In △ABC, M and N are points on the sides AB and AC respectively, such that AM=MB. If ∠AMN=∠ABC , AB=8 cm and CN=6 cm, what is the length of AN?
Answers
Answer:
Hey mate! This is your answer.
Step-by-step explanation:
This question is the proof of midpoint theorem
and can be solved easily .
To Prove: DE || BC and DE = 1212 BC.
Construction: Extend line segment DE to F such that DE = EF.
Proof: In △△ ADE and △△ CFE
AE = EC (given)
∠∠AED = ∠∠CEF (vertically opposite angles)
DE = EF (construction)
hence
△△ ADE ≅≅ △△ CFE (by SAS)
Therefore,
∠ADE = ∠CFE (by c.p.c.t.)
∠DAE = ∠FCE (by c.p.c.t.)
and AD = CF (by c.p.c.t.)
The angles ∠ADE and ∠CFE are alternate interior angles assuming AB and CF are two lines intersected by transversal DF.
Similarly, ∠DAE and ∠FCE are alternate interior angles assuming AB and CF are two lines intersected by transversal AC.
Therefore, AB ∥ CF
So, BD ∥ CF
and BD = CF (since AD = BD and it is proved above that AD = CF)
Thus, BDFC is a parallelogram.
By the properties of a parallelogram, we have
DF ∥ BC
and DF = BC
DE ∥ BC
and DE = 1212BC (DE = EF by construction)
Hence proved.
Hope it helps!
Please mark it as brainliest.
