Let's analyze the image to determine the size of ∠TQS.

The given diagram shows:

• Triangle SQR with angles at S, Q, and R.
• ∠SQR is given as 65°, and ∠S is given as 42°.
• There is an exterior angle TQP which is 40°.

We want to find the measure of ∠TQS.

Step-by-Step Solution:

1. Find the Measure of ∠QSR: In triangle SQR, the sum of the internal angles must equal 180°.

   • ∠S + ∠SQR + ∠QSR = 180°
   • 42° + 65° + ∠QSR = 180°
   • ∠QSR = 180° - 107° = 73°

2. Relate to Exterior Angle TQS: Angle TQS and the interior angle ∠QSR form a linear pair, meaning they are supplementary (they sum to 180°).

   • ∠TQS + ∠QSR = 180°
   • ∠TQS + 73° = 180°
   • ∠TQS = 180° - 73° = 107°

Answer:

The size of ∠TQS is 107°.

Do you want details or have any questions?

Here are 5 related questions to expand on this topic:

1. How do you determine the measure of an unknown interior angle in a triangle?
2. What is the relationship between an exterior angle and its adjacent interior angle?
3. How can we verify if two angles are supplementary in a geometric figure?
4. What are the properties of linear pairs in geometry?
5. How do you apply the triangle angle sum property to different types of triangles?

Tip: Always remember that the sum of the internal angles of any triangle is 180°, and the exterior angle of a triangle equals the sum of the two non-adjacent interior angles.