## Equal Angles

#### Problem

In the square ABCD, M is the midpoint of AB.

A line is drawn through M perpendicular to CM to locate N.

Prove that the size of angle BCM is equal to the size of angle MCN.

#### Solution

We begin by drawing a line through M and parallel to BC to produce E on CN and F on CD.

As right angle triangle CMN is half of a rectangle, CN is one diagonal, and because E is the midpoint of CN, CE = EN = EM.

In which case, triangle CEM is isosceles, `a` = `b` (base angles equal).

Because MF is parallel to BC, `a` = `c` (alternate angles).

Hence `b` = `c`, and we prove that the size of angle BCM is equal to the size of angle MCN.

If AB=4, find the perimeter of triangle CDN.

Problem ID: 196 (21 Dec 2004) Difficulty: 2 Star