Find Angle X for this Octagon! | Step-by-Step Explanation

[debdasmukhopadhyay]

Beautiful. So simple yet so nice. Thanks a lot.

[jeremyacton]

I got the answer by seeing that the octagon sides are either vertical, or horizontal or halfway between, which is 45 degrees. Projecting a horizontal (B to A) and a 'Halfway between' ( C to A) to point A would give 45°.

[mx]

The regular octagon is enclosed in a circle. We can find the central angle OCD easily, which is (360/8)x2 = 90. Angle BAC is the external angle, which is half of the central angle.

[АлексейКаратаев-сз]

Можно проще - продолжим отрезок CD до АВ, отметив точку пересечения Е. СЕ перпендикулярна АВ, а следовательно треугольник АСЕ - прямоугольный, из чего находим угол А как 90 - 45 = 45. Если не очевидно, что СЕ - перпендикуляр, рассмотрим треугольник DBE углы при вершинах D B являются смежными с углами октагона, а стало быть равны 45 градусам. Следовательно угол при вершине Е - 90 градусов

[mikezilberbrand]

Make a vertical line on the right, extend AC and AB. You will see an isosceles right triangle. So X=45.

[philipkudrna]

Before watching and within 1 second: x = 45 degrees. In a regular octagon the corners of the square are „cut off“ at a 45 degree angle (per symmetry: 90/2=45).

[guyhoghton]

If you rotate the octagon by a ¼ turn anticlockwise, side CD lines up with AB through rotational symmetry. Therefore CD is perpendicular to AB and so if you produce CD to meet AB at point E then ΔAEC is a right angled triangle. Since we know that ∠ACE = 45° then x = ∠BAC = 45°.

[fehmidakhader]

I got the answer by joining B and C and thus finding angles of isosceles triangles BDC and ABC.

[procash]

Superb PreMath Guru ji

Great question & beautifully explained solution

Had forgotten all these formulas due to non usage all these years. Now back to school days, thanks to you sir

[devondevon]

I did it differently by forming a triangle CDB with sides 135 degrees 22.5 and 22.5. Since the exterior of the angle to
A and B are 45 degrees given that the total interior angle of the octagon is 1080 (Since 6 triangles can be formed and 180 x 6 =1080 divided by 8 = 135), then x + 45 + 45 + 22.5 + 22.5 =180
x + 135 =180
x =45 answer
x 6= 1080

[sameerqureshi-khcc]

Sir you are not only an excellent math teacher but a very soft spoken guy as well! 😊👍

[paulchapman]

Just from reading the comments it is clear that there are many ways to do math right. The video did a good job of explaining one of them.

[erichumphrey]

With this being a regular octagon, you can extend CD to intersect AB at new point F - that intersection is a right angle. Since C is 45* and F is 90*, 180 - (90+45) = 45*

[Skank_and_Gutterboy]

I started with the approach of finding the sum of internal angles of the octagon, (8-2)*180-deg = 1080-deg. Each internal angle is: 1080-deg/8 = 135-deg. From there, I diverged from the video. First, continue the horizontal line from the top of the octagon to the left. Knowing the internal vertex angle, the angle between this imagined horizontal line and the next counter-clockwise side of the octagon is simply (180-135)degrees = 45-degrees. This angle is equal to x, they're alternate angles because segment AB and the top of the octagon are both horizontal, so they're parallel. Therefore, x = (180-135)degrees = 45-degrees

[theoyanto]

Easy one, but only because of what I've learned on your channel, thanks again 👍🏻

[markbrown]

Extend line segment DB to intersect AC thus creating a 90* right angle isosceles triangle; subtract 90 from 180 and divide the two congruent angles by two, x = 45*.

[williamwingo]

Referring to the diagram as it appears at at 1:12:
Extend line CD above point C. The exterior angle is 360 degrees/8 since it takes eight equal steps to go completely around and wind up pointing in the same direction again. This equals 45 degrees. Thus the vertical angle ACD is also 45 degrees.
Now extend line CD downwards to intersect line AB at a point we'll call J. Angles JDB and JBD are both 45 degrees (by the same reasoning as above) so triangle JDB is a an isosceles right triangle and angle AJD = angle DJB = 90 degrees.
This means triangle AJC is also an isosceles right triangle; and the angle x = 180 - 90 - 45 = 45 degrees.
Thank you, ladies and gentlemen; I'll be here all week.

[johankotze]

Extend AB to E and AC to F such that BE = CF and being also the sides of the octagon. Taking the sides of the diagram as vectors, with EB pointing to 180 you can find the final direction of traversing BA, AC by traversing EB, BD, DC, CF in order. After finding the interior angle, it is easy to see that the change of direction at each vertex is -45 degrees. (A clockwise direction change is negative and a counter clockwise change is positive) Therefore the final direction AC is
180 - 45 - 45 - 45 = 45.

[stephenridenour]

Very good video! Sitting on the sofa, and working the problem in my head, I imagined a line from the ends of each straight leg of the angle to form a triangle. Since such a line would bisect each of the 135 degree angles, I knew that the two angles of the newly created triangle would sum to 135 degrees. 180 - 135 = 45. The method I used would never work on a polygon composed of an odd number of sides.

[toshiyatakanashi]

The intersections E and F of the straight line extending the sides AB and AC to the right and the right side of the regular octagon parallel to the side CD of the regular octagon are extended vertically to obtain the intersections E and F.

From the relationship with the regular octagon, if the perpendicular line is taken from the point E toward the side AF, it becomes the axis of symmetry.

As a result, it is clear that the triangle AEF is an isosceles right triangle.

Therefore, the angle CAB is 45 °.

The argument from symmetry does not require the calculation of angles.