Why are these two angles equal (x=z)?,

[Kamalesh]

Homework Statement

I am trying to understand calculus when doing that I came across this diagram.

Relevant Equations

Can anyone explain why angle x=angle z elaborately?

 

[Delta2]

Hi,

According to the rules of these forums, we cannot give you the full solution to a homework, you ve got to show us your attempt and then we can figure out your mistakes or give you hints towards the solution.

This problem looks like it can be solved by forming various equations containing angles, all such equations sourcing from the fact that the sum of the angles of a triangle equal to 180 degrees. And also it is given that OA=OC hence the angles A and C are equal.

EDIT: I worked through the problem and i seem to get $z=x+\frac{y}{2}$...

 

[neilparker62]

Let the perpendicular from A meet the baseline at B.

Presuming x=z, quadrilateral ACBO will be cyclic (equal angles subtended by chord BC). If this is the case we would have angle OBA = angle OCA . But this is not possible since OBA = 90 degrees but OCA is a base angle of an isosceles triangle and therefore less than 90. Presuming y ≠ 0.

Hence x≠z.

 

[aheight]

It's important to draw as accurate a diagram as possible. The requirement $OA=OC$ implies the triangle is isosceles which is not reflected in your diagram. But once you draw an accurate diagram and angle-chase a bit then you can see the base angles of the triangle are both $(\pi-y)/2$, then $w=\frac{\pi-y}{2}-x$ which implies $z=y/2+x$ so that $z\neq x$.