Angles in Shapes
Basics on the topic Angles in Shapes
Angles of Polygons  Introduction
Angles in twodimensional shapes, such as polygons, are an essential aspect of geometry that lay the foundation for many mathematical and realworld 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 cutout 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 threesided 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 realworld applications.
Frequently Asked Questions about Angles of Polygons
Transcript Angles in Shapes
“The park is closing soon and we have time for ONE more ride, Nia.” “Let’s go on one of the roller coasters!” "Hmmm(...)WHICH ONE?" “ I want to go on the one with the biggest drop!” We can help Nico and Nia figure out which ride to choose by looking at the inside angles created by the roller coaster rides. "Angles in Shapes" All geometric shapes have angles based on their number of sides. A triangle is made up of three angles(...) and when we add up their measurements, it will ALWAYS equal one hundred eighty degrees. Quadrilaterals have FOUR angles and they will ALWAYS total three hundred and sixty degrees. When we look at the outlines of the roller coaster rides, we can see that they make geometric shapes. We can use what we know about angle measurements in shapes to find the measurement of the missing angle at the top of each ride. Let's look at the Obtuse Oblivion. What do you observe about the shape of this ride? The ride is shaped like an obtuse triangle. One angle measures forty degrees, and the other measures thirty degrees. Let’s think about what we know. We know the measure of two angles and the total triangle will measure one hundred eighty degrees. First, solve this by adding the known angles, then SUBTRACTING the sum from one hundred eighty. Fortyplus thirty equals seventy. One hundred eighty minus seventy equals one hundred ten. The missing angle measures one hundred ten degrees. Next, we’ll check out the Right Wrecker! This one looks intense! What do you notice about the shape and angles created by this ride? This shape is a quadrilateral, which means there are four angles and they measure three hundred sixty degrees in total. We are given the angle measurements for THREE of the angles. We have one hundred ten degrees... and thirty. Even though there isn’t a number here, from the box, we know this one is a right angle. What is the measurement of a right angle? It is ninety degrees. To solve for this missing angle, add the known angles and subtract the sum from three hundred sixty. First, we'll add one hundred ten, and ninety and thirty. What is the sum of these angles? The sum of these angles is two hundred, thirty degrees. Now, subtract the sum from three hundred sixty. What is the measure of the missing angle? One hundred thirty degrees. One more ride to consider(...) the Acute Avalanche! This coaster is made up of TWO shapes. Let’s split them apart and solve for the missing angles in each. What is this first shape? This is a parallelogram, which is a type of quadrilateral. We can solve for the missing angle using what we know about parallelograms. A parallelogram has two sets of opposite angles of equal measure. If THIS angle is one hundred twenty degrees,(...) then THIS angle is also one hundred, twenty degrees. If this angle is sixty degrees... what is the measurement of THIS ONE? The opposite angle would also be sixty degrees. To check our answer, add one hundred twenty, (...) sixty, (...) one hundred twenty, and (...) sixty. These angles total three hundred, sixty degrees. Let's look at the EQUILATERAL triangle. What do you notice about the angles? We aren’t given any of the measurements, but we see they are ALL the SAME size. Since a triangle always has a total of one hundred eighty degrees, we can use that to figure out the measurements of the angles. What number can we add three times to equal one hundred eighty degrees? Sixty! Sixty (...) plus sixty (...) plus sixty equals one hundred eighty. Remember, (...) geometric shapes have angles that have set totals. We can solve for missing inside angle measurements in a couple of ways. One way is to find the sum of angles given and subtract that number from the total degrees in the shape. Or use what you know about a shapes' angles to find the missing degrees. “I think Acute Avalanche looks like it has the BEST drop!” “LET’S GO!!” "Nico, are we there yet?" "Yes, we're at the top!" "HANG ON!!!" "WOOOOOO HOOOOO!"
Angles in Shapes exercise

What are the missing angles?
HintsRemember, this box in the bottom right corner means there is a rightangle, 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.
SolutionHere 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°.
 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 rightangle. Rightangles 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?
HintsRemember, the internal angles of a triangle add up to 180°.
Remember, the internal angles of a quadrilateral add up to 360°.
Remember, in a square or parallelogram, opposite angles are equal.
SolutionHere 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 rightangles, meaning that the missing angle must be 90°.
 90° + 90° + 90° + 90° = 360°.
 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°.
 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°.
 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.
HintsRemember, all rightangles 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°.
SolutionHere are the correctly highlighted angles:
Ladder
 The shape made here is a rectangle.
 All angles are rightangles, so are all 90° and are highlighted in green.
 Here is an equilateral triangle.
 60° + 60° = 120°
 180°  120° = 60°
 The missing angle is 60° and is highlighted in blue.
 Here is another equilateral triangle.
 60° + 60° = 120°
 180°  120° = 60°
 The missing angle is 60° and is highlighted in blue.
 The shape here is a rectangle.
 All angles are rightangles so are all 90° and are highlighted in green.
 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.
 Here we have a rightangled triangle.
 One missing angle is a rightangle 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?
HintsDo any of the shapes have rightangles?
Remember, in parallelograms:
 All internal angles add up to 360°
 Opposite angles are equal
All internal angles in a triangle add up to 180°.
SolutionHere 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 rightangled triangle we were given an angle of 15° and a rightangle which is 90°. 90 + 15 = 105. 180  105 = 75 so the angle is 75°. 65° is incorrect.

What is the missing angle?
HintsCan 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°?
SolutionThe 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?
HintsA rightangle is always 90°.
The internal angles of a quadrilateral add up to 360°.
The internal angles of a triangle add up to 180°.
An isosceles triangle has two equal angles.
SolutionThe 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°.
 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.
 One angle is a right angle so is therefore 90°.
 90 + 29 = 119
 180  119 = 61
 Therefore the missing angle is 61°.
 Isosceles triangles have two equal angles.
 180  18 = 162
 162 $\div$ 2 = 81
 Therefore the missing angles are 81° each.