Chapter Test
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Use the External Tangent Congruence Theorem.
See solution.
Let's examine the given diagram.
We can see that JK and LK are tangents to ⊙ C from a common external point K. We want to show that JM is congruent to LM. Let's start by recalling the External Tangent Congruence Theorem.
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External Tangent Congruence Theorem |
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Tangent segments from a common external points are congruent. |
By this theorem, we can conclude that JK and LK are congruent because they are both tangent to ⊙ C from a common external point.
Note that △ CJK and △ CLK are right triangles with a common hypotenuse CK. By the Reflexive Property of Congruence, we know that CK is congruent to itself. We also know that JK ≅ LK. Therefore, by the Hypotenuse Leg Theorem, we can say that △ CLK and △ CJK are congruent triangles. △ CLK ≅ △ CJK Corresponding parts of congruent triangles are congruent. Therefore, we can say that ∠ KCJ and ∠ KCL are congruent angles.
Recall that the measure of an arc is equal to the measure of its central angle. Since the central angles of JM and LM are congruent, by the Congruent Central Angles Theorem, JM and LM are congruent. JM ≅ LM We have shown the congruence of the arcs.
Let's summarize the above process in a flow proof.
