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Rule

Converse Angle Bisector Theorem

If an interior point of an angle is equidistant from the two sides of the angle, then it lies on the angle bisector.

Based on the figure, the following conditional statement holds true.

⎩ ⎪ ⎪ ⎨ ⎪ ⎪ ⎧ \overline{D B} ⊥ A B \overline{D C} ⊥ A C D B = D C \Rightarrow A D bisects ∠ B A C

Proof

Consider $∠ B A C$ and an interior point $D$ that is equidistant from the two sides of the angle.

By the Reflexive Property of Congruence, $\overline{A D}$ is congruent to itself.

It is given that $△ A D B$ and $△ A D C$ are both right triangles that share a hypotenuse, $\overline{A D} .$ Also, $\overline{D B}$ is one of the legs of $△ A D B$ and $\overline{D C}$ is one of the legs of $△ A D C .$

Since the hypotenuse and a leg of

△ A D B

are congruent to the hypotenuse and a leg of

△ A D C,

the triangles are congruent by the Hypotenuse-Leg Congruence Theorem.

△ A D B ≅ △ A D C

Corresponding parts of congruent triangles are congruent. Therefore,

∠ B A D

and

∠ C A D

are congruent angles.

∠ B A D ≅ ∠ C A D

By the definition of an angle bisector,

A D

bisects

∠ B A C .

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Proof