I am ask to write a two column proof. Given that AD is the diameter of circle O, and DC is tangent to O at D. Prove that Triangle ABD ~ (congruent) to triangle ADC
To write a two-column proof, we need to state each step of our reasoning in the left column and provide a justification for each step in the right column. Here's how you can prove that Triangle ABD is congruent to triangle ADC:
Step 1: Triangle ABD is a right triangle.
Justification: If AD is the diameter of circle O, then angle ABD is a right angle by the definition of a diameter.
Step 2: Triangle ADC is a right triangle.
Justification: DC is tangent to circle O at point D, so angle ADC is a right angle by the Tangent-Secant Theorem.
Step 3: Side AD is common to both Triangle ABD and Triangle ADC.
Justification: AD is the diameter of circle O, so it is a common side for both triangles.
Step 4: Side BD is common to both Triangle ABD and Triangle ADC.
Justification: BD is a part of line AD, so it is also a common side for both triangles.
Step 5: Side DC is common to both Triangle ABD and Triangle ADC.
Justification: DC is a part of line AD, so it is a common side for both triangles.
Step 6: Triangle ABD has a right angle at vertex B.
Justification: As stated in Step 1, angle ABD is a right angle.
Step 7: Triangle ADC has a right angle at vertex D.
Justification: As stated in Step 2, angle ADC is a right angle.
Step 8: Triangle ABD and Triangle ADC have two congruent angles and one congruent side.
Justification: Both triangles have a right angle and share side AD.
Step 9: Therefore, by the Angle-Side-Angle (ASA) congruence criterion, Triangle ABD is congruent to Triangle ADC.
Justification: As stated in Step 8, Triangle ABD and Triangle ADC have two congruent angles and one congruent side, satisfying the criteria for ASA congruence.
To write a two-column proof showing that Triangle ABD is congruent to Triangle ADC, we will need to use the information given and relevant geometric properties. Below is a step-by-step guide to constructing the proof:
Step 1: Given the information that AD is the diameter of circle O, and DC is tangent to O at D.
Step 2: Since AD is a diameter, this means that angle ADB is a right angle. (Definition of a diameter)
Step 3: Similarly, since DC is tangent to O at D, angle CDA is a right angle. (Definition of tangent)
Step 4: Using Step 2 and Step 3, we can conclude that angle ADB is congruent to angle CDA because they are both right angles.
Step 5: The side AD is common to both triangles ABD and ADC.
Step 6: By Step 4, angle ADB is congruent to angle CDA.
Step 7: By the Angle-Side-Angle (ASA) postulate, if two angles and the included side in one triangle are congruent to the two corresponding angles and included side in another triangle, the two triangles are congruent.
Step 8: Applying the ASA postulate, we can conclude that triangle ABD is congruent to triangle ADC. (By using the congruent angles and the shared side AD)
Therefore, Triangle ABD is congruent to Triangle ADC.