Geometry can sometimes be a tricky subject to grasp, especially when it comes to understanding interior angles. In this article, we will explore the meaning of interior angles, their types, and provide formulas and examples to help you better comprehend this concept.
What Are Interior Angles?
The term "interior angles" in geometry can be used in two different contexts. First, interior angles are formed when two parallel lines are cut by a transversal. Second, interior angles are formed inside a polygon (a shape). Let's dive into each of these cases to understand interior angles more clearly.
Interior Angles Formed When Two Parallel Lines Are Cut By a Transversal
The angles that lie in the area enclosed between two parallel lines cut by a transversal are also called interior angles. Take a look at the image below:
In the image above, lines L1 and L2 are parallel, and L is the transversal that intersects these lines. Angle 1, angle 4, angle 2, and angle 3 are all examples of interior angles.
What Are the Types of Interior Angles?
Now that we understand the concept of interior angles formed by a transversal, let's explore the different types of interior angles in this context.
Same Side Interior Angles (Co-interior Angles)
Same side interior angles, also known as co-interior angles, are pairs of non-adjacent angles that lie on the same side of the transversal. The sum of two co-interior angles is 180°, making them supplementary. In the image below, angle 1 and angle 4 form a pair of same side interior angles, as do angle 2 and angle 3.
Alternate Interior Angles
Alternate interior angles are angles that lie on opposite sides of the transversal. The Alternate Interior Angles Theorem states that if two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent. In the image above, angle 1 is congruent to angle 3, and angle 2 is congruent to angle 4.
Interior Angles of a Polygon
Moving on from interior angles formed by a transversal, let's explore interior angles in polygons. Interior angles of a polygon refer to the angles that lie inside a polygon formed by its sides. The number of interior angles in a polygon is always equal to the number of sides. Take a look at the image below for an example:
In the image above, you can see the interior angles of a triangle and a pentagon. Notice that the sum of the interior angles of a triangle always adds up to 180°. This holds true for all triangles. Additionally, if we bisect the angles of a triangle, the angle bisectors will meet at a point called the incenter.
Classifying Triangles Based on Interior Angles
Based on their interior angles, triangles can be classified into different types. Let's explore these classifications:
- Acute Triangle: A triangle in which all three interior angles are acute (measuring less than 90°) is called an acute triangle.
- Obtuse Triangle: A triangle that has one interior angle as an obtuse angle (measuring more than 90°) and the other two angles as acute is known as an obtuse triangle.
- Right Triangle: A triangle in which one interior angle is a right angle (measuring exactly 90°) and the other two angles are acute is called a right-angled triangle.
- Equiangular Triangle: A triangle in which all the angles are of equal measure is called an equiangular triangle. If all interior angles are equal and measure 60°, the triangle is also an equilateral triangle.
The Sum of Interior Angles Formula
To understand the concept of the sum of interior angles in a polygon, let's start with the simplest polygon, a triangle. According to the angle sum theorem, the sum of all three interior angles of a triangle is equal to 180°. We can generalize this formula to calculate the sum of interior angles for any polygon with "n" sides.
The sum of interior angles of a polygon can be calculated using the formula:
Sum of interior angles = (n - 2) * 180°In this formula, "S" represents the sum of interior angles, and "n" represents the number of sides of the polygon.
Let's apply this formula to calculate the sum of interior angles for a triangle:
S = (n - 2) * 180° S = (3 - 2) * 180° S = 1 * 180° S = 180°As you can see, the sum of interior angles for a triangle is indeed 180°. We can use the same formula to calculate the sum of interior angles for other polygons as well.
Here is a table illustrating how to use the sum of interior angles formula to calculate the sum of interior angles for different polygons:
| Polygon | Number of Sides (n) | Sum of Interior Angles |
|---|---|---|
| Triangle | 3 | 180° |
| Square | 4 | 360° |
| Pentagon | 5 | 540° |
| Hexagon | 6 | 720° |
| ... | ... | ... |
Finding an Unknown Interior Angle
To find an unknown interior angle of a polygon, we can use the "Sum of Interior Angles Formula." Let's consider an example to understand how this works:
In the above image, we have a quadrilateral with one missing interior angle, denoted as angle x. To find the value of angle x, we can use the sum of interior angles formula for a quadrilateral, which is 360°.
From the given interior angles, we know that two of them are right angles, measuring 90° each. Therefore, we can write the equation:
90° + 90° + 96° + x = 360°
Simplifying the equation, we find:
x = 360° - 90° - 90° - 96° x = 84°
Hence, the missing interior angle x in the quadrilateral is 84°.
Interior Angles of Regular Polygons
When all the sides and angles of a polygon are congruent, we refer to it as a regular polygon. Let's take a look at some examples of regular polygons:
As we know, the formula to calculate the sum of interior angles for any polygon with "n" sides is (n - 2) * 180°. Since all the interior angles of a regular polygon are equal, we can calculate each interior angle by dividing the sum of the angles by the number of sides.
The formula to calculate each interior angle of a regular polygon is:
Each interior angle of a regular polygon = (180° * (n - 2)) / nIn this formula, "n" represents the number of sides of the polygon.
For example, let's calculate the interior angle of a regular hexagon. We know that a hexagon has 6 sides (n = 6). Applying the formula, we get:
Each interior angle of a regular hexagon = (180° * (6 - 2)) / 6 = (180° * 4) / 6 = 120°Therefore, each interior angle of a regular hexagon is equal to 120°.
Here is a table illustrating how to use the formula to calculate the interior angle of regular polygons:
| Regular Polygon | Sides (n) | Sum of Interior Angles (S) | Measure of Each Interior Angle |
|---|---|---|---|
| Triangle | 3 | 180° | 60° |
| Square | 4 | 360° | 90° |
| Pentagon | 5 | 540° | 108° |
| Hexagon | 6 | 720° | 120° |
| ... | ... | ... | ... |
Facts about Interior Angles
- The sum of interior angles in any polygon is always a multiple of 180°.
- The sum of interior angles in a convex polygon is always less than 180° multiplied by the number of sides.
- The sum of interior angles in a concave polygon may exceed 180° multiplied by the number of sides, as it contains one or more angles that fold inward.
- The greater the number of sides in a polygon, the larger the sum of its interior angles.
Conclusion
In this article, we've explored the definitions, types, formulas, and examples related to interior angles in geometry. By understanding the concept of interior angles, you'll be better equipped to solve problems and tackle geometric puzzles. Practice and explore more examples to strengthen your knowledge and confidence in this fascinating field of study.
Solved Examples on Interior Angles
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Five interior angles of an irregular hexagon are: 100°, 130°, 95°, 125°, and 110°. What is the value of its sixth angle?
Solution: We know that the sum of angles of a hexagon is 720°. Let the unknown angle be x.
100° + 130° + 95° + 125° + 110° + x = 720° 560° + x = 720° x = 720° - 560° x = 160°Therefore, the value of the unknown angle is 160°.
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What is the sum of interior angles of a polygon with 15 sides?
Solution: The formula to calculate the sum of interior angles of any polygon with "n" sides is Sum = (n - 2) * 180°. Given, n = 15.
Sum = (15 - 2) * 180° = 13 * 180° = 2340°Therefore, the sum of interior angles of a polygon with 15 sides is 2340°.
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What will be the measure of each angle for a regular 15-gon?
Solution: If the 15-sided polygon is regular, then the measure of each angle will be the total sum of all angles divided by the number of sides. So, the measure of each angle = (n - 2) * 180° / n = 2340° / 15 = 156°. Hence, the measure of each angle in a 15-sided regular polygon is 156°.
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In the figure given below, if m angle 1 = 105°, what is the value of m angle 3?
Solution: In the figure above, angle 1 and angle 3 are alternate interior angles. By the property of alternate interior angles, these angles will be equal. Therefore, the measure of angle 3 will also be 105°.
m angle 3 = 105° -
In the figure given below, if m angle 1 = 100°, what will be the measure of: i) angle 4 ii) angle 3
Solution: i) angle 1 and angle 4 are co-interior angles. The sum of co-interior angles is 180°. Let the measure of angle 4 be x.
100° + x = 180° x = 180° - 100° x = 80°Therefore, the measure of angle 4 is 80°.
ii) angle 1 and angle 3 are alternate interior angles. As per the property of alternate interior angles, they are equal.
m angle 1 = m angle 3 = 100°
Practice Problems on Interior Angles
Here are some practice problems to test your understanding of interior angles:
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Find the missing interior angle in the given quadrilateral:
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Calculate the sum of interior angles for a polygon with 8 sides.
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Determine the measure of each angle for a regular 9-gon.
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In the figure below, if m angle 1 = 110°, find the value of m angle 2:
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In the figure given below, if m angle 1 = 105°, find the measure of angle 4:
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Frequently Asked Questions on Interior Angles
Q1: What are interior angles?
A1: Interior angles are angles formed inside a polygon or between two parallel lines cut by a transversal.
Q2: What is the sum of interior angles in a triangle?
A2: The sum of interior angles in a triangle is always 180°.
Q3: What is the formula to calculate the sum of interior angles in a polygon?
A3: The formula is (n - 2) * 180°, where "n" represents the number of sides of the polygon.
Q4: How are alternate interior angles related to each other?
A4: If two parallel lines are cut by a transversal, the pairs of alternate interior angles are congruent.
Q5: What is an acute triangle?
A5: An acute triangle is a triangle in which all three interior angles measure less than 90°.
Q6: Are the interior angles of a regular polygon congruent?
A6: Yes, the interior angles of a regular polygon are congruent.
Q7: How can we find an unknown interior angle of a polygon?
A7: Use the sum of interior angles formula and the known angles to solve for the unknown angle.
Remember, practice makes perfect. So, be sure to solve more examples and practice problems to strengthen your understanding of interior angles. Geometry can be fascinating once you grasp these concepts. Happy learning!