1.

∆ABC and ∆ADC are two right triangles with common hypotenuse AC. Prove that ∠CAD = ∠CBD.

Answer»

Data: ∆ABC and ∆ADC are right angled triangles having common hypotenuse AC. 

To Prove: ∠CAD = ∠CBD 

Proof: In ∆ABC, ∠ABC = 90° 

∴ ∠BAC + ∠BCA = 90° …………. (i) 

In ∆ADC, ∠ADC = 90° 

∴ ∠DAC + ∠DCA = 90° …………… (ii) 

Adding (i) and (ii), 

∠BAC + ∠BCA + ∠DAC + ∠DCA = 90 + 90 

(∠BAC + ∠DAC) + (∠BCA + ∠DCA) = 180° 

∠BAD + ∠BCD = 180° 

∴ ∠ABC + ∠ADC = 180° 

If opposite angles of a quadrilateral are supplementary, then it is a cyclic quadrilateral. 

∴ ∠CAD = ∠CBD (∵ Angles in the same segment).


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