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 XYZ is​

Answer 1

$\sf \pink{Solution - }$

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"

Thanks.