Midpoints and angles

Geometry Level 4

In △ A B C \triangle ABC , points M M and N N are the midpoints of A C AC and B C BC respectively. It is known that ∠ M A N = 15 ° \angle MAN=\ang{15} and ∠ B A N = 45 ° . \angle BAN=\ang{45}. Find the measure of ∠ A M B . \angle AMB.


The answer is 45.

Sathvik Acharya by Sathvik Acharya , 17, India

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1 solution

W L O G l e t A M = 1. ∠ M A B = 15 + 45 = 6 0 o . ∴ Δ A M N i s S A A . S o l v i n g M N = 0.366025 = 1 2 A B . ∴ A B = 0.73205 i n Δ M A B w h i c h i s S A S . S o l v i n g ∠ A M B = 4 5 o . WLOG~let~AM=1.\\ \angle~MAB=15+45=60^o.\\ \therefore~\Delta~AMN~is~SAA.\\ Solving~MN=0.366025=\frac 1 2 AB.\\ \therefore~AB=0.73205~in~\Delta~MAB~which~is~SAS.\\ Solving~\angle~AMB=\Large~~\color{#D61F06}{45^o}.

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