Angles Of A Pinwheel

Geometry Level 1

In the figure, four line segments A B , C D , E F , G H AB, CD, EF, GH intersect at a single point.

Find the value (in degrees) of the angle sum x + y + w + z . x + y + w + z.


Note : ∠ E F H = 5 5 ∘ \angle EFH = 55 ^ \circ , ∠ A C D = 3 7 ∘ \angle ACD = 37 ^ \circ , and ∠ G H F = 8 5 ∘ \angle GHF = 85 ^ \circ .


The answer is 174.

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Tushar Malik by Tushar Malik , 20, India

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12 solutions

Hon Ming Rou
Sep 6, 2014

2 (180 - 37 - 90) = 106

2 (180 - 85 - 55) = 80

360 - 106 - 80 = 174

16 Helpful 0 Interesting 0 Brilliant 0 Confused

simply good....

Asif Tasin - 5 years, 4 months ago

There is nothing to stop AB from being on the same line as EF or the other way so DC are on the same line as GH

Craig Hall - 4 years, 10 months ago
Vaibhav Maurya
Sep 5, 2014

Too easy question..... Just apply " Sum of all angles of a triangle equals 180° " & " Sum of all angles on a point is 360° " . :p 

                                                                  2 (180 - 37 - 90) = 106
                                                                  2 (180 - 85 - 55) = 80
                                                                  360 - 106 - 80 = 174
7 Helpful 0 Interesting 0 Brilliant 0 Confused

sum of all angles around a point is 360*

Praneeth Reddy - 6 years, 9 months ago

Os triângulos ACDB, são iguais, então os ângulos correspondentes vão ser os mesmos, assim como nos triângulos EGO,OFH( O >>PONTO DE ORIGEM DO SISTEMA), sabendo que a soma dos ângulos internos de um triângulo é 180º: triângulo ACO,ODB >> 180-(37+90)=53º

triângulo EGO,OHF>>180-(85+55)=40º

Sabendo que uma volta inteira é igual a 360º

LOGO, x+y+w+z= 360- (53 X 2)+(40 X 2): 360-106-80=360-186=174º

Resposta : 174º

karen timbo - 6 years, 9 months ago

Could'nt agree with u more...

has kh - 6 years, 7 months ago
Sherilynne Ruiz
Sep 26, 2015

As you can see every triangles and its opposite triangles are the same so you just find the missing angle and then add them all and then deduct it from 360° .. 360-186= 174..

2 Helpful 0 Interesting 0 Brilliant 0 Confused

It's 174.

First, I added the right angle plus 37 then subtracted that from 180. Then did it too the BD yriangle becasue they're vertical angles. Then I added 85 + 55 which I subtracted from 180. Then I did it too EG triangle because of vertical angles. Then subtracted by 180. Then all 4 triangles equaled 186. Then subtracted from 360, which equalled 174.

1 Helpful 0 Interesting 0 Brilliant 0 Confused
Praneeth Reddy
Sep 6, 2014

suppose think point O

then O1+y+O2+z=180 O3+x+O4+w=180 then y+z= 87, x+w=87 as sum of angles of triangle is 180 we can find them. so x+y+w+z = 87+87 = 174

1 Helpful 0 Interesting 0 Brilliant 0 Confused
Adrian Contillo
Jan 1, 2016

I did it this way tho.LoL

180-(90+37)=53

180-(55+85)=40

2(53+40)=186

360-186=174

0 Helpful 0 Interesting 0 Brilliant 0 Confused
Rekarlo Jäger
Oct 30, 2015

I did the hard way lol

180-(90+37)=53

180-(85+55)=40

2(53+40)=186

360-186=174

0 Helpful 0 Interesting 0 Brilliant 0 Confused
Zaher Takieddine
Nov 17, 2014

Write a solution. 180-(85+55)=40 180-(90+37)=53 360-(40+40+53+53)=x+ y+ w+ z=174

0 Helpful 0 Interesting 0 Brilliant 0 Confused
Sagnik Biswas
Oct 8, 2014

Taking the straight line AB z+y=180-(180-85-55)-(180-90-37)=180-40-53=87 As per alternate angles x=z and w=y so similarly x+w=87 Therefore, x+y+w+z=87+87=174

0 Helpful 0 Interesting 0 Brilliant 0 Confused
Adam Coe
Sep 24, 2014

If we take the triangle that has angles H and F with angles of 85 and 55 degrees respectively. Since, the angles in a triangle add up to 180 degrees. We can take

180 - ( H + F ) = 180 - ( 85 + 55 ) = 180 - 140 = 40

If we take the other triangle with angles A and C with 90 and 37 degree angles respectively. From before we know that the three angles of triangle adds up to 180.

180 - ( A + C ) = 180 - ( 90 + 37 ) = 180 - 127 = 53

We also know the angle that is on the other side of the intersection point is the same angle. We also know that a sum of all angles around a point is 360 degrees . We can progress with

360 - 2 ( 40 + 53 ) = 360 - 186 = 174

So

x + y + w + z = 174

0 Helpful 0 Interesting 0 Brilliant 0 Confused
Lilith Jae
Sep 18, 2014

Let's say the centre or intersection point of the 4 lines is O. Angle AOC = angle DOB, which is (180-90-37= 53)°; just as angle HOF = angle EOG, which is (180-85-55=40)°.

Following these angles, it is possible to determine angle w+y+x+z. The question does not ask for each individual angle, therefore (I use < as the angle sign) :

<AOC + <y +<HOF +<z +<DOB +<w +<GOE +<x =360°

53°+<y+40°+<z+53°+<w+40°+<x=360°

<(w+x+y+z)=360°-(53+40+53+40)°

<(w+x+y+z)=(360-186)°

<(w+x+y+z)=174°

0 Helpful 0 Interesting 0 Brilliant 0 Confused
F Gh
Sep 15, 2014

360-(2(90-37)+2(180-(85+55)))= 174

0 Helpful 0 Interesting 0 Brilliant 0 Confused

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