Angles in polygons

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Arclets Explained

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.
problem

Darts and Kites

Age

14 to 16

Challenge level

Explore the geometry of these dart and kite shapes!
problem

Isosceles Meld

Age

11 to 14

Challenge level

Weekly Problem 9 - 2012
What is the angle QPT in this diagram?
problem

Stellar Angles

Age

11 to 14

Challenge level

Weekly Problem 30 - 2013
What is the angle $x$ in the star shape shown?
problem

Pegboard Quads

Age

14 to 16

Challenge level

Make different quadrilaterals on a nine-point pegboard, and work out their angles. What do you notice?
problem

Polygon Cradle

Age

11 to 14

Challenge level

Weekly Problem 18 - 2007
A regular pentagon together with three sides of a regular hexagon form a cradle. What is the size of one of the angles?
problem

Angle Hunt

Age

11 to 14

Challenge level

Weekly Problem 39 - 2010
If you know three lengths and an angle in this diagram, can you find another angle by calculation?
problem

Outside the Nonagon

Age

11 to 14

Challenge level

Weekly Problem 44 - 2010
Extend two of the sides of a nonagon to form an angle. How large is this acute angle?
problem

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.
problem

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