Are these statements always true, sometimes true or never true?

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?

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

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

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

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

How would you move the bands on the pegboard to alter these shapes?

Can you sketch triangles that fit in the cells in this grid? Which ones are impossible? How do you know?

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

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

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

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

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

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

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

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?

Can you work out where the blue-and-red brick roads end?

This LOGO Challenge emphasises the idea of breaking down a problem into smaller manageable parts. Working on squares and angles.

Creating designs with squares - using the REPEAT command in LOGO. This requires some careful thought on angles

Turn through bigger angles and draw stars with Logo.

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

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?

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

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

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?

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

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

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.

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.

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.

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.

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

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.

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. . . .

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?

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

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

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.

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