes that lie in one plane. Triangles are a type of polygons. The sum of all the angles in any triangle is 180º. Triangles can be classified according to the size of its angles. Some examples are :

Acute Triangles
An acute triangle is a triangle whose angles are all acute (i.e. less than 90°). In the acute triangle shown below, a, b and c are all acute angles. Sample Problem 1:
A triangle has angles 46º, 63º and 71º. What type of triangle is this? Answer: Since all its angles are less than 90°, it is an acute triangle.

Obtuse Triangles
An obtuse triangle has one obtuse angle (i.e. greater than 90º). The longest side is always opposite the obtuse angle. In the obtuse triangle shown below, a is the obtuse angle.

Sample Problem 1:
Is it possible for a triangle to have more than one obtuse angle? Solution:
Step 1: Let the angles of the triangle be a, b and c. Let a be the obtuse angle. Step 2: The sum of all the angles in any triangle is 180º. a + b + c = 180º. If a > 90º then b + c must be less than 90º. Therefore, b and c must be acute angles. Answer: No, a triangle can only have one obtuse angle.

The lengths of the sides of triangles is another common classification for types of triangles. Some examples are equilateral triangles, isosceles triangles and scalene triangles.

Equilateral Triangles
An equilateral triangle has all three sides equal in length. Its three angles are also equal and they are each 60º.
Sample Problem 1:
An equilateral triangle has one side that measures 5 in. What is the size of the angle opposite that side? Solution:
Step 1: Since it is an equilateral triangle all its angles would be 60º. The size of the angle does not depend on the length of the side. Answer: The size of the angle is 60º.

Isosceles Triangles
An isosceles triangle has two sides of equal length. The angles opposite the equal sides are also equal.
Sample Problem 1:
An isosceles triangle has one angle of 96º. What are the sizes of the other two angles? Solution:
Step 1: Since it is an isosceles triangle it will have two equal angles. The given 96º angle cannot be one of the equal pair because a triangle cannot have two obtuse angles. (Refer to obtuse triangle above). Step 2: Let x be one of the two equal angles. The sum of all the angles in any triangle is 180°. x + x + 96° = 180°  2x = 84°  x = 42°

Answer: The sizes of the other two angles are 42º each.
Sample Problem 2:
A right triangle has one other angle that is 45º. Besides being right triangle what type of triangle is this? Solution:
Step 1: Since it is right triangle it will have one 90º angle. The other angle is given as 45º. Step 2: Let x be third angle. The sum of all the angles in any triangle is 180º. x + 90º + 45º = 180°  x = 45º

Step 3: Two of the angles are equal which means that it is an isosceles triangle. Answer: It is also an isosceles triangle.

Scalene Triangles
A scalene triangle has no sides of equal length. Its angles are also all different in size.

4.14.1 Right Triangles
Right triangles are triangles in which one of the interior angles is 90o. A 90o angle is called a right angle. Right triangles are sometimes called right-angled triangles. The other two interior angles are complementary, i.e. their sum equals 90o. Right triangles have special properties which make it easier to conceptualize and calculate their parameters in many cases. The side opposite of the right angle is called the hypotenuse. The sides adjacent to the right angle are the legs. When using the Pythagorean Theorem, the hypotenuse or its length is often labeled with a lower case c. The legs (or their lengths) are often labeled a and b.

Either of the legs can be considered a base and the other leg would be considered the height (or altitude), because the right angle automatically makes them perpendicular. If the lengths of both the legs are known, then by setting one of...
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