# Angles in Shapes Rating

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Angles in Shapes
CCSS.MATH.CONTENT.4.MD.C.5

## Angles of Polygons - Introduction

Angles in two-dimensional shapes, such as polygons, are an essential aspect of geometry that lay the foundation for many mathematical and real-world applications. This article explores the basics of angles of polygons, their properties, calculations, and much more.

## Basics of Polygons – Definitions

Let's talk about polygons. They come in two main types: regular polygons and irregular polygons. Polygons are shapes with straight sides. They can be regular (where everything's the same) or irregular (where everything can be different).

• Regular Polygons: Imagine a shape where all the sides are the same length, and all the angles are the same size. That's a regular polygon! These shapes are like perfect team players; everything matches and fits together nicely. If you've ever seen a stop sign, you've seen a regular polygon. It's a perfect fit in many puzzles, designs, and games!

• Irregular Polygons: Now, think of a shape where the sides and angles don't match. That's an irregular polygon! These shapes are like wild adventurers; no two sides or angles need to be the same. They're really useful for drawing things that you find in nature, like mountains or rivers.

Whether you're drawing, building with blocks, or solving fun puzzles, understanding polygons helps you see the world in a whole new way!

## Interior and Exterior Angles in Shapes

There are two types of angles that you can find in shapes.

Interior angles are the angles found inside a polygon. They vary depending on the number of sides.

Exterior angles are the angles formed outside the polygon by extending one side of the polygon. For any polygon, the sum of exterior angles is always 360 degrees.

We will explore these angles in more detail shortly.

### Exploring the Interior Angles of a Shape

Have you ever wondered how the angles inside different shapes add up? Let's explore!

Triangles: Triangles are like a slice of pie with three sides, and the angles inside always add up to 180 degrees. No matter how you change the shape of the triangle, the angles inside will always make a straight line if you put them together. Quadrilaterals: Quadrilaterals have four sides, like a piece of paper or a television screen. If you add up all the corners' angles, you will get 360 degrees, just like a full turn! Why is this useful to know? Well, the total sum of interior angles in different shapes helps us solve puzzles and mysteries. For example, if you know two angles in a triangle, you can find the third one just by subtracting the angles you know from 180! Similarly, if you know three angles in a quadrilateral, you can find the missing angle by subtracting the known angles from 360. ### Exploring the Exterior Angles of a Shape

Have you ever looked at the angles outside a shape like a triangle or a star? These outside angles are called exterior angles, and they have some surprising secrets!

Exterior Angles: These are the angles on the outside edges of shapes, and they're like the little twists or turns between the sides. They're essential in all kinds of different shapes, and they can help us understand how the shape bends and moves.

So why do these outside angles matter? Imagine you have a paper cut-out of a shape, and you want to fold it into a 3D object. Understanding how these angles work helps you figure out how to fold it properly. It's like a map to folding shapes!

While there is some remembering to do when figuring out interior angles of different shapes, exterior angles always add up to 360 degrees, regardless of the shape!

## Common Polygons and Their Angles

Let's talk about some of the commonly found shapes like triangles, squares, pentagons, and hexagons. Did you know that they have angles that add up to special numbers? It's like they have their own rules, and it's so much fun to find out what they are!

Triangles: These three-sided shapes have angles that add up to 180 degrees.

Quadrilaterals: These are like squares or rectangles, and their angles add up to 360 degrees.

Pentagons: With five sides, pentagons have angles that add up to 540 degrees.

Hexagons: Have you ever looked at a beehive's honeycomb? Those are hexagons, and their angles add up to 720 degrees!

And guess what? If a shape has sides that are all the same length, like a perfect square, the angles behave in a special way called congruent. That means they're all the same, like twins!

Shape Number of Sides and Angles Total Sum of Interior Angles Total Sum of Exterior Angles
Triangle 3 180 degrees 360 degrees
Quadrilateral 4 360 degrees 360 degrees
Pentagon 5 540 degrees 360 degrees
Hexagon 6 720 degrees 360 degrees

Understanding these angle rules is like knowing a shape's hidden language. Whether you're drawing pictures, building something, or just looking at the world around you, knowing these angle secrets makes everything more exciting! So next time you see one of these shapes, you'll know a little more about what makes it special.

## Angles of Polygons – Summary

Polygons are 2D shapes categorized as regular (with congruent sides and angles) or irregular. They have interior angles whose sum varies by shape: triangles 180°, quadrilaterals 360°. Exterior angles in all polygons sum up to 360°. Polygons with equal sides also have equal angles. Understanding these angles is fundamental for geometry and real-world applications.

What are angles in shapes?
What is the sum of interior angles in a triangle?
How can I find the measure of one interior angle in a regular polygon?
Why is understanding angles in polygons important?

## Angles in Shapes exercise

Would you like to apply the knowledge you’ve learned? You can review and practice it with the tasks for the video Angles in Shapes.
• ### What are the missing angles?

Hints

Remember, this box in the bottom right corner means there is a right-angle, which is always 90°.

Remember, all of the internal angles of a triangle equal 180°.

Remember, all of the internal angles of a quadrilateral equal 360°.

Remember, opposite angles in parallelograms are equal.

Solution

Here are the missing angles:

• The parallelogram had two missing angles of 75° each.
• The triangle had one missing angle of 90° and one missing angle of 65°.
________________________________________________________

Parallelogram

• The first part of the roller coaster was made of a parallelogram.
• We know that parallelograms are quadrilaterals, so opposite angles are equal and all internal angles add up to 360°.
• We are given two angles of 105°. 105 + 105 = 210.
• We then subtract 210 from 360, which equals 150.
• We then divide 150 by 2, to get 75.
• Therefore each of the missing angles in the parallelogram are 75°.
Triangle
• We know that all internal angles of a triangle add up to 180°.
• We are given one angle of 25°.
• We can then see a square box in the corner of the triangle meaning it is a right-angle. Right-angles always equal 90°.
• We then add 25 + 90 = 115.
• We then subtract 115 from 180, which equals 65.
• The missing angle is 65°.
• ### What are the missing angles?

Hints

Remember, the internal angles of a triangle add up to 180°.

Remember, in a square or parallelogram, opposite angles are equal.

Solution

Here are the missing angles for each shape:

Square

• A square is a quadrilateral, therefore all internal angles must add up to 360°.
• All internal angles of a square are also right-angles, meaning that the missing angle must be 90°.
• 90° + 90° + 90° + 90° = 360°.
Scalene triangle
• All angles are different in a scalene triangle but all internal angles still add up to 180°.
• 80° + 40° = 120°.
• 180° - 120° = 60°.
• The missing angle is therefore 60°.
Parallelogram
• A parallelogram is a quadrilateral, therefore all internal angles add up to 360°.
• We also know that opposite angles are equal.
• 60° + 60° = 120°.
• 360° - 120° = 240°.
• 240° $\div$ 2 = 120°.
• The missing angle is therefore 120°.
Isosceles triangle
• An isosceles triangle has two equal angles and all internal angles add up to 180°.
• We were given the two equal angles of 80°.
• 80° + 80° = 160°
• 180° - 160° = 20°
• The missing angle is 20°.

• ### Find the missing angles.

Hints

Remember, all right-angles are 90°.

Add up the angles given in the triangles and parallelograms and figure out what is missing.

Remember:

• all internal angles in a triangle add up to 180°.
• all internal angles in a parallelogram (a quadrilateral) add up to 360°.

Solution

Here are the correctly highlighted angles:

• The shape made here is a rectangle.
• All angles are right-angles, so are all 90° and are highlighted in green.
Rafters
• Here is an equilateral triangle.
• 60° + 60° = 120°
• 180° - 120° = 60°
• The missing angle is 60° and is highlighted in blue.
Cheese picture
• Here is another equilateral triangle.
• 60° + 60° = 120°
• 180° - 120° = 60°
• The missing angle is 60° and is highlighted in blue.
Rectangular picture frame
• The shape here is a rectangle.
• All angles are right-angles so are all 90° and are highlighted in green.
Parallelogram picture frame
• Opposite angles are equal in a parallelogram.
• 150° + 150° = 300°.
• 360° - 300° = 60°.
• 60° $\div$ 2 = 30°
• The missing angles are therefore both 30° and are highlighted in yellow.
Bookcase
• Here we have a right-angled triangle.
• One missing angle is a right-angle and is 90° so is highlighted in green.
• 60° + 90° = 150°.
• 180° - 150° = 30°.
• The top angle is therefore 30° and is highlighted in yellow.

• ### Can you find the mistakes?

Hints

Do any of the shapes have right-angles?

Remember, in parallelograms:

• All internal angles add up to 360°
• Opposite angles are equal

All internal angles in a triangle add up to 180°.

Solution

Here are the angles that should have been highlighted:

• In the scalene triangle we were given two angles of 48° and 62°. 48 + 62 = 110. 180 - 110 = 70 so the angle was 70° and this is correct.
• In the parallelogram we were given two angles of 125°. 125 + 125 = 250. 360 - 250 = 110. 110 $\div$ 2 = 55 so the angles should have been 55°. The two 45° angles are incorrect.
• In the isosceles triangle we were given two angles of 73°. 73 + 73 = 146. 180 - 146 = 34 so the angle is 34°. 33° is incorrect.
• In the right-angled triangle we were given an angle of 15° and a right-angle which is 90°. 90 + 15 = 105. 180 - 105 = 75 so the angle is 75°. 65° is incorrect.
• ### What is the missing angle?

Hints

Can you remember what all internal angles of a triangle add up to?

The internal angles of a triangle always add up to 180°.

If you add the angles given, what is the difference between that and 180°?

Solution

The missing angle was 35°.

• All internal angles of a triangle add up to 180°.
• 80 + 65 = 145.
• 180 - 145 = 35.
• Therefore the missing angle is 35°.
• ### What are the angles that Nico and Nia saw?

Hints

A right-angle is always 90°.

The internal angles of a triangle add up to 180°.

An isosceles triangle has two equal angles.

Solution

The first roller coaster that Nia and Nico saw had a triangle shape in it. They measured two angles of 72° and 41° so the final angle must be 67°.

• 72 + 41 = 113.
• 180 - 113 = 67
• The missing angle is therefore 67°.
They then saw one that had a parallelogram shape in it. Two angles measured 84° each so the other two must measure 96° each.
• A parallelogram is a quadrilateral so all inside angles add up to 360°.
• 84 + 84 = 168
• 360 - 168 = 192
• 192 $\div$ 2 = 96
• The missing angles are therefore 96° each.
They then saw a roller coaster that had a right-angled triangle in it. One angle measured 29° so the missing angle must measure 61°.
• One angle is a right angle so is therefore 90°.
• 90 + 29 = 119
• 180 - 119 = 61
• Therefore the missing angle is 61°.
Finally they saw a roller coaster with an isosceles triangle. The top angle measured 18° so the bottom two angles must measure 81° each.
• Isosceles triangles have two equal angles.
• 180 - 18 = 162
• 162 $\div$ 2 = 81
• Therefore the missing angles are 81° each.