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Skywalker18
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Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
35%
(medium)
Question Stats:
74% (01:57) correct
26%
(02:35)
wrong
based on 279
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In the figure, JKLMNP is a regular hexagon .Find the measure of \(\angle\) MQN.
A. \(30^{\circ}\)
B. \(45^{\circ}\)
C. \(50^{\circ}\)
D. \(60^{\circ}\)
E. \(75^{\circ}\)
A. \(30^{\circ}\)
B. \(45^{\circ}\)
C. \(50^{\circ}\)
D. \(60^{\circ}\)
E. \(75^{\circ}\)
Attachments
Geometry_Hexagon.PNG [ 461.86 KiB | Viewed 15371 times ]
Skywalker18
vinnisatija
Theres a very simple explanation for this question.
Remember the formula for the sum of internal angles of a polygon.
Sum of angles= 180 X (n-2) ; where n= number of sides
In our case ; 180 ( 6-2) = 720.
Now this is a regular hexagon. All the angles are equal and all the sides are also equal.
therefore Each angle = 120
Now consider triangle LMN
we know angle M=120.
Since all the sides are equal, we know this is an isosceles triangle.
2x+120=180
x=30.
So angle L= N= 30.
Similarly, in triangle KLM,
angle k and M = 30.
From what we have proved above,
Angle QMN 120-30 = 90
Now we have angle QMN+ Angle MQN + angle QNM = 180
90+MQN+30=180
MQN=60
Hope its clear. Let me know through the kudos button
General Discussion
vinnisatija
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Schools: UFL '17 UFlorida '18
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can anyone provide an alternate explanation on this ?
ENGRTOMBA2018
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Skywalker18
An interesting question that requires you to be a bit creative with the information that you have been given.
The given polygon is a REGULAR hexagon --> all sides and all angles are equal. Every interior angle of a hexagon = 120 degrees.
As per the attached image, draw a circle around the given hexagon. Thus , now you get equal arcs LM=MN=NP=JP=JK=KL . As there are 6 arcs and all arcs must subtend a total of 360 degrees at the center of the circle ---> each arc subtends 60 degrees at the center.
Additionally, an angle subtended by any arc of a circle at the center = 2* angle subtended by the same arc on the circumference.
Consider triangle QNM,\(\angle{QNM}\) = angle subtended by LM = 30 degrees (as it is the angle subtended by arc LM on the circumference.)
Similarly, Arcs KJ+JP+PN subtend \(\angle{QMN}\) on the circumference. ---> \(\angle{QMN} = 90 degrees\).
Finally, in triangle QMN, \(\angle{QMN} + \angle {QNM} + \angle {MQN} = 180 degrees\) --->\(\angle {MQN}\) = 180-30-90 = 60 degrees.
D is the correct answer.
Hope this helps.
Attachments
2015-12-05_10-45-47.jpg [ 12.84 KiB | Viewed 14397 times ]
DensetsuNo
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Updated on: 20 May 2016, 07:29
Originally posted by DensetsuNo on 20 May 2016, 04:57.
Last edited by DensetsuNo on 20 May 2016, 07:29, edited 1 time in total.
Last edited by DensetsuNo on 20 May 2016, 07:29, edited 1 time in total.
I think there's an easier way to solve this problem.
As we were saying a regular hexagon has each angle measuring \(120°\).
Now let's look at the image posted above,
the theorem of vertical and adjacent angle pairs says really simply that opposite angles born from the intersection of two lines must be equal.
Therefore if the angle of the hexagon is \(120°\) the opposite angle will be \(120°\) as well,
but we also know that the round angle is 360°, therefore, the sum of the remaining opposite pair must be \(360° - 240° = 120°\),
and since they are opposite as well they have to be equal -> \(\frac{120°}{2}= 60°\)
As we were saying a regular hexagon has each angle measuring \(120°\).
Now let's look at the image posted above,
the theorem of vertical and adjacent angle pairs says really simply that opposite angles born from the intersection of two lines must be equal.
Therefore if the angle of the hexagon is \(120°\) the opposite angle will be \(120°\) as well,
but we also know that the round angle is 360°, therefore, the sum of the remaining opposite pair must be \(360° - 240° = 120°\),
and since they are opposite as well they have to be equal -> \(\frac{120°}{2}= 60°\)
Divyadisha
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Since it is a regular hexagon, angles made at dissection point by all diagonals will be equal.
Total number of angles made by all diagonals= 6
Total angle made at dissection point = 360
Any single angle measurement will b = 360/6= 60
D is the answer
Total number of angles made by all diagonals= 6
Total angle made at dissection point = 360
Any single angle measurement will b = 360/6= 60
D is the answer
Skywalker18
do you have easy solution?
Posts: 3
equilateral hexagon means all sides equal
hexagon has interior angle sum =(n-2)*180 , n= number of sides of a polygon (here) 6
(6-2)*180=720
If all the sides of a polygon are equal, then all its interior angles must be equal.
each angle will be 120
angleKLM=120 and KL & LM are equal forming a isosceles triangleLMK therefoere
angle LKM and angle LMK will be = 30 each
similarily in triangle LMN angle MLN & angle MNL = 30 each
now in triangle LMQ 30+30+ angle LQM=180
LQM=120
anles LQM+MQN =180 supplementry angle
angle MQN=60
D
hexagon has interior angle sum =(n-2)*180 , n= number of sides of a polygon (here) 6
(6-2)*180=720
If all the sides of a polygon are equal, then all its interior angles must be equal.
each angle will be 120
angleKLM=120 and KL & LM are equal forming a isosceles triangleLMK therefoere
angle LKM and angle LMK will be = 30 each
similarily in triangle LMN angle MLN & angle MNL = 30 each
now in triangle LMQ 30+30+ angle LQM=180
LQM=120
anles LQM+MQN =180 supplementry angle
angle MQN=60
D
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