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External Tangent Congruence Theorem

Rule

External Tangent Congruence Theorem

Two tangent segments drawn from a common external point to the same circle are congruent.

Image not found. We apologize, please report this so that we can fix it as soon as possible!File = mljsx_Rules_External_Tangent_Congruence_Theorem_1.svg, id = Rules_External_Tangent_Congruence_Theorem_1
For the above diagram, the following conditional statement holds true.

If and are tangent segments to then

Proof


Consider two triangles.

  • The triangle formed by the radius the segment and the tangent segment
  • The triangle formed by the radius the segment and the tangent segment

These two triangles can be visualized in the diagram.

Image not found. We apologize, please report this so that we can fix it as soon as possible!File = mljsx_Rules_External_Tangent_Congruence_Theorem_2.svg, id = Rules_External_Tangent_Congruence_Theorem_2

Note that and are points of tangency. Therefore, by the Tangent to Circle Theorem, and are right angles. Consequently, and are right triangles.

Image not found. We apologize, please report this so that we can fix it as soon as possible!File = mljsx_Rules_External_Tangent_Congruence_Theorem_3.svg, id = Rules_External_Tangent_Congruence_Theorem_3

Because all radii of the same circle are congruent, it can be said that and are congruent. Moreover, and share the same hypotenuse By the Reflexive Property of Congruence, is congruent to itself.

Image not found. We apologize, please report this so that we can fix it as soon as possible!File = mljsx_Rules_External_Tangent_Congruence_Theorem_4.svg, id = Rules_External_Tangent_Congruence_Theorem_4

Combining all of this information, it can be said that the hypotenuse and one leg of are congruent to the hypotenuse and the corresponding leg of

Therefore, by the Hypotenuse-Leg Theorem, and are congruent triangles. Since corresponding parts of congruent figures are congruent, it can be said that and are congruent.