In triangle ABC, ∠A = 70° and ∠B = 80°. Let D be the incentre of ΔABC (the point where the internal angle bisectors meet). If ∠ACB = 2x° and ∠BDC = y°, what are the values of x and y respectively?

Introduction / Context:
This question combines basic angle sum properties of a triangle with a standard property of the incentre. The incentre is the point where the internal angle bisectors intersect. Once you know all three angles of the triangle, you can find x, and then use the incentre property to find angle ∠BDC.

Given Data / Assumptions:
• Triangle ABC has angles ∠A = 70° and ∠B = 80°.
• Let ∠ACB = 2x°.
• D is the incentre of triangle ABC.
• ∠BDC is denoted by y°.
• The incentre lies inside the triangle and angle ∠BDC is an angle at the incentre formed between the lines BD and CD.

Concept / Approach:
First apply the angle sum property of a triangle: sum of interior angles is 180 degrees. This will give ∠ACB directly and hence x. Next use the known property for the angle formed at the incentre between lines joining the incentre to two vertices: ∠BIC (or here ∠BDC) equals 90° plus half of angle A, since A is the angle between the sides opposite to B and C.

Step-by-Step Solution:
Step 1: Use the angle sum property of triangle ABC: ∠A + ∠B + ∠C = 180°. Step 2: Substitute ∠A = 70°, ∠B = 80° and ∠C = 2x°. Step 3: So 70° + 80° + 2x° = 180° → 150° + 2x° = 180°. Step 4: Subtract 150°: 2x° = 30°, so x = 15°. Step 5: Therefore ∠ACB = 2x° = 2 × 15° = 30°. Step 6: There is a well known property of the incentre: angle between the lines from the incentre to vertices B and C equals 90° plus half of angle A. Step 7: Thus ∠BDC = 90° + (∠A / 2) = 90° + (70° / 2) = 90° + 35° = 125°.

Verification / Alternative check:
Check that all angles are consistent: A = 70°, B = 80°, C = 30° sum to 180°, so triangle ABC is valid. The incentre formula ∠BIC = 90° + A/2 is standard for any triangle, so using A = 70° gives 125°, confirming our value of y.

Why Other Options Are Wrong:
• 15, 130 uses the correct x but an incorrect adjustment for the incentre angle, perhaps using 90° plus half of the wrong angle.
• 35, 40 and 30, 150 either assign the wrong value to x or misapply the incentre angle formula.

Common Pitfalls:
A common error is to confuse the formula for angle at the incentre with the formula for angle at the circumcentre. Another pitfall is miscalculating the remaining angle in the triangle. Always apply the sum of angles in a triangle carefully before using centre related formulas.

Final Answer:
The values are x = 15 and y = 125.