Question

Suppose two adjacent angles are supplementary. Show that if one of them is an obtuse angle, then the other angle must be acute.

Answer 1

In the figure 

∠APQ + ∠BPQ = 180° and ∠APQ is an obtuse angle. 

To prove: ∠BPQ < 90° 

Construction: Draw a perpendicular SP at P 

Proof: In the fig 

∠ APS +∠SPQ + ∠QPS = 180° 

90 + ∠SPQ + ∠QPB = 180° 

∠SPQ + ∠QPB = 180-90 

∠SPQ +∠QPB = 90 

∴∠SPQ < 90° and∠QPB < 90° 

∴∠QPB is an acute angle 

∴∠APQ is an obtuse angle