In the xy-plane, points A, B and C are not on the same line. Is the sl

(1) The ratio of x-intercept to y-intercept of line AB = 1.
It implies that slope of line AB is -1 as slope = difference of y/difference of x, which is (positive) 1.
So, it makes 45∘ angle with X axis. Still, it does not give any information about point C. So insufficient.

(2) The measure of angle ABC is 50 degrees.
This statement doesn't give information about orientation/locations of points ABC. So, insufficient.

(1) + (2)
Let me try to solve it without drawing diagram.
With negative slope -1, line AB makes 45∘ angle with X axis, any line (e.g. BC) making angle less than 45∘ with line AB will have negative slope, similarly line making more than 45∘ angle with AB will turn into positive slope.
In this case BC makes angle 50∘ with AB, hence it has positive slope only (regardless of which quadrants these points lies in). Hence, sufficient.

Ans. C

Given: In the xy-plane, points A, B and C are not on the same line.
Asked: Is the slope of line BC negative?

(1) The ratio of x-intercept to y-intercept of line AB = 1
x/a + y/b = 1
Since a/b = 1
a = b = k
Equation of line AB : x+y=k
Slope of line AB = -1 = tan x ; x = -45
Slope of line BC is unkown
NOT SUFFICIENT

(2) The measure of angle ABC is 50 degrees.
There are 2 cases possible
Case 1 : Slope of line BC >=0
Case 2: Slope of line BC <0
NOT SUFFICIENT

(1) +(2)
(1) The ratio of x-intercept to y-intercept of line AB = 1
x/a + y/b = 1
Since a/b = 1
a = b = k
Equation of line AB : x+y=k
Slope of line AB = -1 = tan x ; x = -45
(2) The measure of angle ABC is 50 degrees.
There are 2 cases possible
Case 1 : Slope of line BC >=0
Case 2: Slope of line BC <0
As it is evident from the image.
NOT SUFFICIENT

IMO E

Stat 1:
as the ratio of both intercept is 1
the line AB would create 45 degree with x axis (considering both positive and negative intercept)
But no information about point C
Insufficient

Stat: 2
ANgle ABC =50 degree
Doesn't help anyway
Slope of BC can be positive or negative

Combined:
Considering both
If BC is above AB, BC would create 95 degree with X axis , Hence Slope of BC positive
Or BC is below AB, BC would create 5 degree with X axis , Hence slope of BC negative
Not Sufficinet

ANS E

In the xy-plane, points A, B and C are not on the same line. Is the slope of line BC negative?

(1) The ratio of x-intercept to y-intercept of line AB = 1
(2) The measure of angle ABC is 50 degrees.

Statement (1) tells us nothing about line BC --> insufficient --> eliminate (a) and (d)
Statement (2) tells us nothing about the relative locations of B and C --> insufficient --> eliminate (b)

Let's assess both statements.
If the x and y intercepts are equal, then the line AB has a slope of -1, and at a 45 degree angle.
We don't know which point on this line represents "A" versus "B", nor do we know which way, clockwise or counter clockwise, we'll need to go from the first line 50 degrees to get the new line.
However, the resultant line (which I recommend drawing to visualize) will have a positive slope no matter what --> both sufficient --> answer is (c)

we need 2 angles to get the confirmed angles only c can give that

(C) Both together are sufficient
Statement 1 implies slope of line AB is 1, implying the angle it makes with the x-axis is 45 degrees
Statement 2 implies angle made by line BC with x-axis is angle made by line AB+/-50 degrees = 45-50 or 45+50 = -5 degrees or 95 degrees
In both cases, slope is negative and hence both statements together are sufficient

Posted from my mobile device

(1) The ratio of x-intercept to y-intercept of line AB = 1
Which means that the intercepts are both of the same sign, negative or positive
In both cases the slope is negative and is equal to 45 degress.
This is insufficient to determine if slope of BC is negative as well.

(2) The measure of angle ABC is 50 degrees.
Insufficient on its own.

(1) + (2)
Any angle greater than 45 degrees is going to make the slope of the resulting line positive.
Therefore we can say that slope if line BC is positive.

Answer: C