Question

wxyz is a quadrilateral whose diagonal intersect each other at the point O such that OW =OX =OZ If angleOWX=50°,find the measures of angleOZW

Answer 1

In ΔOWX,
OW=OX
 ∴ ∠OWX = ∠OXW = ∠50° ( angles opp. to equal sides are equal )
∴ ∠O+∠W+ ∠X = 180 ( angle sum property of a Δ)
∴ 50+50+∠O = 180 => 100+∠O = 180 => ∠O = 80°
In ΔOWZ,
∠ZOW + ∠WOX = 180 ( linear pair )
∠ZOW + 80 = 180 => ∠ZOW = 100
In ΔOZW,
∵ OZ = OW
Let ∠Z= x
∴∠W= x ( angles opp. to equal sides are equal )
∠O+∠Z+∠W = 180( angle sum property of a Δ)
100+2x = 180
2x = 80 => x = 80/2 = 40°
∴ OZW = 40°

                                               (OR)
Cyclic quadrilateral method:

     As OX= OW = OZ, they are on a circle with OW = OZ= OX as the radius. Obviously XZ = diameter. As diameter makes 90° at the circumference,  ∠OWZ = 90-50° = 40°.

   Since  OW = OZ, OWZ is an isosceles triangle  and ∠OZW = 40°