Question

For circle O, m arc CD =125 and m 9 years ago

Answer 1

We are given a figure similar to the one below:



We are also given that mARC CD is \(125^{\circ}\) and m\(\angle{ABC}\) and asked to figure out which two angles amongst the given options have measures equal to \(35^{\circ}\).

Answer 2

In this case, observe that the circle is referred to as "Circle O", which means point O is the center of the circle. Also observe the following:
1. \(\overline{AB}\) is tangent to the circle at point A therefore \(\angle{A}\) is a right triangle.

2. \(\angle{AOD}\) spans ARC DA. If we can find the measure of ARC DA, we will also know the measure of \(\angle{AOD}\).

3. \(\overline{CA}\) goes through point O which means \(\overline{CA}\) is a diameter of the circle. Furthermore, it also tells us that mARC CDA is \(180^{\circ}\). In other words:
mARC CD + mARC DA = \(180^{\circ}\).

Since they give us that mARC CD is \(125^{\circ}\) we can find the measure of arc DA as follows:

125+ mARC DA = \(180\)
mARC DA = \(180 - 125\)
mARC DA = \(55\)

Therefore the measure of arc DA is 55 degrees. Thus the measure of \(\angle{AOD}\) is also \(55^{\circ}\)

Knowing the measures of two interior angles of \(\triangle{AOB}\) enables us to find the measure of the third unknown angle. (Hint, Hint)

Answer 3

@Rachelp, if you come back soon, I can help you with the rest.