a If a point lies on the bisector of an angle, then it is equidistant from the two sides of the angle.
Prove that the point is equidistant from the two sides.
B
b If a point is in the interior of an angle and is equidistant from the two sides of the angle, then it lies on the bisector of the angle.
Prove that the angle is bisected.
A
a See solution.
B
b See solution.
Practice makes perfect
a Let's draw an arbitrary angle, ∠ BAC, that is bisected by a ray, AD. We will also draw perpendicular segments between D and the two sides that creates ∠ BAC.
Proving the theorem
To prove the Angle Bisector Theorem, we have to show that BD=DC. From the diagram, we can identify two right triangles:
△ ABD and △ ADC.
Since AD bisects ∠ BAC, by definition, we know that ∠ BAD≅ ∠ CAD. Additionally, the triangles have another set of congruent right angles, ∠ ABD and ∠ ACD. Finally, from the diagram we can also see that the triangles share a side, AD. Therefore, by the Reflexive Property of Congruence, we know this side is congruent in our triangles as well.
Since two angles and a non-included side of △ ABD are congruent to two angles and a non-included side of △ ADC, we can claim by the AAS Congruence Theorem that
△ ADB ≅ △ ADC.
Note that BD and DC are congruent corresponding sides and therefore we can say that
BD=DC.
Let's show this as a two-column proof.
Statement
Reason
1.
& AD bisects ∠ BAC & DB⊥ AB, DC⊥ AC
1.
Given
2.
∠ BAD≅ ∠ CAD
2.
Definition of angle bisector
3.
∠ ABD and ∠ ACD are right angles
3.
Definition of perpendicular lines
4.
∠ ABD≅ ∠ ACD
4.
Right Angles Congruence Theorem
5.
AD≅ AD
5.
Reflexive Property of Congruence
6.
△ ADB ≅ △ ADC
6.
AAS Congruence Theorem
b In this part, we are supposed to prove that a point lies on the bisector of an angle if it's equidistant from the two sides of the angle. Let's draw a diagram that illustrates this.
△ ABD and △ ADC.
We can see that one leg of the right triangles are congruent. We also see that the triangles share their hypotenuses, AD. By the Reflexive Property of Congruence, we know these sides are congruent. Therefore, we can use the HL Congruence Theorem to prove the triangles are congruent:
△ ABD ≅ △ ADC.
Since ∠ BAD and ∠ CAD are corresponding angles, we can say that ∠ BAD≅ ∠ CAD. Finally, we use the definition of an angle bisector to claim that AD bisects ∠ BAC. Let's show this as a two-column proof.
Statement
Reason
1.
& DC ⊥ AC, DB ⊥ AB & BD=CD
1.
Given
2.
∠ ABD and ∠ ACD are right angles
2.
Definition of perpendicular lines
3.
△ ABD and △ ACD are right triangles
3.
Definition of a right triangle
4.
BD≅ CD
4.
Definition of congruent segments
5.
AD≅ AD
5.
Reflexive Property of Congruence
6.
△ ADB ≅ △ ACD
6.
HL Congruence Theorem
7.
∠ BAD ≅ ∠ CAD
7.
Corresponding parts of congruent triangles are congruent
