Write a two column proof. Start with stating the given statements and the statement that we will prove.
Statements
Reasons
1.
BD⊥ AC and
△ ABC is an isosceles with base AC
1.
Given
2.
∠ BDA and ∠ BDC are right angles.
2.
Definition of a Right Angle
3.
∠ BDA ≅ ∠ BDC
3.
All right angles are congruent
4.
AB ≅ BC
4.
Definition of Isosceles Triangle
5.
∠ BAD ≅ ∠ BCD
5.
Isosceles Triangle Theorem
6.
△ BAD ≅ △ BCD
6.
Angle-Angle-Side Theorem
7.
∠ ABD ≅ ∠ CBD
7.
Congruent parts of the congruent triangles are congruent
8.
BD bisects ∠ ABC
8.
Definition of Angle Bisector
Practice makes perfect
Let's start with showing the given information on the diagram.
To show that BD bisects ∠ ABC, let's write a two column proof. As always, we will start with stating the given statements and the statement that we will prove.
Given: BD⊥ AC and
△ ABC is an isosceles with base AC
Prove: BD bisects ∠ ABC
Using the given information BD⊥ AC, we will write the second step by the Definition of a Right Angle.
2. Definition of a Right Angle
∠ BDA and ∠ BDC are right angles.
Since we know that all right angles are congruent, we can write the third step.
3. All right angles are congruent
∠ BDA ≅ ∠ BDC
Considering the given information △ ABC is isosceles with base AC, let's write the next step by the Definition of an Isosceles Triangle.
4. Definition of an Isosceles Triangle
AB ≅ BC
The Isosceles Triangle Theorem states the following.
If two sides of a triangle are congruent,
then the angles opposite
those sides are congruent.
In this case, the fifth step of the proof can be written by this theorem.
5. Isosceles Triangle Theorem
∠ BAD ≅ ∠ BCD
Looking at the diagram, we see that two angles and the nonincluded side of △ BAD are congruent to two corresponding angles and the nonincluded side of △ BCD. Thus, the next step can be written by the Angle-Angle-Side Theorem.
6. Angle-Angle-Side Theorem
△ BAD ≅ △ BCD
Since the congruent parts of the congruent triangles are congruent, let's write the seventh step.
7. CPCTC
∠ ABD ≅ ∠ CBD
Finally, using the Definition of Angle Bisector, we can complete the proof.
8. Definition of Angle Bisector
BD bisects ∠ ABC
Combining these steps, let's construct the two column proof.
Statements
Reasons
1.
BD⊥ AC and
△ ABC is an isosceles with base AC
1.
Given
2.
∠ BDA and ∠ BDC are right angles.
2.
Definition of Right Angle
3.
∠ BDA ≅ ∠ BDC
3.
All right angles are congruent
4.
AB ≅ BC
4.
Definition of Isosceles Triangle
5.
∠ BAD ≅ ∠ BCD
5.
Isosceles Triangle Theorem
6.
△ BAD ≅ △ BCD
6.
Angle-Angle-Side Theorem
7.
∠ ABD ≅ ∠ CBD
7.
Congruent parts of the congruent triangles are congruent
