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

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

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

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

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

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

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

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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?

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

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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.

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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

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

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

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

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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?

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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.

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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. . . .

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.

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

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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?

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

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

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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.

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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.

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

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

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Turn through bigger angles and draw stars with Logo.

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