**In right triangle ABC, right angled at C, M is the mid-point of hypotenuse AB. C is joined to M and produced to a point D such that ****DM = CM. Point D is joined to point B (see Figure). Show that:**

**(i)**

**(ii)** is a right angle.

**(iii)**

**(iv)**

**Solution (i)**

In and

** (Given)**

** (Vertically Opposite Angles)**

MC=MD ** (Given)**

Therefore, by SAS rule,

**Solution (ii)**

From** solution (i)**, we know that

Therefore, ** (Corresponding parts of congruent triangles are equal)**

But, these are alternate interior angles which means that

means that ** (Co-Interior Angles)**

**Solution (iii)**

In and , we have

BD=CA **(Corresponding parts of congruent triangles, and )**

** (Proved in solution (ii))**

BC=CB ** (Common)**

Therefore, by SAS rule,

**Solution (iv)**

From solution** (iii)**, we know that

Therefore, AB=DC **(Corresponding parts of congruent triangles are equal)**

** (It is given that MD=CM)**