Chord geometry problem

Yes it’s homework. No it’s not mine. I am trying to help my son with geometry and this one has me stumped.

The diagram is a circle with four chords inscribed, see link. The problem says, in its entirety:In the diagram, m<CAB = 35 and m<ACB = 95. What are the measures of angles <EDC and <DEC, respectively?
The < symbol is an approximation for the “angle” symbol.

Point C is not specified to be the center of the circle and I can see no information that could allow to conclude that it is.

I deduced that

ABC = 50
<EDC + <DEC = 85

But I don’t know enough about the rules of chords to establish any other relationships. I have not been in the 9th grade for a very long time.

Since the angle <DEC faces the arc DA, but also the angle <DBA faces the arc DA, then those two angles are equivalent. They are equivalent because they are both inscribed angles. I think you can go from there with the sum of the angles in a triangle equaling 180º.

Ironically, I have just spent the last 6 hours re teaching myself this chapter of a geometry book for my part time private tutoring job. An experience that can only be described as hell.

Makes sense to me. And I can’t believe the good fortune of my timing :slight_smile:

I didn’t know that inscribed angles defining the same arc were the same angle.Thank you!

Also, in the special case where the arc is a semicircle, all such angles are 90 degrees.

ETA: In any problem like this when you’re stuck with angles in a circle, sketch in the centre, and a radius to each point on the circumfererence. Isosceles triangles will pop up all over the place (because all radii are equal in length) and soon you will be able to identify equal angles.