Question

Find the unknown angle x.
circle theorem

Answer 1

`\sf\\1.\ \angle SCD=\angle DST\ \ \ [\textsf{The angle between a chord and a tangent through one of the}\\\textsf{}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \textsf{end points is equal to the angle in the alternate segment.]}\\or,\ \angle DST=80^o`

`\sf\\2.\ \angle DST+\angle DTS+\angle TDS=180^o\ \ \ [\textsf{Sum of angles of triangle is 180}^o.]\\or,\ 80^o+\angle DTS+60^o=180^o\\or,\ \angle DTS=40^o`

`\sf\\3.\ \angle DTS=\angle TED\ \ \ [\textsf{The angle between a chord and a tangent through one of the}\\\textsf{}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \textsf{end points is equal to the angle in the alternate segment.]}\\or,\ \angle TED=40^o`

`\sf\\4.\ \angle TDE+\angle TED+\angle DTE=180^o\ \ \ [\textsf{Sum of angles of triangle is }180^o.]\\or,\ x^o+40^o+37^o=180^o\\or,\ x^o=103^o`

In case you didn't understand what I told about the alternate segment theorem, please look at the images.

The angles that are equal are marked with same color.