Can Thales' Theorem Help Me Solve My Trapezoid Equation?

[lornick]

I have the above equation to try and complete and I have no idea how to do it I have worked out all the angles for A, B, C, and D accept the ones within the diagonal line E and F. Any hints?

 

[MarkFL]

I don't see an equation you are trying to complete, but what do we know about the opposite interior angles of a cyclic quadrilateral?

 

[lornick]

Sorry don't have an equation, my wording mistake, just trying to calculate the angle of E and F. What I have worked out is the angles of A, B, C, and D. A=70 B=110 C=120 D=60. But have no idea how to calculate E and F. If I turn C and D into a right angle on the outside of the trapezoid I get 30 and if I turn B and A into a right angle on the outside of the trapezoid I get 20. But not sure how to compute this into my E and F. Also I forgot to mention that A and D are parallel lines to B and C. Forgot to draw in the arrows on the line.Thank you for any help in solving this.

 

[MarkFL]

Since $\overline{AD}\parallel\overline{BC}$, and the trapezoid is cyclic, we know it must be isosceles. So we know $\angle A=60^{\circ}$, if $\measuredangle C=120^{\circ}$.

Thus, we know:

$$\angle E+\angle F=60^{\circ}$$

If we label the two angles at vertex $B$ as $G$ and $H$, we then know:

$$\angle G+\angle H=120^{\circ}$$

Can you find two more equations involving these 4 angles? Hint: add the interior angles of the two triangles making up the trapezoid...:D

Also, to simplify matters, $G$ and $H$ must share a special relationship to $E$ and $F$. ;)

 

[lornick]

[QUOTE Since $\overline{AD}\parallel\overline{BC}$, and the trapezoid is cyclic, we know it must be isosceles. So we know $\angle A=60^{\circ}$, if $\measuredangle C=120^{\circ}$.

I had changed the original degrees and didn't realize I had changed the triangles to equal sides. My triangles are not level only the horizontal lines are level, which leaves me with one obtuse angle and one acute angle. :S so sorry I confused matters, but I didn't want to get into trouble for my question from my teachers. Or am I confused lol.

 

[I like Serena]

Hi lornick! Welcome to MHB! :)

Thales' theorem may help you.

Thales' theorem: if AC is a diameter, then the angle at B is a right angle.

 

[lornick]

Hi lornick! Welcome to MHB! :)

Thales' theorem may help you.

Thales' theorem: if AC is a diameter, then the angle at B is a right angle.

Oh nooo I can't believe it was that simple, I have been trying to calculate it as a acute and obtuse angle (hence my issues). The time when you are doing maths and you start to bash your head against a brick wall, only to realize that the answer was right in front of you lol. Thank you sooooooooo much!