In the diagram below,  A, B, C and  D are four points on a circle.

X is the point of intersection of the lines  AC and  BD. 

M is the midpoint of  AD.

XM is perpendicular to  AD.

Prove that triangles  AXB and  DXC are congruent.
 

Congruent triangles have three equal angles and three equal sides.
You do not need to show that all of them are equal to prove congruence, you can look for one of the five standard tests of congruence. These are SAS, SSS, ASA, AAS, RHS.

Use the information given in the question to look for corresponding angles and sides that are equal.

Identify any corresponding sides that are equal between triangles AXB  and DXC.

As  M is the midpoint of  AD, AM = MD.
Triangles  AMX and DMX  share the side  MX, which meets AD at right angles so by the rule of congruency SAS, triangles AMX and DMX are congruent triangles.

 
The sides AX and DX are equal. 
Triangles  AXM and  CXM are congruent
 

Identify any corresponding angles that are equal between triangles AXB  and DXC.

Vertically opposite angles are equal.

Angle AXB = angle DXC 
Vertically opposite angles

Angles in the same segment are equal.

Angle ABX = angle DCX 
Angles subtended from the same arc are equal

The statement that if two corresponding angles and one side are the same then the two triangles are congruent must be made. This is the AAS property (angle, angle, side)

Write which angles or sides are equal out clearly in the proof, making sure to give the reasons why they are equal.

AX  = XD  as triangles AMX and DMX are congruent.
Angle AXB  = angle DXC as they are vertically opposite angles.
Angle ABX  = angle DCX as they are angles in the same segment.
  By the standard test AAS the triangles are congruent