In the figure, JKLMNP is a regular hexagon .

Question

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}

rohan89 · Accepted Answer

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

vinnisatija · Answer

can anyone provide an alternate explanation on this ?

ENGRTOMBA2018 · Answer

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.

DensetsuNo · Answer

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 -> \(\frac{120°}{2}= 60°\)

Divyadisha · Answer

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

mangamma · Answer

do you have easy solution?

ABHINAVSAXENA1611 · Answer

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

bumpbot · Answer

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In the figure, JKLMNP is a regular hexagon .

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In the figure, JKLMNP is a regular hexagon .

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