Steply

The alternate segment theorem

Geometry · 20 problems, easy to harder

▶ Watch the first one solved

How we do it

  1. Use the rule
  2. Use the angle fact
  3. Check

The steps for Tangent at A, angle between the tangent and AB 65°, angle ABC 70°…. Every solution names its steps, so you always know what comes next.

Worked examples

Tangent at A, angle between the tangent and AB 65°, find the angle ACB Easy

The question

Question: the line A T touches the circle at A, the angle between it and the chord A B is 65°: find angle A C B.

1 Use the rule

The angle between the tangent and the chord A B is 65°.

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 65°.

angle ACB = 65°

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 Check

Let's check: both angles stand on the same chord A B, from opposite sides of it.

So they are equal: 65°.

So it is right!

65° = angle TAB = angle ACB

Let's look back.

First: use the rule.

Last: check.

Answer
Careful: the angle in the OTHER segment, across the chord

▶ Watch it solved

Tangent at A, angle between the tangent and AB 65°, angle ABC 70°… Medium

The question

Question: the line A T touches the circle at A, the angle between it and the chord A B is 65°, and angle A B C is 70°: find angle B A C.

1 Use the rule

The angle between the tangent and the chord A B is 65°.

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 65°.

angle ACB = 65°

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 65 minus 70 is 45°.

angle BAC = 180° − 65° − 70° = 45°

3 Check

Let's check: on the straight tangent line at A, the angle on the other side is 70°, by the same rule.

65 plus 45 plus 70 is 180°: a straight line.

So it is right!

on the tangent at A: 65° + 45° + 70° = 180°

Let's look back.

First: use the rule.

Then: use the angle fact.

Last: check.

Answer
angle BAC = 180° − 65° − 70° = 45°

▶ Watch it solved

Tangent at A, angle between the tangent and AB 47°, angle ABC 66°… Harder

The question

Question: the line A T touches the circle at A, the angle between it and the chord A B is 47°, and angle A B C is 66°: find angle B A C.

1 Use the rule

The angle between the tangent and the chord A B is 47°.

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 47°.

angle ACB = 47°

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 47 minus 66 is 67°.

angle BAC = 180° − 47° − 66° = 67°

3 Check

Let's check: on the straight tangent line at A, the angle on the other side is 66°, by the same rule.

47 plus 67 plus 66 is 180°: a straight line.

So it is right!

on the tangent at A: 47° + 67° + 66° = 180°

Let's look back.

First: use the rule.

Then: use the angle fact.

Last: check.

Answer
angle BAC = 180° − 47° − 66° = 67°

▶ Watch it solved

Practice

Try each one, then open the answer. Stuck? Open the problem to watch every step.

Easy

  1. tangent at A, angle between the tangent and AB 65°, find the angle ACB
    Answer
    Careful: the angle in the OTHER segment, across the chord
  2. tangent at A, angle between the tangent and AB 50°, find the angle ACB
    Answer
    Careful: the angle in the OTHER segment, across the chord
  3. tangent at A, angle between the tangent and AB 72°, find the angle ACB
    Answer
    Careful: the angle in the OTHER segment, across the chord
  4. tangent at A, angle between the tangent and AB 38°, find the angle ACB
    Answer
    Careful: the angle in the OTHER segment, across the chord
  5. tangent at A, angle between the tangent and AB 55°, find the angle ACB
    Answer
    Careful: the angle in the OTHER segment, across the chord
  6. tangent at A, angle between the tangent and AB 44°, find the angle ACB
    Answer
    Careful: the angle in the OTHER segment, across the chord
  7. tangent at A, angle between the tangent and AB 61°, find the angle ACB
    Answer
    Careful: the angle in the OTHER segment, across the chord

Medium

  1. tangent at A, angle between the tangent and AB 65°, angle ABC 70°, find the angle BAC
    Answer
    angle BAC = 180° − 65° − 70° = 45°
  2. tangent at A, angle between the tangent and AB 50°, angle ABC 60°, find the angle BAC
    Answer
    angle BAC = 180° − 50° − 60° = 70°
  3. tangent at A, angle between the tangent and AB 40°, angle ABC 75°, find the angle BAC
    Answer
    angle BAC = 180° − 40° − 75° = 65°
  4. tangent at A, angle between the tangent and AB 58°, angle ABC 64°, find the angle BAC
    Answer
    angle BAC = 180° − 58° − 64° = 58°
  5. tangent at A, angle between the tangent and AB 35°, angle ABC 80°, find the angle BAC
    Answer
    angle BAC = 180° − 35° − 80° = 65°
  6. tangent at A, angle between the tangent and AB 70°, angle ABC 45°, find the angle BAC
    Answer
    angle BAC = 180° − 70° − 45° = 65°
  7. tangent at A, angle between the tangent and AB 62°, angle ABC 53°, find the angle BAC
    Answer
    angle BAC = 180° − 62° − 53° = 65°

Harder

  1. tangent at A, angle between the tangent and AB 47°, angle ABC 66°, find the angle BAC
    Answer
    angle BAC = 180° − 47° − 66° = 67°
  2. tangent at A, angle between the tangent and AB 52°, angle ABC 83°, find the angle BAC
    Answer
    angle BAC = 180° − 52° − 83° = 45°
  3. tangent at A, angle between the tangent and AB 68°, angle ABC 57°, find the angle BAC
    Answer
    angle BAC = 180° − 68° − 57° = 55°
  4. tangent at A, angle between the tangent and AB 33°, angle ABC 91°, find the angle BAC
    Answer
    angle BAC = 180° − 33° − 91° = 56°
  5. tangent at A, angle between the tangent and AB 75°, angle ABC 75°, find the angle BAC
    Answer
    angle BAC = 180° − 75° − 75° = 30°
  6. tangent at A, angle between the tangent and AB 42°, angle ABC 48°, find the angle BAC
    Answer
    angle BAC = 180° − 42° − 48° = 90°