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In the diagram above, AB is the diameter and CD is a chord such that AB is perpendicular to CD. Therefore, the following congruences hold true.
EC ≅ ED and BC ≅ BD
Consider the segments OC and OD. Since AB and CD are perpendicular segments, then ∠ OEC and ∠ OED are right angles. Therefore, △ OCE and △ ODE are right triangles.
Next, since two radii of a circle are congruent, then OC and OD are congruent. Furthermore, by the Reflexive Property of Congruence, OE is congruent to itself. OC ≅ OD and OE ≅ OE Therefore, the hypotenuse and one leg of △ OCE are congruent to the hypotenuse and the corresponding leg of △ ODE.
By the Hypotenuse Leg Theorem, △ OCE and △ ODE are congruent triangles. △ OCE ≅ △ ODE Since corresponding parts of congruent triangles are congruent, then the other legs of the triangles are also congruent.
EC ≅ ED
The proof for the first part of the statement has been completed.
Now, to show the congruence of the arcs BC and BD, the properties of congruent right triangles will be considered. Since corresponding parts of congruent triangles are congruent, it can be said that ∠ COE and ∠ DOE are congruent angles. ∠ COE ≅ ∠ DOE Therefore, by the Congruent Central Angles Theorem, BC and BD are congruent.
BC ≅ BD
This completes the proof.
