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3.2. Sum of the Measures of the Exterior Angles of a Polygon
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Today, we will start by discussing what exterior angles are. Does anyone know how they differ from interior angles?
I think exterior angles are the angles outside the polygon when a side is extended.
Exactly! When you extend a side of a polygon, the angle formed outside is an exterior angle. Can anyone give an example of a polygon and say how many exterior angles it has?
A triangle has three exterior angles!
Great! And if we think about a pentagon, how many exterior angles does that have?
Five exterior angles.
Right! Now, here’s a memory aid: remember the phrase 'exterior equals 360'! This helps us remember that the total of all exterior angles of any polygon is always 360°. Can someone summarize what we discussed?
Exterior angles are outside the polygon, and their total measure is always 360°.
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Now that we understand exterior angles, let’s learn how to calculate unknown angles. If I tell you the first angle is 90°, the second is 50°, and the third is 110°, how can we find the unknown fourth angle?
We can add the known angles and subtract from 360°.
Exactly! So we calculate it like this: 90° + 50° + 110° = 250°. To find the unknown angle, we do 360° - 250°.
That means the fourth angle is 110°!
Good work! Always remember this formula when calculating unknown angles. Who can summarize what we just learned?
To find an unknown exterior angle, add known angles and subtract from 360°.
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Create a free accountFor this activity, I want you each to draw a polygon on the floor with chalk. As you walk around and turn at each vertex, measure your turns. What do you observe?
When I walked around my pentagon, all the turns I made added up to 360°!
Me too! It’s the same for my hexagon.
Excellent observations! This is a clear demonstration of how the sum of exterior angles is always 360° for any polygon. Who can remind us why this is significant?
It helps us understand the shape and size of polygons, especially when calculating angles.
Exactly right! Now let’s summarize today’s lessons.
Exterior angles add up to 360°, and we can calculate any unknown angles using this rule.
Overview
Medium Summary
Exterior angles of a polygon provide useful insights into the nature of interior angles and help in understanding polygon geometry. This section demonstrates that regardless of the number of sides a polygon has, the total measure of its exterior angles remains constant at 360°.
Detailed Summary
Sum of the Measures of the Exterior Angles of a Polygon
In this section, we explore the concept of exterior angles of polygons. An exterior angle is formed when you extend one side of the polygon. The section highlights that when you walk around a polygon and measure each turn you make (the exterior angle), the sum of these angles amounts to 360°. For instance, if you take a pentagon, as you walk along its edges and turn at each vertex, you will find that the cumulative measure of the turns is 360°. This relationship holds true for any polygon, regardless of the number of sides.
The section includes examples and problems prompting students to calculate unknown exterior angles using the established formula. Moreover, it emphasizes activities such as drawing polygons to physically demonstrate the concept and its universal applicability. This understanding lays the foundation for further exploration into polygon properties in geometry.
Similar Question: (Have to draw image manually by teacher)
Example : Find the number of sides of a regular polygon whose each exterior angle has a measure of 60°.
Solution: Total measure of all exterior angles = 360°
Measure of each exterior angle = 60°
Therefore, the number of exterior angles = .
The polygon has 6 sides.
Reference YouTube Videos
Audio Book
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Create a free accountOn many occasions a knowledge of exterior angles may throw light on the nature of interior angles and sides.
Detailed Explanation
Exterior angles are the angles formed when a side of a polygon is extended. They can provide insights into the properties of the polygon's interior angles. Understanding exterior angles helps us grasp how they relate to the overall shape of the polygon.
Examples & Analogies
Think of a door frame as a polygon. When you open a door, the angle it makes with the wall extends outward, which represents an exterior angle. Observing how these angles behave when the door is opened at different positions can help us understand the shape of the room.
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Create a free accountDO THIS: Draw a polygon on the floor, using a piece of chalk. In the figure, a pentagon ABCDE is shown.
Detailed Explanation
To visualize and understand exterior angles, one can physically draw a polygon. By labeling the vertices, one can begin to visualize the turning angles that occur as you navigate from one vertex to another.
Examples & Analogies
Imagine walking along a track in a park that forms a pentagon. Each time you turn a corner, you essentially create an exterior angle. This real-life movement helps to connect the abstract concept of angles with a physical experience.
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Create a free accountThus, m∠1 + m∠2 + m∠3 + m∠4 + m∠5 = 360°. This is true whatever be the number of sides of the polygon.
Detailed Explanation
When you walk around the polygon, the sum of all the exterior angles equal 360 degrees, regardless of how many sides the polygon has. It's important to note that this property is consistent for all polygons.
Examples & Analogies
Think of a race track. No matter how many bends (sides) the track has, when you complete a full lap, you would have turned a total of 360 degrees, similar to how exterior angles add up in a polygon.
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Create a free accountExample 1: Find measure x in Fig 3.3. Solution: x + 90° + 50° + 110° = 360° (Why?) x + 250° = 360° x = 110°.
Detailed Explanation
In this example, we are provided with the measures of some exterior angles and tasked with finding the missing angle, x. By setting up the equation, we recognize that the total equals 360 degrees, enabling us to solve for x.
Examples & Analogies
Imagine calculating the remaining distance on your journey around a park's trail by knowing how far you've already walked around it. By understanding what's left based on a total (360 degrees), you can easily find the remaining distance.
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Create a free accountTRY THESE: 1. What is the sum of the measures of its exterior angles x, y, z, p, q, r? 2. Is x = y = z = p = q = r? Why? 3. What is the measure of each? (i) exterior angle (ii) interior angle.
Detailed Explanation
These exploration questions encourage students to apply their understanding of exterior angles to regular polygons, where all sides and angles are equal. Students must think critically about the relationships between angles and how they can derive the measures of each exterior angle.
Examples & Analogies
Consider a perfectly shaped pizza. If it’s sliced into equal pieces, each slice represents an exterior angle. Discussing the same angle for each piece helps to reinforce the idea of equality in a regular polygon.
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Create a free accountExample 2: Find the number of sides of a regular polygon whose each exterior angle has a measure of 45°. Solution: Total measure of all exterior angles = 360°. Measure of each exterior angle = 45°. Therefore, the number of exterior angles = 360/45 = 8. The polygon has 8 sides.
Detailed Explanation
Here, we are using the insight that the sum of all exterior angles is constant (360 degrees). By knowing the measure of a single exterior angle, we can easily find out how many sides the polygon has.
Examples & Analogies
Imagine arranging chairs in a circle for a party. If each gap between chairs represents an exterior angle, and you know how much space each chair occupies (like knowing the angle), you can calculate how many chairs you can fit around without exceeding your space.
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Key Concepts
Examples
Step-by-step examples to apply the section's ideas and test your understanding.
In a pentagon, if you measure each exterior angle at every vertex, the total will be 360°.
If three angles of a polygon are known as 100°, 80°, and 60°, then the unknown angle can be found by 360° - (100° + 80° + 60°) = 120°.
Memory Aids
Interactive tools to help you remember key concepts