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
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??
xyz
Thanks
Well, considering that triangle XYZ is isosceles and XY = XZ, it means that angle YXZ is also equal to angle XYZ.
Now, let's focus on angle XYW. Since XW = WY, we have two equal sides. According to the Law of Sines, the ratio of the length of a side to the sine of its opposite angle is constant.
But here's the catch - since XW = WY, that means that the length of the side XY is equal to the length of the side YW. And since we know that angle YXW is 180 degrees (a straight angle), it means that the sum of angles XYW and YXW must be 180 degrees as well.
So, using all of this information, we can conclude that angle XYW is... drumroll, please... 90 degrees!
Why did the math book go to therapy? Because it had way too many problems!
To find the measure of angle XYW in the given diagram, we can use the properties of isosceles triangles and the fact that the lengths XY, XZ, and YZ are equal.
1. Start by drawing the given diagram with triangle XYZ, where XY = XZ.
2. Extend the side YZ to point W such that XW = WY = YZ. This creates an equilateral triangle XWY.
3. Since triangle XYW is isosceles (XY = XW), angle XYW is congruent to angle XWY.
4. In an equilateral triangle, all angles are congruent, so angle XWY is 60 degrees.
5. Therefore, the measure of angle XYW is also 60 degrees.
In summary, the measure of angle XYW in the given diagram is 60 degrees.