Hexagon problem

KrazyFK

Suppose $ABCDEF$ is a hexagon with all sides equal and opposite angles equal. Does it follow that it is regular? Explain your answer.

**Posted:** Thu Nov 05, 2009 8:41 am

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AndrewTom

I've always thought that a regular polygon had all its sides (and angles equal). Am I missing the point?

**Posted:** Thu Nov 05, 2009 9:40 am

isabella2296

I think what KrazyFX is saying is that if a hexagon has all sides equal but only opposite angles equal, does it follow that all the angles are equal, therefore making the hexagon regular.

**Posted:** Thu Nov 05, 2009 9:44 am

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AndrewTom

Thanks, I misunderstood. I should have realised that someone like KrazyFK would have a problem worthy of attack.

Pete Hein:

Problems worthy of attack
Prove their worth by hitting back.

**Posted:** Thu Nov 05, 2009 9:47 am

minime1235813 · New Member Offline Joined: 05 Nov 2009 Posts: 4

No, Imagine it like the Following, but visualize it with equal sides. (I couldnt figure out the program... First time using it)

[/img]

**Posted:** Thu Nov 05, 2009 1:07 pm

fishythefish

minime1235813 is correct.

Contradictory example:

$AB=BC=CD=DE=EF=FA$ ( $=1$ if you want), but

$m\angle ABC=m\angle DEF=119^{\circ}$
$m\angle BCD=m\angle EFA=121^{\circ}$
$m\angle CDE=m\angle FAB=120^{\circ}$

**Posted:** Thu Nov 05, 2009 3:55 pm

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minime1235813 · New Member Offline Joined: 05 Nov 2009 Posts: 4

Its actually kinda funny, in math the last year, We were taught this to be true until I showed my teacher (-:

**Posted:** Fri Nov 06, 2009 1:22 pm

Arrange your tan · Riemann Hypothesis Offline Joined: 08 Nov 2007 Posts: 383

Or, make it an hourglass shape:

D\--------------/C
....\............../
......\.........../
......E\......../B
......../........\
......./...........\
....../..............\
...F/--------------\A

$m\angle A = m\angle C = m \angle D = m \angle F$

$m\angle B = m\angle E$ . . . (They are reflex angles. That is, they are each greater than 180 degrees.)

Or "flip out" the sides and make angle B and angle E both obtuse for a
different type, again having two different numbers of degrees of angles
inside the hexagon.

**Posted:** Fri Nov 06, 2009 9:24 pm

fishythefish

I like that solution, because you can draw it on the board without being overly concerned that you measured the angles correctly, as I would have had to.

**Posted:** Sat Nov 07, 2009 2:06 pm

_________________
I am NOBODY. NOBODY is PERFECT. Therefore, I am PERFECT. Ninja

There are 3 types of people in the world. Those who can count and those who can't. maybe

ATTENTION ALL CALCOHOLICS!!! Know your limits. Don't drink and derive. Rotfl