The difference between two supplementary angles is 20°. If the smaller of…

Two angles are called supplementary when the sum of their measures equals 180°. When two supplementary angles also differ by a fixed amount, both angles can be found by solving a pair of simple linear equations built from their sum and their difference.

1. Let the smaller angle be p° and the larger angle be q°. Since the two angles are supplementary, p + q = 180°.

2. The two angles differ by 20°, so q − p = 20°.

3. Adding the two equations: (p + q) + (q − p) = 180° + 20°, which gives 2q = 200°, so q = 100°.

4. Subtracting the difference equation from the sum equation: (p + q) − (q − p) = 180° − 20°, which gives 2p = 160°, so p = 80°.

5. Substituting p = 80° into the given expression: 3p − 50° = 3(80°) − 50° = 240° − 50° = 190°.

Cross-check: p + q = 80° + 100° = 180° (the angles are indeed supplementary) and q − p = 100° − 80° = 20° (matching the given difference), confirming p = 80° is correct.

So, the value of 3p − 50° is 190°.