$A, B, C$ and $D$ lie on the circumference of a circle, and when the line segments $AB$ and $DC$ are extended they meet at $X$, outside the circle.

Prove that $△\mathrm{ACX}$, is similar to $△\mathrm{DBX}$.

$A, B, C, D, E$ and $F$ lie on the circumference of a circle, with $AB = BC$ and $DF$ perpendicular to $BE, AE, BE$ and $CE$ meet $DF$ at $X, Y$ and $Z$ respectively. Prove that $\angle \mathrm{EXY} = \angle \mathrm{EZY}$

.

Find the size of marked angles.

Find the size of marked angles.

Find the size of marked angles.

Justify your answers.

Find the size of marked angles.

Justify your answers.

For each statement, write down the converse. Determine which statements, and which of their converses, are true.

If $a = 7$ then $3a-2 = 19$.

For each statement, write down the converse. Determine which statements, and which of their converses, are true.

 If $b=a$ then $a-b=0$.

For each statement, write down the converse. Determine which statements, and which of their converses, are true. If it is a bird then it has wings.