# Angles

On this page we have a variety of activities that look at angles along a line, interior and exterior angles. If the activity requires an applet to work, just click on the image and you will be taken directly to the web page required.

**Calculate Angles**

A short activity looking at angles along a line and around a point and finding connections between the two.

What do we know about a and b? How do we know this?

Which angle fact might you need to use when answering this question?

Which angles are already given? How can we use this to calculate unknown angles?

**Triangle Angle Theorems**

Interact with the applet below for a few minutes. which looks at the sum of interior angles in a triangle. Then, answer the questions that follow.

1) What geometric transformations took place in the applet above? 2) When working with the triangle’s interior angles, did any of these transformations change the measures of the blue or green angles? 3) From your observations, what is the sum of the measures of the interior angles of *any triangle? * 4) When working with the triangle’s exterior angles, did any of these transformations change the measures of the green or gray angles? 5) From your observations, what is the sum of the measures of the exterior angles of *any triangle? *

**Angles in a Triangle**

This is a practical activity that allows pupils to self discover what the angles in a triangle add up to, moving onto quadrilateral as well.

What’s the same and what’s different about the four types of triangle?

What do the three interior angles add up to? Would this work for all triangles?

Does the type of triangle change anything?

Does the size of the triangle matter?

The same can be done for a quadrilateral as well

**Vertically Opposite Angles**

This activity gives pupils an opportunity ti discover that opposite angles are equal rather than just being told the fact, make sure they do this activity as few time to find out this is always the case.

What sentences can we write about vertically opposite angles in relation to other angles?

How can we find the missing angle?

Is there more than one way to find this angle?

#### Angles around a point

There are 360^{0} in a full turn, from this following diagram, what can you conclude about the angles in the polygons drawn?

**Investigation – Angles in polygons**

Here is a lovely worksheet that you can give to pupils to allow them to find about rules and facts about interior angels in polygons. Its really nice as it is accessible to all abilities.

EXTENSION QUESTION: Can you write a general rule relating the number of sides of a polygon to its interior angle sum?

#### 30^{0}, 60^{0}, 90^{0} triangles

A nice activity that requires pupils to split polygons into 30, 60 90 triangles and then use this to work out the size of the angles in the polygons drawn. The first sheet is splitting into 2 triangles.

Now have a go at these ones which you need to split into 3 triangles

Finally 4 triangles

**First Letter Prompts**

Can you work out what all these Maths facts are? HINT: the first on is vertically opposite angles are always equal. Really nice activity to consolidate their learning.

**Exterior Angles of Polygons**

Click the “▶” button. Observe the angles turned by the car at the corners. Drag the green points. Check the “Show Measurement” box to see the measurements of the angles. Once pupils have had a chance to play around with this applet, then start asking questions like the ones below.

**1. **What is a + b + c + d + e, the total angles turned by the car?

** **2. a, b, c, d, e are called the exterior angles of the convex pentagon.** **

3. In general, what is the sum of all exterior angles of a convex polygon?

4. How would you modify the result if the pentagon is NOT convex?

**Descending Angles**

A short problem to ask your pupils as it really gets them thinking about angels in triangles

The six angles of two different triangles are listed in decreasing order.

The list starts: 115°, 85°, 75°, 35°

What is the smallest angle in the list?

**Polygon Cradle**

The figure shows a regular pentagon *PQRST* together with three sides *XP*, *PR*, *RU* of a regular hexagon with vertices *PRUVWX*. What is the size of angle *SRU*?

**Pentagon Ring**

A short problem that allows pupils to investigate pentagons and tessellation.

Equal regular pentagons are placed together to form a ring.

The diagram shows the first three pentagons.

How many pentagons are needed to complete the ring?

**Knowledge Organiser**

A simple worksheet that pupils can put in their books and refer back to when tackling angles problems. Can you get your pupils to create their own?

**First Letter Prompts**

**Descending Angles**

Since the three angles in each triangle must add up to 180 degrees, 115 cannot be in the same triangle as 85 or 75, so 85 and 75 must be in the same triangle. 75+85=160 so the remaining angle in this triangle is 180 – 160 = 20 degrees.

This leaves the other two angles in the list together (115 and 35) which sum to give 150, so the other angle not included in the list is 180 – 150 = 30 degrees.

Therefore the complete list is 115, 85, 75, 35, 30, 20, so the smallest angle is 20 degrees.

**Polygon Cradle**

48∘

As *P**Q**R**S**T* is a regular pentagon, each of its internal angles is 108∘. The internal angles of the quadrilateral *P**R**S**T* add up to 360∘ and so by symmetry ∠*P**R**S*=∠*R**P**T*=12(360∘−2×108∘)=72∘. Each interior angle of a regular hexagon is 120∘, so ∠*P**R**U*=120∘.

Therefore ∠*S**R**U*=∠*P**R**U*−∠*P**R**S*=120∘−72∘=48∘.

**Pentagon Ring**

Each pentagon has a turn of 36^{o} to pack against its neighbour.

Therefore it will take 10 (x36^{o}) turns to complete a full-turn, i.e. 10 pentagons