I have an illustration in my book of a quadrilateral inside a circle. The quadrilateral is ABCD; AB is a vertical line and then AC and BD are the same length, but AC slopesd downward and BD upwards. Finally, there's CD; it's not vertical, as it slopes upward from D to C and is shorter than AB. The question is how are angles BAD and BCD related?

I think it has something to do with triangles, but I can't quite find the relation.
I know this is probably really confusing, but I tried drawing a picture of it on paint and when I went to post it, it said I cn't post websites; please help me, I really don't get it.

If a quadrilateral inside a circle has its 4 vertices on the same circle, then it is a cyclic quadrilateral.

One of the properties is that opposite angles add up to 180º

I think this is the relationship you are looking for according to your description.

No worries, I'll try my best to help you understand the relationship between angles BAD and BCD in the quadrilateral ABCD inside a circle.

To understand their relationship, let's break down the problem step by step. It's great that you've already identified triangles as being relevant here.

1. Start by drawing the given quadrilateral ABCD with AB as a vertical line, AC and BD as the same length but sloping in different directions, and CD sloping upwards and shorter than AB.

2. Since ABCD is inside a circle, we can assume it is a cyclic quadrilateral. This means that all its vertices lie on the circumference of the circle.

3. Now, focus on triangle ABC. Since AC and BD are the same length, we can conclude that triangle ABC is an isosceles triangle. In an isosceles triangle, the base angles (angles between the equal sides) are congruent.

4. Angle BAC is equal to angle BCA (or angle ABC). This is because in an isosceles triangle, the base angles are equal.

5. Moving on, let's consider triangle BCD. We know that CD is shorter than AB and slopes upward from D to C. As a result, angle BCD (the angle opposite the shorter side) must be smaller than angle BAC (the angle opposite the longer side).

Therefore, we can conclude that angle BAD is larger than angle BCD in the quadrilateral ABCD inside the circle.

Remember, drawing a diagram can greatly help visualize the problem. If you have access to other platforms, such as photo-sharing platforms or online whiteboards like Google Jamboard, you can draw and share the diagram there to help in future explanations.