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Without using a protractor, can you verify that every angle of the vacant quadrilateral in the adjacent figure is a right angle?

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In the square ABCD the shaded triangles are right angled and are the same.

In ∆LBM, 

m∠BLM + m∠BML + m∠LBM = 180° …. (Sum of the measures of the angles of a triangles is 180° ) 

∴ m∠BLM + m∠BML + 90° = 180° 

∴ m∠BLM + m∠BML = 90° …. (i) 

Now, ∆LBM and ∆LAP are same. 

∴ m∠BML = m∠ALP …. (ii) 

∴ m∠BLM + m∠ALP = 90° …. IFrom (i) and (ii)l 

Now, m∠ALP + m∠PLM + m∠BLM = 180° …. (The measure of a straight angle is 180°) 

∴ m∠ALP + m∠BLM + m∠PLM = 180° 

∴ 90° + m∠PLM = 180° 

∴ m∠PLM = 180°- 90° = 90° 

∴ m∠PLM is a right angle. 

Similarly, we can prove that the other angles of the vacant quadrilateral are right angles.

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