Step by step
The question
Question: the line A T touches the circle at A, the angle between it and the chord A B is 33°, and angle A B C is 91°: find angle B A C.
1 Use the rule
The angle between the tangent and the chord A B is 33°.
The alternate segment theorem: it equals the angle at C, on the other side of the chord, in the alternate segment.
So angle A C B is 33°.
angle ACB = 33°
Careful!
Not the angle at B, next to the tangent angle.
The equal angle is across the chord, at C.
Careful: the angle in the OTHER segment, across the chord
2 Use the angle fact
The three angles of triangle A B C add up to 180°.
180 minus 33 minus 91 is 56°.
angle BAC = 180° − 33° − 91° = 56°
3 Check
Let's check: on the straight tangent line at A, the angle on the other side is 91°, by the same rule.
33 plus 56 plus 91 is 180°: a straight line.
So it is right!
on the tangent at A: 33° + 56° + 91° = 180°
Let's look back.
First: use the rule.
Then: use the angle fact.
Last: check.
Answer
angle BAC = 180° − 33° − 91° = 56°