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A and B are the endpoints of a line segment. Segment AB is crossed through point C by another line segment with endpoints D and E. If ∠ACD > 90?, and the sum of ∠ACE and ∠BCD is x, then which of the following must be true?
A. x < 90
B. x > 90
C. 90 < x < 180
D. x < 180
E. x > 180
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A. x < 90
B. x > 90
C. 90 < x < 180
D. x < 180
E. x > 180
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Posts: 10
Ans will be B
since in the below diagram c is the point of intersection. If ∠ACD > 90 which means ∠BCD is less than 90 because the angle of a straight line is 180 degrees. for eg. if ∠ACD is 110 degrees then ∠BCD is max 70 degrees.
since in the below diagram c is the point of intersection. If ∠ACD > 90 which means ∠BCD is less than 90 because the angle of a straight line is 180 degrees. for eg. if ∠ACD is 110 degrees then ∠BCD is max 70 degrees.
∠ACD = ∠ECB (Opposite angles)
∠ACE = ∠BCD (Opposite angles)
Total angle around a point = 360
Let ∠ACD = ∠ECB = a
∠ACE + ∠BCD = x
2a+x = 360
We can test the minimum and maximum values for a to get the range
Minimum a = 91
2*91 + x = 360
x = 360-182 = 178
Maximum a = 179
2*179+x = 360
358+x = 360
x = 360-358 = 2
X<180
Answer D
∠ACE = ∠BCD (Opposite angles)
Total angle around a point = 360
Let ∠ACD = ∠ECB = a
∠ACE + ∠BCD = x
2a+x = 360
We can test the minimum and maximum values for a to get the range
Minimum a = 91
2*91 + x = 360
x = 360-182 = 178
Maximum a = 179
2*179+x = 360
358+x = 360
x = 360-358 = 2
X<180
Answer D
Attachments
angles.png [ 10.42 KiB | Viewed 2030 times ]
Bunuel
Here is another way of looking at this:
Consider the diagram shared by Sr1994 above.
∠ACE and ∠BCD are vertically opposite angles so they will be equal. Since their sum is x, each of them will be x/2.
Also, angles ACD and BCD form a straight angle and since ACD is more than 90, ∠BCD will be less than 90.
So all we can say is that x/2 < 90
x < 180
Answer (D)
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