# Angles, Polygons and Geometrical Proof - Stage 3

### Square It

##### Stage: 3 and 4 Challenge Level:

Players take it in turns to choose a dot on the grid. The winner is the first to have four dots that can be joined to form a square.

### Square Coordinates

##### Stage: 3 Challenge Level:

A tilted square is a square with no horizontal sides. Can you devise a general instruction for the construction of a square when you are given just one of its sides?

### Triangles in Circles

##### Stage: 3 Challenge Level:

How many different triangles can you make which consist of the centre point and two of the points on the edge? Can you work out each of their angles?

### Subtended Angles

##### Stage: 3 Challenge Level:

What is the relationship between the angle at the centre and the angles at the circumference, for angles which stand on the same arc? Can you prove it?

### Right Angles

##### Stage: 3 Challenge Level:

Can you make a right-angled triangle on this peg-board by joining up three points round the edge?

##### Stage: 3 Challenge Level:

A game for 2 or more people, based on the traditional card game Rummy. Players aim to make two `tricks', where each trick has to consist of a picture of a shape, a name that describes that shape, and two properties of the shape.

### Shapely Pairs

##### Stage: 3 Challenge Level:

A game in which players take it in turns to turn up two cards. If they can draw a triangle which satisfies both properties they win the pair of cards. And a few challenging questions to follow...

### Property Chart

##### Stage: 3 Challenge Level:

A game in which players take it in turns to try to draw quadrilaterals (or triangles) with particular properties. Is it possible to fill the game grid?

### Semi-regular Tessellations

##### Stage: 3 Challenge Level:

Semi-regular tessellations combine two or more different regular polygons to fill the plane. Can you find all the semi-regular tessellations?

##### Stage: 3 Challenge Level:

What can you say about the angles on opposite vertices of any cyclic quadrilateral? Working on the building blocks will give you insights that may help you to explain what is special about them.

### Which Solids Can We Make?

##### Stage: 3 Challenge Level:

Interior angles can help us to work out which polygons will tessellate. Can we use similar ideas to predict which polygons combine to create semi-regular solids?

### Opposite Vertices

##### Stage: 3 Challenge Level:

Can you recreate squares and rhombuses if you are only given a side or a diagonal?

##### Stage: 3 Challenge Level:

How many questions do you need to identify my quadrilateral?

##### Stage: 3 Challenge Level:

We started drawing some quadrilaterals - can you complete them?

### Of All the Areas

##### Stage: 4 Challenge Level:

Can you find a general rule for finding the areas of equilateral triangles drawn on an isometric grid?

### Estimating Angles

##### Stage: 2, 3 and 4 Challenge Level:

How good are you at estimating angles?

### Tilted Squares

##### Stage: 3 Challenge Level:

It's easy to work out the areas of most squares that we meet, but what if they were tilted?

### Hidden Squares

##### Stage: 3 Challenge Level:

Can you find the squares hidden on these coordinate grids?

### Triangle in the Corner

##### Stage: 3 Short Challenge Level:

A triangle is shaded within a regular hexagon. Can you find its area?

### Funky Hexagon

##### Stage: 3 Short Challenge Level:

Can you find the value of x in this diagram?

### Perimeter in a Hexagon

##### Stage: 3 Short Challenge Level:

Can you find the perimeter of this triangle inscribed in a hexagon?

### Tricky Tessellations

##### Stage: 3 Short Challenge Level:

Can you work out the fraction of the tiles that are painted black in this pattern?

### Descending Angles

##### Stage: 3 Short Challenge Level:

Given four of the angles in two triangles, can you find the smallest angle overall?

### Clock Face Angles

##### Stage: 3 Short Challenge Level:

The time is 20:14. What is the smaller angle between the hour hand and the minute hand on an accurate analogue clock?

### Overlapping Beer Mats

##### Stage: 3 Short Challenge Level:

Can you find the area of the overlap when these two beer mats are placed on top of each other?

### Clock Angle

##### Stage: 3 Short Challenge Level:

What is the angle between the the hands of a clock at 8:24?

### Two Triangles

##### Stage: 3 Short Challenge Level:

Prove that the angle marked $a$ is half the size of the angle marked $b$.

### Inner Rectangle

##### Stage: 3 Short Challenge Level:

If the shape on the inside is a rectangle, what can you say about the shape on the outside?

### No Rights

##### Stage: 3 Short Challenge Level:

Prove that the angle bisectors of a triangle can never meet at right angles.

### Perimeter Puzzle

##### Stage: 3 Short Challenge Level:

If four copies of this triangle are joined together to form a parallelogram, what is the largest possible perimeter of the parallelogram?

### Diagonal Side

##### Stage: 3 Short Challenge Level:

Can you work out the area of a square drawn on a diagonal?