Alternate Segment Theorem

Worked example

1. 1

   $\angle TAB=58^{\circ }$ — The angle between the tangent TA and the chord AB. Both lines start at A, the point where the tangent touches the circle.

2. 2

   $C\text{ is in the alternate segment}$ — Chord AB cuts the circle in two. C is on the far side of AB from the 58°, so it is in the other, alternate, segment.

3. 3

   $\angle ACB=58^{\circ }$ — The angle at C is made by lines to the two ends of the same chord, A and B. Alternate segment theorem: it equals the tangent–chord angle.

Worked example

1. 1

   $\angle TAB=70^{\circ }\text{ uses chord }AB$ — First name the chord. The 70° sits between the tangent and chord AB.

2. 2

   $\angle ACB=70^{\circ }$ — Angle ACB has arms to A and B, so it stands on chord AB, and C is across AB from T. Alternate segment theorem.

3. 3

   $\angle SAC=45^{\circ }\text{ uses chord }AC$ — Now the other side of the tangent. The 45° sits between the tangent and chord AC.

4. 4

   $\angle ABC=45^{\circ }$ — Angle ABC has arms to A and C, so it stands on chord AC, and B is across AC from S. Alternate segment theorem.

5. 5

   $70+45=115$ — Add the two angles at A that sit on the tangent line.

6. 6

   $\angle BAC=180-115=65^{\circ }$ — SAT is a straight line, and angles on a straight line add up to 180°.

7. 7

   $65+70+45=180$ — Check: the three angles of triangle ABC add up to 180°, so the answers agree.

Worked example

1. 1

   $\angle ACB=112÷2=56^{\circ }$ — Pick any point C on the big arc, across AB from T. The angle at the centre is twice the angle at the circumference, so halve 112.

2. 2

   $\angle TAB=56^{\circ }$ — C is in the alternate segment, so by the alternate segment theorem the tangent–chord angle is also 56°.

Worked example

1. 1

   $OA=OB$ — Both are radii, so triangle OAB is isosceles and its base angles at A and B are equal.

2. 2

   $180-112=68$ — Angles in a triangle add up to 180°. The two base angles share what is left.

3. 3

   $\angle OAB=68÷2=34^{\circ }$ — Equal base angles, so each gets half.

4. 4

   $\angle OAT=90^{\circ }$ — A tangent meets the radius at 90°.

5. 5

   $\angle TAB=90-34=56^{\circ }$ — Chord AB lies between the radius OA and the tangent, so take the 34° away from the 90°.

Worked example

1. 1

   $\angle ACB=64^{\circ }$ — The tangent TA and the chord AB make 64°, and C is across AB from it. Alternate segment theorem: angle ACB is also 64°.

2. 2

   $\angle BAC=64^{\circ }$ — BA = BC, so triangle ABC is isosceles. Its base angles, at A and C, are equal.

3. 3

   $64+64=128$ — Add the two base angles you now know.

4. 4

   $\angle ABC=180-128=52^{\circ }$ — Angles in a triangle add up to 180°, so the third angle is what is left.

5. 5

   $\angle ADC=180-52=128^{\circ }$ — ABCD is a cyclic quadrilateral, and B and D are opposite corners. Opposite angles of a cyclic quadrilateral add up to 180°.