P, Q and R are the three angles of a triangle. If P − Q = 20° and Q − R = 26°, then ∠ P is equal to:

Correct — A. Rule: the three angles of a triangle add to 180°.

Express both other angles through Q:
P = Q + 20°, R = Q − 26°

(Q + 20) + Q + (Q − 26) = 180
3Q − 6 = 180 → 3Q = 186 → Q = 62°

P = 62 + 20 = 82° → option (a)
Check: 82 + 62 + 36 = 180 ✔

Why the others are wrong

• (b)
  46° — 46° is 20 + 26, the total gap between P and R, not an angle of the triangle. Take P = 46° and the other two become 26° and 0°, which is not a triangle.
• (c)
  52° — 52° fails the sum test: P = 52° forces Q = 32° and R = 6°, adding to 90° rather than 180°. With the gaps fixed, any P below 82° leaves the triangle short.
• (d)
  92° — 92° overshoots. P = 92° gives Q = 72° and R = 46°, a total of 210°, so the largest angle has been pushed 10° too far.

Two differences and one total make three equations in three unknowns, but you never need three variables — write everything in terms of the middle angle Q.

P is 20° above Q and R is 26° below it, so the sum is 3Q − 6. Setting that equal to 180° is the entire question.

The angles come out as 82°, 62° and 36°: one triangle, no ambiguity, and the differences confirm themselves at 20° and 26°.

Key facts

• The interior angles of any triangle total 180°.
• P − Q = 20° and Q − R = 26° together give P − R = 46°.
• The solution here is P = 82°, Q = 62° and R = 36°.

Common traps

• Adding 26° instead of subtracting it when writing R in terms of Q.
• Answering with Q = 62° when the question asked for P.
• Treating 20° and 26° as angles rather than as differences.

No directly related past PYQ was found.