Question

the figure, line t is tangent to circle A at point B. Point C lies on line t outside of the circle.





Question
Prove, by contradiction, that the radius, segment A B, of circle A is perpendicular to the tangent line, t. Move options to the lines to complete the proof.

Response area with 8 blank spaces
Assume that point C lies on line t such that the point C is not equal to the point B and segment A C is perpendicular to Line t.


By the definition of perpendicular lines,
Blank space 1 empty
​must be a right angle. Then,
Blank space 2 empty
must be acute because the angles in a triangle add to 180º. This means
Blank space 3 empty
is less than
Blank space 4 empty
because larger sides are opposite larger angles in a triangle. However, by the definition of a tangent line, point
Blank space 5 empty
lies on the circle and point
Blank space 6 empty
lies outside the circle. This means that the length of
Blank space 7 empty
is less than
Blank space 8 empty
.

Answer 1

Assume that point C lies on line t such that the point C is not equal to the point B and segment AC is perpendicular to Line t.

By the definition of perpendicular lines,
Blank 1: ∠ACB
must be a right angle. Then,
Blank 2: ∠ABC
must be acute because the angles in a triangle add to 180º. This means
Blank 3: AC
is less than
Blank 4: AB
because larger sides are opposite larger angles in a triangle. However, by the definition of a tangent line, point
Blank 5: B
lies on the circle and point
Blank 6: C
lies outside the circle. This means that the length of
Blank 7: AB
is less than
Blank 8: AC.