Triangles and Pythagorean Theorem
4.14 TRIANGLES
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
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