Exercise 46 Page 536

We want to prove the External Tangent Congruence Theorem.

External Tangent Congruence Theorem

Tangent segments from a common external point are congruent.

We will prove that $\overline{SR}\cong \overline{ST},$ given that $\overline{SR}$ and $\overline{ST}$ are tangent to $\odot P.$

To do so, we will use congruent triangles. Let's start by drawing $\overline{PR},$ $\overline{PT},$ and $\overline{PS}.$

Having drawn these segments, we have two triangles, $△SRP$ and $△STP.$ From here, we will continue by showing that these triangles are congruent. As we can see, $\overline{PR}$ and $\overline{PT}$ are the radii of $\odot P,$ and $R$ and $T$ are their corresponding points of tangency. Let's consider the Tangent Line to Circle Theorem.

Tangent Line to Circle Theorem

In a plane, a line is tangent to a circle if and only if the line is perpendicular to a radius of the circle at its endpoint on the circle.

With this theorem, we can say that $\angle PRS$ and $\angle PTS$ are right angles.

Therefore, we can conclude that $△SRP$ and $△STP$ are right triangles. Next, recall that radii of a circle are congruent. Therefore, $\overline{PR}$ and $\overline{PT}$ are congruent segments.

We can also see that $△SRP$ and $△STP$ share the same hypotenuse $\overline{PS}.$ By the Reflexive Property of Congruence, $\overline{PS}$ is congruent to itself.

Combining all this information, we can say that the hypotenuse and one leg of $△SRP$ are congruent to the hypotenuse and one leg of $△STP.$ Therefore, by the Hypotenuse-Leg Theorem, we have that $△SRP$ and $△STP$ are congruent triangles. Finally, because corresponding parts of congruent triangles are congruent, we can conclude that $\overline{SR}\cong \overline{ST}.$

Let's summarize the above process in a flow proof.