Construct your proof considering the fact that the radii of a circle are congruent.
See solution.
Practice makes perfect
We will show that AC≅ BC and AD≅ BD, given that ⊙ O with diameter ED ⊥ AB at C.
Let's begin by finding a relation between △ ACO and △ BCO. Since ED ⊥ AB, by the definition of a right angle, ∠ ACO and ∠ BCO are right angles.
Therefore, by the definition of a right triangle, △ ACO and △ BCO are right triangles. Now, let's remember that the radii of a circle are congruent.
Since both AO and BO are radii of ⊙ O, they are also congruent.
AO≅ BO
We can also see that △ ACO and △ BCO share the same leg CO. By the Reflexive Property of Congruence, CO is congruent to itself.
CO≅ CO
Combining all of this information, we can see that the hypotenuse and one leg of △ ACO are congruent to the hypotenuse and the corresponding leg of △ BCO. Thus, by the Hypotenuse Leg (HL) Theorem, △ ACO is congruent to △ BCO.
Since Corresponding Parts of Congruent Triangles are Congruent (CPCTC), we can prove the first part of the statement.
AC ≅ BC
Next, again by CPCTC, we can tell that ∠ COA is congruent to ∠ BOC.
∠ COA ≅ ∠ BOC
Now, to prove the second part of the statement, we will recall Theorem 12-4.
