Proving angles are equal in a triangle in a circle.

[markosheehan]

View attachment 6094

hi can someone help me work this out. i think it has something to do with the exterior angle of a triangle is equal to the sum of the interior angles but i can't work it.

 

[Evgeny.Makarov]

Prove that $\angle CAB=\angle CBA$ by connecting these angles to measures of arcs using the inscribed angle theorem and its corollary about tangent lines.

 

[markosheehan]

how do you know the arcs AC and CB are the same

 

[mathlearn]

how do you know the arcs AC and CB are the same

Are you familiar with the Alternate Segment theorem, which states that

The angle between the tangent and chord at the point of contact is equal to the angle in the alternate segment.

View attachment 6099

So now do you see a similarity between the angle marked in blue in the above diagram & the angle marked in blue in your diagram.

$\therefore$ It can be said that $\angle CAT = \angle CBA$ using alternate segment theorem

Now what can be said about the triangle CBA using it's angles?

 

[markosheehan]

It's isosceles