Q. 10

# Two straight lines parallel to the sides BC and BA respectively through two vertices A and C of ΔABC meet at D. Let’s prove that ∠ABC = ∠ADC.

Given: ΔABC, line l||BC, line m||BA, line l and line m are meeting at point D

The figure for the given question is as shown below, Now in the given figure BA is parallel to line m with AC as tranversal line, so

BAC = ACD……….(i) (as they form is alternate interior angles)

Similarly, BC is parallel to line l with AC as tranversal line, so

ACB = CAD……….(ii) (as they form is alternate interior angles)

Now consider ΔABC,

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

So in this case,

ABC + BAC + ACB = 180°

Substituting the values from equation(i) and (ii), we get

ABC + ACD + CAD = 180°

ABC = 180° - ACD - CAD…………(iii)

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

So in this case,

Equating equation (iii) and equation (iv), we get

Hence proved

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