Given : isosceles triangle ABC with vertex Angle A and with angle bisector AD
Prove: AD is the median
Use 2 column proof
since AB = AC, use ASA to prove congruence of triangles ADB and ADC
That means that BD=DC, hence AD is the median from A to BC.
Statement | Reason
-------------------------------------------------------------
1. Triangle ABC is isosceles | Given
2. Angle A = Angle B | Definition of isosceles triangle
3. Angle ADC = Angle ADB | Angle bisector divides angle in two congruent angles
4. Triangle ADC is congruent to triangle ADB | Angle-side-angle congruence
5. AD = AD | Reflexive property of equality
6. AD bisects side BC | Definition of angle bisector
7. AD is a median of triangle ABC | Definition of median
Sure! Here's a two-column proof to demonstrate that AD is the median of the isosceles triangle ABC.
| Statements | Reasons |
|----------------------------------------|------------------------------------------------------------------------------------------------|
| 1. Triangle ABC is an isosceles triangle with vertex angle A. | Given |
| 2. AD is the angle bisector of angle A. | Given |
| 3. ∠CAD ≅ ∠BAD | Angle bisector definition |
| 4. AC ≅ BC | Definition of an isosceles triangle |
| 5. ∠CAD ≅ ∠BAD and AC ≅ BC | Congruent parts of congruent triangles are congruent (CPCTC) |
| 6. Triangle ACD ≅ Triangle BCD | Angle-side-angle (ASA) congruence postulate |
| 7. AD ≅ AD | Reflexive property of equality |
| 8. Triangle ABD ≅ Triangle ACD | Side-angle-side (SAS) congruence postulate |
| 9. AB ≅ AC | Corresponding parts of congruent triangles are congruent (CPCTC) |
| 10. AD is the median of triangle ABC | Definition of a median |
In this proof, we start by stating that triangle ABC is an isosceles triangle with vertex angle A. We are also given that AD is the angle bisector of angle A. From the angle bisector definition, we can conclude that ∠CAD ≅ ∠BAD. Additionally, since triangle ABC is isosceles, we know that AC ≅ BC.
Using the congruent parts of congruent triangles are congruent (CPCTC), we can conclude that ∠CAD ≅ ∠BAD and AC ≅ BC. Then, by the angle-side-angle (ASA) congruence postulate, we can conclude that triangle ACD ≅ triangle BCD.
Using the reflexive property of equality, we can state that AD ≅ AD. Then, using the side-angle-side (SAS) congruence postulate, we can conclude that triangle ABD ≅ triangle ACD.
Finally, by the corresponding parts of congruent triangles are congruent (CPCTC), we can state that AB ≅ AC. This shows that AD is the median of triangle ABC, as defined by the median being a line segment connecting a vertex of the triangle to the midpoint of the opposite side.
Hence, we have proven that AD is the median of triangle ABC.