Prove that the tangents to a circle at the endpoints of a diameter are parallel. State what is given, what is to be proved, and your plan of proof. Then write a two-column proof. Hint draw a DIAGRAM with the points labeled. Can some one please check for me

Given : ____________________________
To prove: _______________________________
Plan: ______________________________________

Given:
Lines AB and CD such that the lines are tangent to the circle with the center O at points B and D.
Segment BD is the diameter of the circle.
Prove:
Line AB is parallel to Line CD.
Plan:
Apply the parallel line property:
Two lines that are perpendicular to the same line are parallel.

Statements:
1) Line AB and Line CD are tangent to the circle
2) Segment BD is the diameter of the circle
3) Segment OB and Segment OD are the radii
4) Line AB is perpendicular to Segment OB; Line CD is perpendicular to Segment OD
5) Line AB is perpendicular to Segment BD; Line CD is perpendicular to Segment BD
6) Line AB is parallel to Line CD

Reasons:
1) Given
2) Given
3) Definition of a diameter
4) If a line is tangent to a circle, then the line is perpendicular to the radius drawn to the point of tangency.
5) Substitution property
6) Two lines that are perpendicular to the same line are parallel.

Your proof is excellent. I agree with all steps and reasons given. By "sustitution property" in statement # 5, iI assume you mean the reason is the same as #4

Yes, you are correct. The reason for statement #5 should be the same as statement #4, which is the property that if a line is tangent to a circle, then the line is perpendicular to the radius drawn to the point of tangency. Thank you for catching that.

Thank you for reviewing the proof. You are correct, the reason for statement #5 should indeed be the same as statement #4. The substitution property is not applicable in this case. I apologize for the error.

Here is the revised version of the two-column proof:

Given:
1) Lines AB and CD are tangent to the circle at points B and D respectively.
2) Segment BD is the diameter of the circle.

To prove:
Line AB is parallel to Line CD.

Plan:
Apply the property that states if a line is tangent to a circle, then the line is perpendicular to the radius drawn to the point of tangency.

Statements:
1) Line AB and Line CD are tangent to the circle.
2) Segment BD is the diameter of the circle.
3) Segment OB and Segment OD are radii of the circle.
4) Line AB is perpendicular to Segment OB (By the property mentioned in the plan).
5) Line CD is perpendicular to Segment OD (By the property mentioned in the plan).
6) Line AB is perpendicular to Segment BD (Segment BD is a diameter which passes through point B on Line AB).
7) Line CD is perpendicular to Segment BD (Segment BD is a diameter which passes through point D on Line CD).
8) Line AB is parallel to Line CD (Parallel lines have the same perpendicular slope).

Reasons:
1) Given.
2) Given.
3) Definition of a diameter.
4) If a line is tangent to a circle, then the line is perpendicular to the radius drawn to the point of tangency.
5) If a line is tangent to a circle, then the line is perpendicular to the radius drawn to the point of tangency.
6) Segment BD is a diameter, and Line AB intersects it at a right angle.
7) Segment BD is a diameter, and Line CD intersects it at a right angle.
8) Parallel lines have the same perpendicular slope.

This proof demonstrates that the tangents to a circle at the endpoints of a diameter are indeed parallel, based on the given information and the properties of tangents to circles.

Find the area of the larger segment whose chord is 8" long in a circle with an 8" radius.

Line AC is tangent to circle D at point B. Line HF is tangent to circle D at point E. Segments DB an DE are radii of circle D. Segment BC is congruent to segment EF.