SOLUTION: Given:Triangle ABC, M and N are the midpoints of AB and AC respectively. Prove: triangle AMN is similar to triangle ABC . Write the formal proof please

Question 761907: Given:Triangle ABC, M and N are the midpoints of AB and AC respectively. Prove: triangle AMN is similar to triangle ABC . Write the formal proof please
Answer by ramkikk66(644) (Show Source): You can put this solution on YOUR website!
Look at the diagram for the problem below (may not be exactly to scale)

Definition: Two triangles are said to be similar if their angles are equal.
Step 1:
We will use the "mid-point theorem" which states that: The line connecting the mid-points of two sides of a triangle, is parallel to the 3rd side.
So, since M is the mid-point of AB and N the mid-point of AC, MN (which connects them) is parallel to the 3rd side which is BC. MN || BC.
Step 2:
Since MN and BC are parallel, and AB and AC are like transversals to the parallel lines,
Angle AMN = Angle ABC, and angle ANM = angle ACB.
Step 3:
Now if we compare the angles of the two triangles ABC and AMN
-> angle A is common to both
-> Angle AMN = Angle ABC
-> Angle ANM = Angle ACB.
So the 2 triangles have all the 3 angles congruent. Hence by definition, triangles ABC and AMN are similar.
Hope this helps.
:)

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