ABCD is a cyclic quadrilateral, ABX is a straight line and PQ is a tangent to the circle at A�Angle °CBX 85= , angle °BAQ 55= and angle °CAD 42=

To find the unknown angles in the cyclic quadrilateral ABCD, we can use the properties of cyclic quadrilaterals and the relationships between the angles.

Step 1: Break Down the Problem

We know the following angles:

• $\angle CBX = 85^\circ$
• $\angle BAQ = 55^\circ$
• $\angle CAD = 42^\circ$

We need to find other angles related to the cyclic quadrilateral and tangent line properties.

Step 2: Relevant Concepts

1. In a cyclic quadrilateral, opposite angles sum to $180^\circ$.
2. A tangent to a circle forms a right angle with the radius at the point of contact. Therefore, the angle between the radius $OA$ (where $O$ is the center of the circle) and the tangent line $PQ$ (which touches the circle at point $A$) is $90^\circ$.
3. Use the exterior angle theorem which states that the angle formed outside the quadrilateral is equal to the sum of the opposite interior angles.

Step 3: Analysis and Detail

1. Calculate $\angle CAB$: From the tangent property at point $A$: $\angle CAB + \angle BAQ = 90^\circ$ Therefore, $\angle CAB = 90^\circ - 55^\circ = 35^\circ$

2. Calculate $\angle ABC$: Using the property of cyclic quadrilaterals, $\angle ABC + \angle CAD = 180^\circ$ So, $\angle ABC = 180^\circ - 42^\circ = 138^\circ$

3. Finding $\angle ACD$: Since $\angle CBX$ is given as $85^\circ$, and applying the property of opposite angles in cyclic quadrilaterals, $\angle ACD = \angle CBX = 85^\circ$

Step 4: Verify and Summarize

Now we have:

• $\angle CAB = 35^\circ$
• $\angle ABC = 138^\circ$
• $\angle ACD = 85^\circ$
• $\angle CAD = 42^\circ$

Final Answer

The angles in the cyclic quadrilateral ABCD, calculated through tangent and cyclic properties, are:

• $\angle CAB = 35^\circ$
• $\angle ABC = 138^\circ$
• $\angle ACD = 85^\circ$
• $\angle CAD = 42^\circ$