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Broad Topics > Angles, Polygons, and Geometrical Proof > Angle properties of polygons ### Always, Sometimes or Never? Shape

##### Age 7 to 11 Challenge Level:

Are these statements always true, sometimes true or never true? ### Walking Round a Triangle

##### Age 5 to 7 Challenge Level:

This ladybird is taking a walk round a triangle. Can you see how much he has turned when he gets back to where he started? ##### Age 11 to 16 Challenge Level:

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

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

Why does this fold create an angle of sixty degrees? ### 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? ### Triangles and Petals

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

An equilateral triangle rotates around regular polygons and produces an outline like a flower. What are the perimeters of the different flowers? ### Terminology

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

Given an equilateral triangle inside an isosceles triangle, can you find a relationship between the angles? ### Transformations on a Pegboard

##### Age 7 to 11 Challenge Level:

How would you move the bands on the pegboard to alter these shapes? ### Name That Triangle!

##### Age 7 to 11 Challenge Level:

Can you sketch triangles that fit in the cells in this grid? Which ones are impossible? How do you know? ### Triangle or No Triangle?

##### Age 5 to 7 Challenge Level:

Here is a selection of different shapes. Can you work out which ones are triangles, and why? ### Angles Inside

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

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

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

Take a look at the photos of tiles at a school in Gibraltar. What questions can you ask about them? ### Angles in Three Squares

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

Drawing the right diagram can help you to prove a result about the angles in a line of squares. ### 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? ### 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 Rings

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

Join pentagons together edge to edge. Will they form a ring? ### 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? ##### Age 16 to 18 Short Challenge Level:

Can you work out where the blue-and-red brick roads end? ### LOGO Challenge 4 - Squares to Procedures

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

This LOGO Challenge emphasises the idea of breaking down a problem into smaller manageable parts. Working on squares and angles. ### Logo Challenge 3 - Star Square

##### Age 7 to 16 Challenge Level:

Creating designs with squares - using the REPEAT command in LOGO. This requires some careful thought on angles ### First Forward Into Logo 9: Stars

##### Age 11 to 18 Challenge Level:

Turn through bigger angles and draw stars with Logo. ### First Forward Into Logo 7: Angles of Polygons

##### Age 11 to 18 Challenge Level:

More Logo for beginners. Learn to calculate exterior angles and draw regular polygons using procedures and variables. ### Tessellating Hexagons

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

Which hexagons tessellate? ##### Age 14 to 16 Challenge Level:

Make five different quadrilaterals on a nine-point pegboard, without using the centre peg. Work out the angles in each quadrilateral you make. Now, what other relationships you can see? ### Triangle Pin-down

##### Age 7 to 11 Challenge Level:

Use the interactivity to investigate what kinds of triangles can be drawn on peg boards with different numbers of pegs. ### Triangles All Around

##### Age 7 to 11 Challenge Level:

Can you find all the different triangles on these peg boards, and find their angles? ### Cartesian Isometric

##### Age 7 to 11 Challenge Level:

The graph below is an oblique coordinate system based on 60 degree angles. It was drawn on isometric paper. What kinds of triangles do these points form? ### Fred the Class Robot

##### Age 7 to 11 Challenge Level:

Billy's class had a robot called Fred who could draw with chalk held underneath him. What shapes did the pupils make Fred draw? ### Convex Polygons

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

Show that among the interior angles of a convex polygon there cannot be more than three acute angles. ### Pentakite

##### Age 14 to 18 Challenge Level:

ABCDE is a regular pentagon of side length one unit. BC produced meets ED produced at F. Show that triangle CDF is congruent to triangle EDB. Find the length of BE. ### Bow Tie

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

Show how this pentagonal tile can be used to tile the plane and describe the transformations which map this pentagon to its images in the tiling. ### Dodecawhat

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

Follow instructions to fold sheets of A4 paper into pentagons and assemble them to form a dodecahedron. Calculate the error in the angle of the not perfectly regular pentagons you make. ### Arclets Explained

##### Age 11 to 16

This article gives an wonderful insight into students working on the Arclets problem that first appeared in the Sept 2002 edition of the NRICH website. ### Getting an Angle

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

How can you make an angle of 60 degrees by folding a sheet of paper twice? ### Pent

##### Age 14 to 18 Challenge Level:

The diagram shows a regular pentagon with sides of unit length. Find all the angles in the diagram. Prove that the quadrilateral shown in red is a rhombus. ### Tricircle

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

The centre of the larger circle is at the midpoint of one side of an equilateral triangle and the circle touches the other two sides of the triangle. A smaller circle touches the larger circle and. . . . ### Floored

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

A floor is covered by a tessellation of equilateral triangles, each having three equal arcs inside it. What proportion of the area of the tessellation is shaded? ### A Sameness Surely

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

Triangle ABC has a right angle at C. ACRS and CBPQ are squares. ST and PU are perpendicular to AB produced. Show that ST + PU = AB ### No Right Angle Here

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

Prove that the internal angle bisectors of a triangle will never be perpendicular to each other. ### Darts and Kites

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

Explore the geometry of these dart and kite shapes! ### Golden Triangle

##### Age 16 to 18 Challenge Level:

Three triangles ABC, CBD and ABD (where D is a point on AC) are all isosceles. Find all the angles. Prove that the ratio of AB to BC is equal to the golden ratio. ### At a Glance

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

The area of a regular pentagon looks about twice as a big as the pentangle star drawn within it. Is it?