Determining tangent lines: angles

Solve two problems that apply properties of tangents to determine if a line is tangent to a circle.

Problem 1

Segment start overline, O, C, end overline is a radius of circle O.
Note: Figure not necessarily drawn to scale.
Is line tangent to circle O?
Choose 1 answer:
Choose 1 answer:

A start color red, l, i, n, e, end color red that is tangent to a circle at a particular point is perpendicular to the start color gray, r, a, d, i, u, s, end color gray at that point.
The interior angles of a triangle sum to 180, degree. Let's check whether angle, A, C, O is a right angle.
mCAO+mAOC+mACO=18032+58+mACO=180mACO=90\begin{aligned} \pink{\text{m}\angle CAO} + \blue{\text{m}\angle AOC} + \text{m}\angle ACO&= 180^\circ \\ \pink{32^\circ} + \blue{58^\circ} + \text{m}\angle ACO&= 180^\circ\\ \text{m}\angle ACO&= 90^\circ\\ \end{aligned}
Angle angle, A, C, O is a right angle.
Yes, is tangent to circle O, because start overline, A, C, end overline is perpendicular to start overline, O, C, end overline.

Problem 2

Segment start overline, B, C, end overline is a diameter of circle O.
Note: Figure not necessarily drawn to scale.
Is line tangent to circle O?
Choose 1 answer:
Choose 1 answer:

A start color red, l, i, n, e, end color red that is tangent to a circle at a particular point is perpendicular to the start color gray, r, a, d, i, u, s, end color gray at that point. Thus, it is also perpendicular to the diameter at the same point.
The interior angles of a triangle sum to 180, degree. Let's check whether angle, A, C, B is a right angle.
mBAC+mABC+mACB=18051+49+mACB=180mACB=80\begin{aligned} \pink{\text{m}\angle BAC} + \blue{\text{m}\angle ABC} + \text{m}\angle ACB &= 180^\circ \\ \pink{51^\circ} + \blue{49^\circ} + \text{m}\angle ACB &= 180^\circ\\ \text{m}\angle ACB &= 80^\circ\\ \end{aligned}
Angle angle, A, C, B is not a right angle.
No, is not tangent to circle O, because start overline, A, C, end overline is not perpendicular to start overline, B, C, end overline.