Abc is a triangle where ad bisects angle a and d is the midpoint of bc prove that triangle is isosceles
In triangles adb and adc,
angle adb = angle adc
bd = dc
ad = ad
So, the triangles are congruent, meaning
ab = ac so abc is isosceles.
I've seen this mistakes in a lot of pages. But you should know S.S.A. axiom doesn't prove triangles are congurent..
To prove that triangle ABC is isosceles, we need to show that two sides of the triangle are equal in length.
Given:
- Triangle ABC with AD as the angle bisector of angle A.
- Point D is the midpoint of BC.
Proof:
1. We know that AD is the angle bisector of angle A. Therefore, angle BAD is congruent to angle CAD. (Angle Bisector Theorem)
2. Since D is the midpoint of BC, we have BD = CD. (Given)
3. Now, let's consider triangle ABD and triangle ACD.
- Angle BAD = Angle CAD (by the Angle Bisector Theorem)
- Angle ABD = Angle ACD (corresponding angles)
4. By angle-side-angle (ASA) congruence criterion, to prove congruence between triangles ABD and ACD, we need to prove two corresponding sides and the included angle are congruent.
5. We have:
- AD is a common side shared by both triangles.
- BD = CD (given)
6. Now, since angle BAD is congruent to angle CAD (proved in step 3), we can conclude that triangle ABD is congruent to triangle ACD by SAS congruence criterion.
7. Consequently, AB = AC (corresponding sides of congruent triangles).
8. Therefore, triangle ABC is isosceles since it has two equal sides, AB and AC.
By going through the above steps, we have proven that triangle ABC is isosceles based on the given conditions and the properties of triangles.