Question

in the figure, line t is tangent to circle a at point b. point c lies on line t outside of the circle. prove, by contradiction, that the radius (overline{ab}) of circle a is perpendicular to the tangent line, t. move options to the lines to complete the proof. assume that point c lies on line t such that (c
eq b) and (overline{ac}perp t). by the definition of perpendicular lines, ______ must be a right - angle. then ____ must be acute because the angles in a triangle add to (180^{circ}). this means ____<____ because larger sides are opposite larger angles in a triangle. however, by the definition of a tangent line, point ____ lies on the circle and point ____ lies outside the circle. this means that the length of ____<______. this proves, by contradiction, that (overline{ab}perp t). (angle abc) (angle acb) (ab) (ac) (a) (b) (c)

Explanation:

Step1: Identify right - angle

By the definition of perpendicular lines, $\angle ACB$ must be a right angle.

Step2: Determine other angle

Then $\angle ABC$ must be acute because the angles in a triangle add to $180^{\circ}$.

Step3: Apply angle - side relationship

Since larger sides are opposite larger angles in a triangle, and $\angle ACB>\angle ABC$, we have $AB < AC$. But point $B$ lies on the circle and point $C$ lies outside the circle. In a circle, the radius is the shortest distance from the center to the circle. If we assume $AC\perp t$ instead of $AB\perp t$, we get a contradiction as the distance from the center $A$ to the tangent should be the radius (the shortest distance).

Answer:

1. $\angle ACB$
2. $\angle ABC$
3. $AB$
4. $AC$
5. $B$
6. $C$
7. $AB$
8. $AC$