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- We are starting off with triangle ABC
- When we see the triangle we know that length
- of AB = AC. This is a triangle with 2 sides of equal length
- We call this an isosceles triangle
- This means two sides are equal to each other.
- What i want to prove here is that these angles
- the angles opposite to the sides that are equal
- are the same. For side AC the opposite angle is angle ABC
- and for side AB the opposite angle is angle ACB
- I want to prove angle ABC = angle ACB
- So we have a lot of information here
- and the first step is to construct two triangles
- to construct two triangles, set up another point D here.
- Let us say D is the mid point of line BC
- this means BD = DC.
- Let me draw segment AD. Now we have constructed
- two triangles, triangle ABD & triangle ADC
- In these two triangles, AB = AC, BD = DC and they share
- the side AD.We know that triangle ABD = traingle ACD
- we know this because all sides of these triangles are equal
- What is useful about this information is if the sides are equal
- then their angles are equal.We have proved our result.
- angle ABC = angle ACB
- If we have an isosceles triangle with two sides equal
- then their base angles or angle opposite to these sides
- are equal.
- Now can we make the other statement, if the
- base angles are equal then the opposite sides are
- of teh same length. Let us construct another triangle
- this triangle is called ABC, and am going to start
- with the idea that angle ABC = angle ACB
- what we want to do is to prove
- that sides AB = side AC
- or length of AB = length of AC
- We need to have two triangles to prove this.
- Like we did earlier let us construct two triangles
- This time instead of defining the other point as mid point
- I am going to define D as the point that goes
- straight down from A
- This point intersects BC at a right angle
- Then in the two triangles these angles will be 90 degrees
- What is interesting about this is
- i have constructed AD such that
- AD is perpendicular to BC
- So over here we an angle and side in common
- We know these triangles are same by AAS method
- that is Angle Angle Side ( AAS)
- triangle ABD = triangle ACD
- If we know two triangles are equal, then
- every corresponding angle or side of the two triangles are same
- So we know sides AB = AC
- Now we have proved that if two angles are equal,
- then their opposite sides are of same length
- So side BD = side DC , side AB = side AC
- So point D is not only the mid point ,
- it is the place where AD bisects BC at 90 degrees