Q17 of 39 Page 150

In ΔXYZ, ∠XYZ = 90° and XY = 1/2 XZ; Lets prove that ∠YXZ = 60°.

Given: ΔXYZ, ∠XYZ = 90° and


To prove: ∠YXZ = 60°


The figure for the above question is as shown below,



Given ΔXYZ is a right - angled triangle, now applying the properties of the triangle,


We know



Now when θ = ∠XZY, XY is the opposite side and XZ is the hypotenuse, then the above equation becomes,



Given



Equating equation (i) and (ii), we get



But,


Hence θ = 30°


Or ∠XZY = 30°….(iii)


Now in ΔXYZ,


We know in a triangle the sum of all three interior angles is equal to 180°.


So in this case,


∠XYZ + ∠XZY + ∠YXZ = 180°


Substituting given value and value from equation (iii) in above equation we get


90° + 30° + ∠YXZ = 180°


⇒ ∠YXZ = 180° - 90° - 30°


⇒ ∠YXZ = 60°


Hence proved


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