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

[@pranavamali05]

Thnx for such brilliant logical question

[@PreMath]

So nice of you Pranav Thank you for your continued love and support! Cheers! You are awesome 😀

[@watleythewizard2381]

I didn’t know the trick with the angles of the intercepting arcs, but once explained, it’s as logical and as straightforward as can be.

[@NitinKumar-yl9mu]

Angle substended by arc at centre and circumference par nhi banega

[@henkbekker3300]

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

[@PreMath]

Very well done. Thank you for your feedback! Cheers! You are awesome Henk😀

[@Zollaho]

Yes, that's the shortest method. Solved it mentally in 1 minute.

[@chrisrowland2255]

Much easier to solve this way.

[@sandanadurair5862]

Very short approach

[@watleythewizard2381]

Yeah, that’s a good method. But, I wouldn’t hav3 learned about intercepting angles on an arc. Now I know both. Thanks Henk.

[@pinklady7184]

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

[@PreMath]

Glad to hear that! Thank you for your nice feedback! Cheers! You are awesome 😀

[@paulc3960]

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

[@PreMath]

Excellent! Thank you for your feedback! Cheers! You are awesome Paul😀

[@timeonly1401]

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.

[@PreMath]

Super Thank you for your feedback! Cheers! You are awesome 😀

[@geoninja8971]

I did it this way too.... took a few minutes to spot I'm not embarrassed to say.... :)

[@sameerqureshi-kh7cc]

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

[@PreMath]

Thank you Sameer dear! You are the best. Stay blessed😀

[@МаксимКинзибаев]

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

[@billymorris3265]

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.

[@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°

[@PreMath]

Great job Thank you for your feedback! Cheers! You are awesome Hassan😀

[@MrPaulc222]

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.

[@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.

[@GillAgainsIsland12]

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.

[@DB-lg5sq]

27,5

[@BharatKumar-ou2cv]

this was a very easy question even for those who have not prepareed for IOQM

[@henkbekker3300]

By the way, my comment is just to show another method. Science, in every form or shape, is just fun!! Nothing against premath, on the contrary!!

[@PreMath]

Great point! Thank you for your feedback! Cheers! You are awesome Ken.😀

[@biaohan4358]

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.

[@946towguy2]

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.

[@jonathancapps1103]

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°.

[@henkbekker3300]

Oh, this is not a contest. I see comments like “I did it in 40 seconds”. For me, Its just to show there are other ways to get to a solution. Nothing more. Lets not get carried away. Just look at this as a challenge and give your solutions. Nobody is better, nobody is worse.

[@AbhishekK-hn4gd]

really excellent question to test intelligence level of elementary standard student.

[@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°.

[@davidr1781]

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.

[@PreMath]

Thank you for your feedback! Cheers! You are awesome David😀

[@MathsOnlineVideos]

Nicely done! Thank you very much.

[@PreMath]

Glad you liked it! Thank you for your continued love and support! Cheers! You are awesome Siyanda. Stay blessed 😀

[@tayyihcheung7450]

It's pretty straightforward: x+alpha=95 and x=alpha+40. Solving for x yields x=(95+40)/2=135/2=65.5

[@Yora21]

Alternative solution: The two red lines form four angles. 95° to the left, so also 95° on the right. That leaves 170° for up and down, so 85° each. The tip of arrowhead shape bounded by the red lines forms a diamond shape with the angles 40° at the tip and 95° at the opposite end. A four-sided shape has internal angles of 360°. Minus 40° minus 95° leaves us with 225° for the top and bottom angle, so 112.5° each. Now we have two angles for the triangle formed by the alpha angle and the 40° angle, which are 40° and 112.5°. A triangle has internal angles of 180. Minus 40° minus 112.5° leaves 27.5° for alpha. As the final step, we know that the bottom angle of the two crossed red lines is 85°, and also that alpha is 27.5° Together that makes 112.5°. Subtract that from the 180° for a triangle, and that leaves x with 67.5°. The weakness of this approach is that it assumes the whole shape is symmetrical, but does not prove it. If it were not the case, I think the solution would not work.

[@kdaniel8721]

Good form dictates writing the =signs in a straight line.

[@gamingpro6771]

I have another solution on this problem: First of all I rewrite this problem as, Let A,B,C,D are concyclic;AC and BD intersects at point P and AB and CD intersects at point Q,such that

[@vidyadharjoshi5714]

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

[@vishalmishra3046]

Sum of angles of quadrilateral = 360 and the same on a line = 180. Therefore, 40 + 95 + 2 x (180 - x) = 360. Therefore, x = (95 + 40) / 2 = 67.5 deg. *Easy. Right* ?

[@ramray9922]

Sir u r lord Shiva for us. Kindly carry on all ur superb problems. Take care of ur health. If possible then kindly present us some tricky geometry problems of K.C.Nag.

[@PreMath]

Thank you for your continued love and support! Cheers! You are awesome Ram. Stay blessed 😀

[@Константин-л2к3э]

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

[@kinghass3842]

What is the name of the program that is used to write like this and the solution?

[@kanonmai]

I think this is the best way to start angle chasing geometry on olympiad lvl

[@ΚΑΝΑΒΟΣΜΑΝΩΛΗΣ]

12 sec for this one...for calculations.(+,-) Have a great day! and merry Christmas!!!

[@millipro1435]

easier than I thought 👍😂

[@PreMath]

Bravo Thank you for your feedback! Cheers! You are awesome Milli.😀

[@mihaiio82]

95°=(arc x + arc a)/2 ; 40°=(arc x - arc a)/2; arc x = 135° -> x° = arc x /2 = 135°/2 = 67.5°

[@PreMath]

Thank you for your feedback! Cheers! You are awesome Mihai.😀

[@mohanbhaipatel7266]

The Sum of the Quadrilateral will be (180-X),+ (180-X)+ 95 + 40 = 360 hence 2X= 135 hence X = 67.5

[@NitinKumar-yl9mu]

Angle substended by arc at centre and circumference par nhi banega

[@pa_videos5618]

I drew line between bottom two points, then the angles of the black and red triangles are 70 and 42.5 leaving 27.5 for alpha, then x=180-27.7-85 = 67.5 since angle opposite x is 180-95,

[@andrewrao634]

That only works if those triangles are isosceles, which you can't guarantee. However, you can say that the base angles of those 2 triangles sum to 140 and 85, and add to 225, which must be the same as the sum of the base angles of the other 2 triangles on that base, each of whose apex angle is x, so 2x + 225 = 2 × 180, which gives the correct answer.

[@truthandreason8394]

40 is equal to half of big arc minus half of small arc. x is half o f big arc, a is half of small arc so x-a = 40, x+a = 95 => 2x = 135 => x = 67.5

[@maisseraboudjema]

I solved this equation with another method : We have : 180° - x + 40 + 180° - ( 85° + x ) = 180° So : 315° - x - x = 180° - 2x = 180° - 315° - 2x = - 135 x = - 135 / - 2 x = 67.5°

[@sandanadurair5862]

Using exterior angle property X= alpha+40. ....1 X+alpha = 95.....2 From 1 X-alpha. =40.....3 Adding2 &3 2x = 135 X= 67.5

[@PreMath]

Excellent Sandanadurai Thank you for your feedback! Cheers! You are awesome 😀

[@tomapyne4790]

Please sir I want you solve this question if (m+n)^2=m^2+n^2, what is the value of (3^m)^n you have contributed great to improve myself may Allah elevate higher Ameen

[@redeyexxx1841]

Answer would be 1 is suppose.

[@vidushijoshi3921]

Big fan sir

[@PreMath]

Thank you for your continued love and support! Cheers! You are awesome Anita Joshi. Stay blessed 😀 Love and prayers from the USA!

[@susennath6035]

Excelent

[@PreMath]

Thank you Susen😀

[@kiranvootori8101]

Once you know alpha 27.5°, x=27.5+40 =67.5°(exterior angle theorem)

[@SonuSharma-le4ie]

I solved it by equation x = 95° - x° + 40° (Ext. Angle Property)

[@theophonchana5025]

#inscribedangle #congruentangles #intercept

[@формалинстрептоцид]

67,5°

[@マルデムリ]

360=95+(180-x)+40+(180-x)

[@theophonchana5025]

#exteriorangle #exteriorangletheorem

[@subhashinijanakiraman1018]

Please name it as A,B,C,D in order to avoid confusion

[@ihorlifanchuk5444]

1)95°+40°= 135°; 2)135°/2=67,5°✓

[@PreMath]

Excellent Ihor Thank you for your feedback! Cheers! You are awesome 😀

[@nihalverma1554]

67.5 degree

[@PreMath]

Super Nihal Thank you for your feedback! Cheers! You are awesome 😀

[@antmonyanedrawamos2311]

x=a+40.,,95=a+x., from both a is removed my subtracting one equation from the other then 2x= 95 +40, therefore x = 135/2,=67.5

[@elkanahgray9810]

because.... 95+(reflex angle 265)= 2theta angle + 40 +( reflex angle 265)=360.....therefore 360 -( reflex 265)=2theta angle+40=95.....therefore theta angle =27.5.....then x=95-27.5'=67.5

[@poppyaustin7315]

Done

[@VolkGreg]

a = x - 40 = 95 - x x = 67.5

[@PreMath]

Super Greg Thank you for your feedback! Cheers! You are awesome 😀

[@mumtazrasul8263]

67.5 °

[@PreMath]

Super Mumtaz👍

[@theophonchana5025]

x + alpha = 95° x + 27.5° = 95° x = 67.5°

[@dhrubajyotidaityari9240]

Ext angle 95=x+x-40 x=67½

[@michaelavishay8172]

2*(180-X)+95+40=360

[@Egzegezan]

It's evidently not 95 degrees there.

[@Ivan-Matematyk]

x = 95 - a and x = 40 +a. Adding, we obtain 2x = 135. Thus x = 67,5. End!!!

[@mahalakshmiganapathy6455]

Very nice

[@PreMath]

Glad to hear that! Thank you for your feedback! Cheers! You are awesome Mahalakshmi dear😀

[@theophonchana5025]

alpha + alpha + 40° = 95° 2 alpha + 40° = 95° 2 alpha = 55° alpha = 27.5°

[@hughvera3569]

one way another way from the fig 0:01 x=y+40 and x+y=95 x=67.5

[@tejlaitonjam9681]

I solve it by looking in the thumbnail only

[@davidmam]

This seems overly complex. Each of the two described triangles are isoceles with a shared base. So the black triangle has base angles of 70 degrees, and the red one 42.5. In the triangle with the angle x you have one base angle at 70 and one at 42.5. The angle x is therefore 180 -42.5 -70 = 67.5 degrees. Far simpler than the other solutions.

[@theophonchana5025]

#sum #anglesum #sumofangles

[@オモドウ-b4p]

x＋(x−40)＝95 x＝67.5

[@Gargaroolala]

I couldn’t solve it cos I duno the fact about the angles in inscribed angles

[@PreMath]

No worries Garrick. We are all lifelong learners Thank you for your feedback! Cheers! You are awesome 😀

[@Gargaroolala]

@@xxrgxxcasco don’t worry. All is in good taste. I agree sometimes I just don’t see it, but I still do enjoy problem solving. Sometimes I do feel the solution presented was more confusing by my own solution lol.

[@anestismoutafidis4575]

X=67,5

[@МаликШамилов-ы6р]

Легко доказать чтр х среднее арифметическое углов 40 и95 градусов

[@constantinfedorov2307]

x+y=95; x-y = 40;

[@theophonchana5025]

x = alpha + beta + gamma

[@MAREKROESEL]

A B C P D E

[@bienminh3168]

L

[@eddewan307]

The angle labeled 95 degrees is obviously LESS than 90 degrees! Bogus illustration.

[@jakkima1067]

Класс!

[@PreMath]

You are awesome Jakhima.😀

[@jakkima1067]

@@PreMath .in fact, I'm from Russia and I don't understand English well. but there are no language barriers for mathematics, so I understand you perfectly;)

[@jakkima1067]

@@PreMath . and further. thanks for the detailed explanations! please continue and good luck!

[@СемёнВикторович-о6о]

To much easy. For school children.

[@PreMath]

Thank you for your feedback! Cheers! You are awesome 😀