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# In ∆ ABC above, what is the sum of p + q + r + s + t + u ?

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In ∆ ABC above, what is the sum of p + q + r + s + t + u ?  [#permalink]

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06 Sep 2017, 04:40
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15% (low)

Question Stats:

92% (01:23) correct 8% (01:21) wrong based on 51 sessions

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In ∆ ABC above, what is the sum of p + q + r + s + t + u ?

(A) 180
(B) 225
(C) 270
(D) 360
(E) 450

Attachment:

2017-09-06_1535_002.png [ 11.77 KiB | Viewed 1282 times ]

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Re: In ∆ ABC above, what is the sum of p + q + r + s + t + u ?  [#permalink]

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06 Sep 2017, 04:49
Find : p + q + r + s + t + u

Let the angles of center triangle formed inside big triangle be x, y and z
=> x+y+z=180
Now as Sum of angles on 1 side of straight line = 180

=> p+u+x=180
=>q+r+y=180
=>s+t+z=180
p+q+r+s+t+u = 540 -(x+y+z) = 540-180 =360

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Re: In ∆ ABC above, what is the sum of p + q + r + s + t + u ?  [#permalink]

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06 Sep 2017, 05:32
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Attachment:

2017-09-06_1535_002.png [ 12.87 KiB | Viewed 984 times ]

There are 3 smaller triangles(∆ADE,∆BDF and ∆CEF) each which has a third angle(unmarked) - A,B and C
A + B + C = 180 degree(sum of the three angles of the triangle ABC total 180)

Similarly each of the smaller triangles also has a sum of angles equal to 180.
p + q + r + s + t + u + A + B + C = 540
p + q + r + s + t + u = 540 - 180 = 360(Option D)
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Re: In ∆ ABC above, what is the sum of p + q + r + s + t + u ?  [#permalink]

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06 Sep 2017, 09:16
1
Bunuel wrote:

In ∆ ABC above, what is the sum of p + q + r + s + t + u ?

(A) 180
(B) 225
(C) 270
(D) 360
(E) 450

Attachment:
2017-09-06_1535_002.png

A + B + C = 180
[180-(r+s) ] + [180-(q+p)] + [180 - (t+u)] = 180
3(180) - 180 = r+s +p+q+t+u
2(180) = r+s +p+q+t+u
=360
option D
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Re: In ∆ ABC above, what is the sum of p + q + r + s + t + u ?  [#permalink]

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06 Sep 2017, 11:54
1
Bunuel wrote:

In ∆ ABC above, what is the sum of p + q + r + s + t + u ?

(A) 180
(B) 225
(C) 270
(D) 360
(E) 450

Attachment:
The attachment 2017-09-06_1535_002.png is no longer available

Attachment:

triangles111.png [ 31.82 KiB | Viewed 839 times ]

A, B, and C are interior angles of the larger ∆ ABC

So A + B + C = 180

(A + B + C) can be used to find p + q + r + s + t + u

Angles in each small triangle (yellow, purple, green) also sum to 180.

From diagram, write the equations -- and add them

r + s + A = 180
p + q + B = 180
t + u + C = 180____________________
r + s + A + p + q + B + t + u + C = 540, or

p + q + r + s + t + u + (A + B + C) = 540, or

p + q + r + s + t + u + 180 = 540

p + q + r + s + t + u = 360

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Re: In ∆ ABC above, what is the sum of p + q + r + s + t + u ?  [#permalink]

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11 Sep 2017, 10:54
Bunuel wrote:

In ∆ ABC above, what is the sum of p + q + r + s + t + u ?

(A) 180
(B) 225
(C) 270
(D) 360
(E) 450

Attachment:
2017-09-06_1535_002.png

We see that the sum of the interior angles is 180. Let’s say each angle within the triangle is 60 degrees. Then s + t = r + q = p + u = 180 - 60 = 120.

So, p + q + r + s + t + u = 360.

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Re: In ∆ ABC above, what is the sum of p + q + r + s + t + u ?   [#permalink] 11 Sep 2017, 10:54
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# In ∆ ABC above, what is the sum of p + q + r + s + t + u ?

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