Question

I am supposed to find the value of 'a' but I don't know why we add a third line. Can someone please explain to me how to solve it and why?

Answer 1

m < a = 95 degrees.

Explanation:

We need to add a line which passes through point C and is parallel to AB.

Let this line be BF.

Now m < ABC = m < BCF = 35 degrees ( alternate angles).

and m < FCD = 180 - 120 = 60 degrees (same side angles add up to 180 degrees).

So m < a = m < BCF + m < FCD = 35 + 60 = 95 degrees.

Answer 2

Explanation:

It seems easiest to relate the angles if we can take advantage of the fact that alternate interior angles where a transversal crosses parallel lines are congruent. We can use this fact a couple of ways:

1. draw line CF to the right from point C parallel to AB and DE. Then angle BCF is 35°, matching angle CBA.

Angles FCD and CDE are supplementary, being same-side angles where transversal CD crosses parallel lines CF and DE. Hence angle FCD is 180° -120° = 60°.

Angle C is the sum of angles BCF and FCD, so is 35° + 60° = 95°. In short, ...

a° = 95°

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2. We can extend lines BC and ED so they meet at point G, forming triangle CGD. The angle at G is an alternate interior angle with angle B where transversal BG crosses parallel lines AB and GE. Hence angle G is 35°.

Angle CDG is the supplement to angle CDE, so is 180° -120° = 60°. And angle a° is the sum of opposite interior angles CDG and CGD, so is ...

a° = ∠CDG + ∠CGD = 60° +35°

a° = 95°