4. Show that the diagonals of a square are equal and bisect each other at right angles.

Consider triangles DAB and CBA, AD = BC [Sides of a• square] AB is mmon. 1 2 C 0 L DAB = L CBA -A DAB A CBA BD = AC and L l = L 2 [90° each] [SAS] A 3 B [CPCT] •..(i) [CPCT] Proving as above we can show L3 = L4. '• ...(ii) Also, L 2 = L 3 [•: AB = BC, angles opposit.e to equal sides are equal.] .. L 1= L 4 [From (i), (ii), (iii)] Consider triangles AOD and COD, AD = DC OD is common. 1 [Sides of a square] QUADRILATERALS 111 L l = L 4 :. A AOD A COD .. OA = OC Similarly, we can show that OB:OD and L AOD = LCOD Also, L AOD + LCOD = 180° => 2L AOD = 180° [Proved above] [SAS] •..(iv) [CPCT] .••(v) [CPCT] [Linear pair] [Using (v)] => L AOD = 90° ...(vi) Hence, diagonals are equal and bisect each other at right angles.