Find the area of an equilateral triangle having altitude of hcm Prove that if a transversal intersects two parpllel - Maths - Lines and Angles

Question

Find the area of an equilateral triangle having altitude of
hcm
Prove that if a transversal intersects two parpllel lines the
each pair of alternate interior angles are equal
If a transversal intersects two lines in such a way that a pair
of alternate interior angles are equal, then the two lines are
parallel
If a transversal intersects two parallel lines then each pair
of consecutive interior angles are supplementary

Pronay · Accepted Answer

1)

Given: Î”ABC is an equilateral triangle AD is perpendicular to BC $AD=h cm$ $∠ BAD=180 - ∠ ADB - ∠ ABC =180 - ∠ ADC - ∠ ACB = ∠ CAD$ In Î”ABD and Î”ACD AB=AC $∠ BAD= ∠ CAD$ AD = AD Therefore Î”ABD $≅$ Î”ACD BD=DC (CPCT) Therefore ${AB}^{2} = {AD}^{2} + {BD}^{2} (pythagorus theorem) \Rightarrow {AB}^{2} = h^{2} + \frac{{BC}^{2}}{2} \Rightarrow {AB}^{2} = h^{2} + \frac{{AB}^{2}}{2} (since AB = BC= AC) \Rightarrow \frac{{AB}^{2}}{2} = h^{2} \Rightarrow AB= \sqrt{2} h$ $Area of Δ ABC = \frac{\sqrt{3}}{4} {AB}^{2} = \frac{\sqrt{3}}{4} {(\sqrt{2} h)}^{2} = \frac{\sqrt{3}}{2} h^{2} {cm}^{2}$