What is a + b + c + d?

Nice question!!

I can see some triangles that include the necessary angles (a, b, c and d)
So, let's add some extra angles, x and y:

Since x, 30 and y are on a line, we know that x + y + 30 = 180
This means that x + y = 150


Now let's examine some hidden triangles. Here's one:

Since angles in a triangle add to 180 degrees, we can write: a + c + x + 30 = 180
Subtract 30 from both sides to get: a + c + x = 150

Here's another hidden triangle

Since angles in a triangle add to 180 degrees, we can write: b + d + y + 30 = 180
Subtract 30 from both sides to get: b + s + y = 150

We now have two equations:
a + c + x = 150
b + s + y = 150

ADD them to get: a + b + c + d + x + y = 300
Since x + y = 150, we can write: a + b + c + d + 150 = 300
Subtract 150 from both sides to get: a + b + c + d = 150

Answer:

Cheers,
Brent

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Hi Brent
Sorry for using your image.
I am using it to arrive at answer using different approach.
I will try to explain using Theorem: An exterior angle of a triangle is equal to the sum of the opposite interior angles.


x=b+d
y= a+c

and x+y+30 =180.
Putting x and y value in above equation.
we get
b+d+a+c+30=180
a+b+c+d= 150

b+d+30+X=180
a+c+30+Y=180
30+X+Y=180
so a+b+c+d=150

a + c + 30 + x = 180
-> x = 180 - a - c

b + d + 30 + 180 - x = 180
-> b + d - x = -30
-> b + d + a + c -180 = -30
-> a + b + c + d = 150
ANSWER D

No problem (and nice solution!)

Cheers,
Brent
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150 - (b+d) = a+c
a + b +c +d = 150

IMO D
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I've just imagined that the figure kind of "spread its wings" and became triangle (the 30 angle rise up and the peak turned around its axis).
We know that the sum of angles in triangle = 180. All we need is to substruct 30.
The answer is D.

Posted from my mobile device

Let the two angles adjacent to the 30-degree angle be e (on the left) and f (on the right). Notice that e + f = 180 - 30 = 150.

Now, using the two biggest triangles we can form, we have:

a + c + 30 + e = 180

and

b + d + 30 + f = 180

Adding these two equations together, we have:

a + b + c + d + e + f + 60 = 360

a + b + c + d + e + f = 300

But e + f = 150, so a + b + c + d = 150 also.

Answer: D
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