Questions: If the angle ABD is 60 degrees, what is angle BAC? Show all your work.
Transcript text: If the $\angle A B D$ is 60 degrees, what is $\angle B A C$ ? Show all your work.
Solution
If ∠ABD is 60 degrees, what is ∠BAC?
Given
∠ABD=60∘
Since B, D, and C are collinear,
∠ABC=180∘.
Thus,
∠DBC=180∘−∠ABD=180∘−60∘=120∘
In △ABC, AD is the altitude, and D is the midpoint of BC, therefore
AD is also the median, angle bisector, and perpendicular bisector of BC.
Since AD is the median,
BD = CD.
Since AD is an altitude,
AD is perpendicular to BC. Therefore ∠ADB=∠ADC=90∘.
Since AD is the angle bisector,
∠BAD=∠CAD. Let ∠BAC=x. Then ∠BAD=∠CAD=2x.
Consider △ABD. We know ∠ABD=60∘ and ∠ADB=90∘. Therefore
∠BAD+∠ABD+∠ADB=180∘2x+60∘+90∘=180∘2x+150∘=180∘2x=30∘x=60∘