The lines m, l, and n intersect at point Q. What are the mea

The lines m, l, and n intersect at point Q. What are the measures of the six angles formed by these lines?

All the same color angles are equal. Thus we need to find the measures of say z, w, and v, while knowing that \(z+w+v=180\).

(1) The measure of z is one half the square of the measure of x --> \(z=\frac{x^2}{2}\) --> \(z=\frac{w^2}{2}\). Not sufficient.

(2) v + w = 130 --> \(z=50\) (from \(z+w+v=180\)). Not sufficient.

(1)+(2) \(z=50\) --> \(w=10\) (from \(z=\frac{w^2}{2}\)) and \(v=120\). Sufficient.

Answer: C.

The lines m, l, and n intersect at point Q. What are the measures of the six angles formed by these lines?

(1) The measure of z is one half the square of the measure of x.

Keep in mind that we have lots of vertical angles here...

z = 1/2 x^2 which doesn't tell us a whole lot because we don't know what z or x are. Insufficient.

(2) v + w = 130
if v + w = 130 then x + y = 130 as well because they are opposite angles. z + t are vertical angles and they equal the remainder of angles. z + t = 360 - (x + y) - (v + w) --> z + t = 360 - 130 - 130. z + t = 100 z = t because they are vertical angles. so z, t = 50 But we don't know the rest of the measures. Insufficient.

1 +2) If z = 1/2 x^2 then 50 = 1/2 x^2 --> 100 = x^2 --> x = 10. The opposite angle to x, w = 10 as well.

x + y = 130 --> 10 + y = 130 y = 120. y and it's opposite angle v = 120. Sufficient.

C

(1) z = 1/2x². No more info about x or z, IS.
(2) v+w= 130, hence t=180-130 = 50. But v could be 30 and w 100 or 90 and 40. IS.

Together: z = t = 50 = 1/2x² => x = w = 10 ==> y = v = 180 - 50 - 10 = 120. SUFF.

C

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