Find Angle X in this Compound Shape | Step-by-Step Tutorial

[pranavamali]

Thnx for such brilliant logical question

[pinklady]

I love your thumbnails. They make me guess solutions before I click your thumbnail.

[henkbekker]

Well, the angle opposite of 95 is also 95. The four sided figure with the angle of 40 has a total of 360 degrees. Since the two unknown angles are identical,
x = 180- (360-95-40)/2 and that is 67, 5 degrees

[paulc]

Thank you for video. I did it by different way. The angle opposite to 40 is 95, therefore with the quadrilstetal, both angle touching the circle is (360 - 95 - 40)/2 = 112.5.
X is adjacent angle of 112.5, so X = 180 - 112.5 = 67.5

[timeonly]

Reading the comments, I'm impressed with all the many ways to do this problem! Everyone give yourselves a hand. :-)

[玉皮踊る]

x+α=95, 40+α=x.
40+α=x → x-α=40.
Thus, 2x=95+40=135.
The value of x is 67.5.

[МаксимКинзибаев]

О! К сумме углов треугольника добавилось сумма углов четырёхугольника! Гениально!!!

[billymorris]

I solved this before watching with a simpler, yet, more limited approach. This method will serve superior to my previous one in 3D applications. Thanks for sharing this.

[sameerqureshi-khcc]

Sir you are a living legend! Love from Pakistan 😊👍🌹

[GillAgainsIsland]

The exterior 40 degree angle is one half the difference between the major and minor arc it intercepts. The 95 degree angle is one half the sum of the arcs the chords creating it intercept. From that you can easily calculate the major arc to be 135 degrees. X is just the inscribed angle which is half of 135 degrees, therefore 67.5 degrees.

[biaohan]

There is actually a pretty easy solution. If you connect the midpoint and the point outside the circle you can use two times the exterior angle theorem to find that 2x=40+95 degrees.

Also we can use the degree of the arcs to solve it. If we set the big arc on the left side to be A and the small arc on the right side to be B then we have the 40 degree angle equals (A-B)/2 while the 95 degree angle equals (A+B)/2 from basic arc and angle relationship knowledge. Then we can easily solve A=40+95 degrees while x=A/2.

[calculas]

thanks to your videos, I tried to solve this by myself and i got it right. I hope i don't need to struggle in class 7 next year. Also have a good day.

[MrPaulc]

Excellent. I got the same answer with a very different method. Where the lines cross within the circle you have two pairs of angles at 95 and 85 degrees (one of the 95's is given). The right hand 95 degree angle is one angle of a kite, and the 40 degree angle is the right end angle of the kite. 95 + 40 = 135. Angles of the kite add up to 360 degrees, so the two remaining angles are each 112.5 degrees ((360-135)/2). One of these is an angle on a straight line with x, so 180 - 112.5 = 67.5=x.

[HassanLakiss]

Thank you for a nice geometry question.
With respect, x=α+40⇒α=x-40
But x+α=95
⇒x+x-40=95, 2x=95+40=135
x=135/2=67.5°

[OrenLikes]

Other angle on circle that share the same arc of 95° and 40° is also X.
Complimentary angles of X are both X' -> X'=180°-X -> X=180°-X'.
Opposite of 95° is also 95°.
Quadrilateral's sum of angles is 360°.
->
360°=95°+40°+2X' -> 360°=135°+2X' -> 2X'=225° -> X'=112.5° -> X=180°-X' -> X=180-112.5° -> X=67.5°.

[towguy]

Much simpler way:
Opposite angles of intersecting lines are congruent, so 95 degrees on both sides.
Extend line from intersection to apex of outside 40 degree angle. Measurements are 1/2 of before, so 47.5 and 20 degrees.
Add 20 to 47.5 to get x=67.5

To get the rest: Subtract 20 from 47.5 to get alpha=27.5
Supplementary angle of 95 is 85, so beta angle of triangle is 85.
A triangle has 180 degrees, so you can check by adding 85+47.5+27.5=180.

[davidr]

x = (40+95) / 2 = 67.5

because angle between black lines, 40 = +x -95 +x, where -95 is vertically opposite to the given 95 ; and x is in both triangles as these are similar triangles.

[Константин-лкэ]

You should have add that the circle exterior angle is equal big chord minus small chords. So, after you have found alpha chord as 27.5, the X chord is then 40 = X - A, i.e. X = A+40 = 67.5

[jonathancapps]

So, haven't watched this yet, but my intuitive approach:

Imagine a line that bisects both the 40° and the 95° angles. The 2 lines that make the angle x are then 95/2° and 40/2° on either side of this bisecting line.

95/2° +40/2° = 47.5° + 20° = 67.5°.

[vidyadharjoshi]

Draw a Line opp 40 deg angle joining the 2 points on the circle vertically. So this will form two triangles one having 40 deg angle. Since these 2 triangles are isosceles triangles so the 2 angles on 40 deg triangle will be 70 deg each. The other triangle will have 95 deg. as one angle the other 2 angles will be 42.5 each ( 2*42.5 + 95 = 180 ) So angle on the other side of line of X will be 42.5+70 = 112.5 so X = 67.5