The Great Angle Chase There are many different pathways that lead to the same destination. Here is just one of them …

Aim: To write in the size of every angle. J I L M N K O S P Q R T U G H 25 100 35

Look for the angle between a tangent and a chord. Locate the angles in the alternate segment. J I L M N K O S P Q R T U G H 25 100 35 25 25

J I L M N K O S P Q R T U G H 25 100 35 25 25

Look for angles in the same segment. J I L M N K O S P Q R T U G H 25 100 35 25 35 25

J I L M N K O S P Q R T U G H 25 100 35 25 35 25

Look for angles standing on equal chords. J I L M N K O S P Q R T U G H 25 100 35 25 25 25 35 25

J I L M N K O S P Q R T U G H 25 100 35 25 25 25 35 25

Look for right angles: G U P T K L J M I O H N Q S R 25 100 35 The angle between a tangent and a radius The angle in a semicircle 25 65 25 65 40 25 35 25

G U P T K L J M I O H N Q S R 25 65 35 25 65 25 40 25 100 35

Look for triangles: G U P T K L J M I O H N Q S R Angle sum 25 65 100 35 Angle sum 25 65 55 25 55 65 40 25 35 40 75 25

Look for triangles: G U P T K L J M I O H N Q S R Angle sum 25 65 100 35 Angle sum 25 65 55 25 55 65 40 75 25 120 35 40 75 25

Look for triangles: G U P T K L J M I O H N Q S R Angle sum 25 100 35 25 Angle sum Exterior angle 65 55 25 125 55 65 40 75 25 120 35 40 75 25

Look for triangles: G U P T K L J M I O H N Q S R Angle sum 25 100 35 25 Angle sum Exterior angle 65 55 60 25 125 55 65 40 75 25 120 35 40 75 25

Look for vertically opposite angles and angles at a point. J I L M N K O S P Q R T U G H 25 100 35 25 65 120 60 55 125 60 25 120 125 55 65 40 105 75 75 60 25 105 80 120 120 100 60 80 35 40 75 25

And there’s another triangle. J I L M N K O S P Q R T U G H 25 100 35 25 65 120 60 55 15 125 60 25 120 125 55 65 40 105 75 75 60 25 105 80 120 120 100 60 80 35 40 75 25

And then another pair of angles in the same segment. J I L M N K O S P Q R T U G H 25 100 35 25 65 120 60 55 15 125 60 25 120 125 55 65 40 105 75 75 60 25 105 80 120 120 100 60 80 35 40 75 25 15

Some more right angles. G U P T K L J M I O H N Q S R 25 100 35 25 65 40 120 60 55 15 125 60 25 120 125 55 65 40 105 75 75 60 25 105 80 120 120 100 60 80 35 40 75 25 75 15

Aha! A cyclic quadrilateral. J I L M N K O S P Q R T U G H 25 100 35 25 65 40 120 60 55 15 125 60 25 120 125 55 65 40 105 75 75 60 25 105 80 120 120 100 60 80 35 40 80 75 25 75 15

More angles at a point. G U P T K L J M I O H N Q S R 25 100 35 25 65 40 120 60 55 15 125 60 25 120 125 55 65 40 105 75 75 60 25 105 80 120 120 100 60 80 100 35 40 80 80 75 100 25 75 15

G U P T K L J M I O H N Q S R 25 100 35 25 65 40 120 60 55 15 125 60 25 120 125 55 65 40 105 75 75 60 25 105 80 120 120 100 60 80 100 35 40 80 80 75 100 25 75 15

Another angle between a tangent and a chord. J I L M N K O S P Q R T U G H 25 100 35 25 65 40 120 60 55 15 125 60 25 120 125 55 65 40 105 75 75 60 25 105 80 120 120 100 60 80 100 35 40 80 80 75 100 40 25 75 15

A cyclic quadrilateral. J I L M N K O S P Q R T U G H 25 100 35 25 65 40 120 60 55 15 125 60 25 120 125 55 65 40 105 75 75 60 25 105 80 120 120 100 60 80 100 35 40 80 80 75 100 40 65 25 75 15

And we’re done! G U P T K L J M I O H N Q S R 25 100 35 25 65 40 120 60 55 15 125 60 25 120 125 55 65 40 105 75 75 60 25 105 80 120 120 100 60 80 100 35 40 80 80 75 100 40 65 25 75 15