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Answer:
See Below.
Step-by-step explanation:
In the given figure, O is the center of the circle. Two equal chords AB and CD intersect each other at E.
We want to prove that I) AE = CE and II) BE = DE
First, we will construct two triangles by constructing segments AD and CB. This is shown in Figure 1.
Recall that congruent chords have congruent arcs. Since chords AB ≅ CD, their respective arcs are also congruent:
Arc AB is the sum of Arcs AD and DB:
Likewise, Arc CD is the sum of Arcs CB and DB. So:
Since Arc AB ≅ Arc CD:
Solve:
The converse tells us that congruent arcs have congruent chords. Thus:
Note that both ∠ADC and ∠CBA intercept the same arc Arc AC. Therefore:
Additionally:
Since they are vertical angles.
Thus:
By AAS.
Then by CPCTC:
