Question

In the diagram triangle xyz is isosceles with xy=xz.Also,point w is on xz so that xw=wy=yz.The measure of angle xyw is

Answer 1

We have 3 isosceles triangles
In triangle xyw, xw = yw, as stated
mark angle x = angle xyw = a
in triangle xyz, xy = xz, as stated
mark angle xyz = angle xzy = b

in triangle wyz, yw = yz, as stated
mark angle wyz = b-a

also angle ywz is an exterior angle in triangle xyw,
so angle ywz = 2a

so back to triangle wyz,
b = 2a, isosceles triangle
and
2a + b-a + b = 180°
a + 2b = 180
a + 2(2a) = 180
5a = 180
a = 36°
then b = 72°

and angle xyw = 36°

aside:
This particular configuration is considered the "golden triangle"
note that cos 36° = (1+√5)/4
which is 1/2 the Golden Ratio.

Also see "Golden Ratio" and "golden rectangle"

Isn't Math wonderful??

Answer 2

xyz

Answer 3

Good

Answer 4

Thanks