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If AB and AC are tangent segments to ⊙O, then AB≅AC.
Consider two triangles.
These two triangles can be visualized in the diagram.
Note that B and C are points of tangency. Therefore, by the Tangent to Circle Theorem, ∠B and ∠C are right angles. Consequently, △ABO and △ACO are right triangles.
Because all radii of the same circle are congruent, it can be said that OB and OC are congruent. Moreover, △ABO and △ACO share the same hypotenuse OA. By the Reflexive Property of Congruence, OA is congruent to itself.
Combining all of this information, it can be said that the hypotenuse and one leg of △ABO are congruent to the hypotenuse and the corresponding leg of △ACO.
Therefore, by the Hypotenuse-Leg Theorem, △ABO and △ACO are congruent triangles. Since corresponding parts of congruent figures are congruent, it can be said that AB and AC are congruent.
AB≅AC