Introduction to interior angle sum theorem.
Any plane closed figures formed by straight lines are called polygons. According to the number of lines used to form the polygon, the polygon gets its name. If there are three lines, it called a triangle. To get a closed figure, we need minimum three lines. If there are four lines in the polygon, it is called as a quadrilateral. We know that the sum of the interior angles of a triangle is 1800. Similarly the sum of the interior angles of a quadrilateral is 3600.
A plane figure bounded by straight lines is called a rectilinear figure. A closed plane figure, bounded by at least three line segments is called a polygon. A polygon is named by the number of sides in it, as given below:
1. Convex Polygon: If each angle of a polygon is less than 1800; the polygon is called a convex polygon.
2. Concave Polygon: If at least one angle of a polygon is greater than 1800; it is called a concave polygon.
Unless otherwise stated, a polygon means a convex polygon.
In a polygon of 'n' sides, the sum of the interior angles is equal to (2n – 4) right angles. This is the interior angle sum theorem. Is this topic parts of a circle hard for you? Watch out for my coming posts.
Example Problems on Interior Angle Sum Theorem:
Ex 1: Find the sum of the interior angles of a polygon of 10 sides.
Solution: The sum of the interior angles with n sides = (2n = 4) `xx ` 90.
Here n = 10 sides, = (2 (10) – 4) `xx` 90.
= (20 - 4) x 90
= 16 `xx ` 90.
= 14400.
Ex 2: In a pentagon ABCDE; AB is parallel to ED and angle B = 1400. Find the angles C and D, if |_C: |_D = 5: 6.
Solution : Since, `[AB]/[ED]`` implies` |_A + |_E = 1800.
Given: |_C : |_D = 5 : 6 `implies` if |_C = 5x, |_D = 6x.
Now. |_A + |_B + |_c + |_D + |_E = (2 `xx` 5 – 4)` xx` 900.
`implies ` (|_A + |_E) + 1400 + 5x + 6x = 5400
`implies ` 1800 + 1400 + 5x + 6x = 5400
That is 11x = 5400 – 3200 `implies ` 11x = 2200 and x = `220^0/ 11` = 200.
Therefore |_C = 5x = 5` xx` 200 = 1000 and |_D = 6x = 6 `xx` 200 = 1200.
Ex 3: The sum of the interior angles of a polygon is five times the sum of its exterior angles. Find the number of sides in the polygon.
Solution: Let the number of sides by n.
Given: The sum of the interior angles of the polygon
= 5` xx` the sum of its exterior angle.
Therefore (2n – 4) `xx ` 900 = 5 `xx` `360^0`
(2n - 4) = 5 x 4 [ since 360 / 90 = 4]
2n - 4 = 20
On solving, we get: 2n = 24 and n = 12.
Therefore the required number of sides in the polygon = 12.
Practice Problems on Interior Angles Sum Theorem.
1. The angle of a pentagon are in the ratio 4: 8: 6: 4: 5. Find each angle of the pentagon.
[Ans: 80, 160, 120, 80 and 100]
2. In a polygon there are 5 right angles and the remaining angles are equal to 1950 each.
Find the number of sides in the polygon.
[Ans: 11]
Any plane closed figures formed by straight lines are called polygons. According to the number of lines used to form the polygon, the polygon gets its name. If there are three lines, it called a triangle. To get a closed figure, we need minimum three lines. If there are four lines in the polygon, it is called as a quadrilateral. We know that the sum of the interior angles of a triangle is 1800. Similarly the sum of the interior angles of a quadrilateral is 3600.
A plane figure bounded by straight lines is called a rectilinear figure. A closed plane figure, bounded by at least three line segments is called a polygon. A polygon is named by the number of sides in it, as given below:
1. Convex Polygon: If each angle of a polygon is less than 1800; the polygon is called a convex polygon.
2. Concave Polygon: If at least one angle of a polygon is greater than 1800; it is called a concave polygon.
Unless otherwise stated, a polygon means a convex polygon.
In a polygon of 'n' sides, the sum of the interior angles is equal to (2n – 4) right angles. This is the interior angle sum theorem. Is this topic parts of a circle hard for you? Watch out for my coming posts.
Example Problems on Interior Angle Sum Theorem:
Ex 1: Find the sum of the interior angles of a polygon of 10 sides.
Solution: The sum of the interior angles with n sides = (2n = 4) `xx ` 90.
Here n = 10 sides, = (2 (10) – 4) `xx` 90.
= (20 - 4) x 90
= 16 `xx ` 90.
= 14400.
Ex 2: In a pentagon ABCDE; AB is parallel to ED and angle B = 1400. Find the angles C and D, if |_C: |_D = 5: 6.
Solution : Since, `[AB]/[ED]`` implies` |_A + |_E = 1800.
Given: |_C : |_D = 5 : 6 `implies` if |_C = 5x, |_D = 6x.
Now. |_A + |_B + |_c + |_D + |_E = (2 `xx` 5 – 4)` xx` 900.
`implies ` (|_A + |_E) + 1400 + 5x + 6x = 5400
`implies ` 1800 + 1400 + 5x + 6x = 5400
That is 11x = 5400 – 3200 `implies ` 11x = 2200 and x = `220^0/ 11` = 200.
Therefore |_C = 5x = 5` xx` 200 = 1000 and |_D = 6x = 6 `xx` 200 = 1200.
Ex 3: The sum of the interior angles of a polygon is five times the sum of its exterior angles. Find the number of sides in the polygon.
Solution: Let the number of sides by n.
Given: The sum of the interior angles of the polygon
= 5` xx` the sum of its exterior angle.
Therefore (2n – 4) `xx ` 900 = 5 `xx` `360^0`
(2n - 4) = 5 x 4 [ since 360 / 90 = 4]
2n - 4 = 20
On solving, we get: 2n = 24 and n = 12.
Therefore the required number of sides in the polygon = 12.
Practice Problems on Interior Angles Sum Theorem.
1. The angle of a pentagon are in the ratio 4: 8: 6: 4: 5. Find each angle of the pentagon.
[Ans: 80, 160, 120, 80 and 100]
2. In a polygon there are 5 right angles and the remaining angles are equal to 1950 each.
Find the number of sides in the polygon.
[Ans: 11]
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