# Angles, Polygons and Geometrical Proof - Stage 3

### Square It

##### Age 11 to 16 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

##### Age 11 to 14 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

##### Age 11 to 14 Challenge Level:

Can you find triangles on a 9-point circle? Can you work out their angles?

### Subtended Angles

##### Age 11 to 14 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

##### Age 11 to 14 Challenge Level:

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

##### Age 11 to 14 Challenge Level:

A game for 2 or more people, based on the traditional card game Rummy.

### Shapely Pairs

##### Age 11 to 14 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

##### Age 11 to 14 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

##### Age 11 to 16 Challenge Level:

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

##### Age 11 to 16 Challenge Level:

Draw some quadrilaterals on a 9-point circle and work out the angles. Is there a theorem?

### Which Solids Can We Make?

##### Age 11 to 14 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

##### Age 11 to 14 Challenge Level:

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

##### Age 11 to 14 Challenge Level:

How many questions do you need to identify my quadrilateral?

##### Age 11 to 14 Challenge Level:

We started drawing some quadrilaterals - can you complete them?

### Of All the Areas

##### Age 14 to 16 Challenge Level:

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

### Estimating Angles

##### Age 7 to 14 Challenge Level:

How good are you at estimating angles?

### Tilted Squares

##### Age 11 to 14 Challenge Level:

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

### Hidden Squares

##### Age 11 to 14 Challenge Level:

Can you find the squares hidden on these coordinate grids?

### Polygon Rings

##### Age 11 to 14 Challenge Level:

Join pentagons together edge to edge. Will they form a ring?

### Star Polygons

##### Age 11 to 14 Challenge Level:

Draw some stars and measure the angles at their points. Can you find and prove a result about their sum?

### Polygon Pictures

##### Age 11 to 14 Challenge Level:

Can you work out how these polygon pictures were drawn, and use that to figure out their angles?

### An Equilateral Triangular Problem

##### Age 11 to 14 Challenge Level:

Take an equilateral triangle and cut it into smaller pieces. What can you do with them?

### Triangle in the Corner

##### Age 11 to 14 Short Challenge Level:

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

### Funky Hexagon

##### Age 11 to 14 Short Challenge Level:

Can you find the value of x in this diagram?

### Perimeter in a Hexagon

##### Age 11 to 14 Short Challenge Level:

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

### Tricky Tessellations

##### Age 11 to 14 Short Challenge Level:

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

### Descending Angles

##### Age 11 to 14 Short Challenge Level:

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

### Clock Face Angles

##### Age 11 to 14 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

##### Age 11 to 14 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

##### Age 11 to 14 Short Challenge Level:

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

### Two Triangles

##### Age 11 to 14 Short Challenge Level:

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

### Inner Rectangle

##### Age 11 to 14 Short Challenge Level:

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

### No Rights

##### Age 11 to 14 Short Challenge Level:

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

### Perimeter Puzzle

##### Age 11 to 14 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

##### Age 11 to 14 Short Challenge Level:

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

##### Age 11 to 14 Short Challenge Level:

Find the sum of all of the angles denoted by letters in this diagram

### Angles Inside

##### Age 11 to 14 Challenge Level:

Draw some angles inside a rectangle. What do you notice? Can you prove it?

### Fraction of a Square

##### Age 11 to 14 Short Challenge Level:

What fraction of this square is shaded?

### Square Within a Square Within...

##### Age 11 to 14 Short Challenge Level:

What fraction of this square is shaded?