Sum The Grey Angles

Relevant wiki: Triangles - Angle Sum

Notice that the three triangles are joined at the base. If we call those 3 unshaded angles $x,y,z,$ we know that $x+y+z = 180$ since they form a line.

The sum of all of the angles in the 3 triangles is $3\cdot 180 = 540,$ and the sum of the unshaded angles is $180.$ Thus, the sum of the grey angles is $540-180 = 360.$

The overall shape is a hexagon. The sum of all corner angles is 720. In this case 2 straight angles are missed out in the 1st and 3rd triangles. Hence, the required sum is $720 - 2\times 180 = 360.$

Let the angles of the 1st triangle $a,b$ , the 2nd $c,d$ , the 3rd $e,f$ , and let the remaining angles of the 1st,2nd,3rd as $x,y,z$ . Now we want to find $a+b+c+d+e+f$ .

We see that $a+b+x=180 ^{\circ}$ (1) because the sum of angles of a triangles is 180 degrees.

This also comes with $c+d+y=180^{\circ}$ (2) and $e+f+z=180^{\circ}$ (3) for same reason.

We also see that $x+y+z=180^{\circ}$ (4) because the measurement of a straight angle is 180 degrees.

So you can use these information to find the gray angles.

Let's go ahead and sum (1), (2), and (3) we'll get...

$a+b+x+c+d+y+e+f+z = 180^{\circ}+180^{\circ}+180^{\circ} = 540^{\circ}$

Substituting (4) and we get $a+b+c+d+e+f+180^{\circ} = 540^{\circ}$

Therefore $a+b+c+d+e+f = \boxed{360^{\circ}}$

Let the three un-shaded angles around the common vertex be X, Y and Z from left to right.

In the left most triangle, the sum of shaded angles = Y + Z

In the middle triangle, the sum of shaded angles = X + Y

(In the above case, use exterior angle theorem and use the fact that vertically opposite angles are equal.)

In the right most triangle, the sum of shaded angles = X + Z

Therefore, sum of all shaded angles = 2(X + Y + Z)

Now, X + Y + Z = 180

sum of all shaded angles = 2(X + Y + Z) = 360

Quite Simple,

Using: 1) Angle Sum Property says that sum pf all interior angles of a triangle is 180. 2) Angles on one side of a straight line will always add to 180 degrees.

Sum of all angles of the 3 triangles = 3*(180) {Using 1} Sum of the pending non gray interior angles of the 3 triangles = 180 {Using 2}

So, the Sum of gray angles = 3(180) - 180 = 2(180)
= 360

the base line of the two triangles and the base line of the inverse triangle are parallel, the shaded angels on the inverse triangles are equal to the non-shaded angels on the base line, so all shaded angels are for two triangles >>>> angles Sum =360

Suppose 60*6=360

Need to say that the base of the middle triangle is paralel with the basis of the other two triangles!

This is not the most robust method, but one of the quickest. I took the basic geometric rule that the sum of interior angles of a polygon is equal to 180(n-2), where n is the number of sides. So the sum of angles is 720, but there are two places where part of that sum is left out of the red. Fortunately, both of those places are straight lines on the interior, so they are each 180 degrees and add up to 360. Subtract 360 from 720 and you get 360. The answer is 360. The drawback to this approach is that you must know and understand the base formula for the interior angles of a polygon, and there are similar questions which may have a larger complication than simply subtracting 2(180) from the total.

the upside down triangle is irrelevant, all you need to know is that there are two triangles, since each one has 180°, the sum of both triangles is 360°

All the triangles are placed in the base line so the line angle is 180° The of triangle angle is 540° We subtract 540-180 =360° Therefore the sum of the triangle is 360°

The three angles in the bottom form one 180° angle.. Sum of 3 angles in one triangle is equal to 180°. We have 3 triangles here, so from here it's easy to get the formula: 3n-n => 180*3-180=360

If we take the three angles that are not included, we can see simply they equal 180 degrees. Because there are three triangles, we can say that the sum is equal to 3(180)-180, or 2*180. This is 360 degrees.

See all the triangles with the same height while keeping the same angle ratio... the figure become a quadrilateral and apply:

"The interior angles of a simple quadrilateral ABCD add up to 360°."

"

Here three triangles are there .required sum of angles is equal to 3 (180)-straight angle180.then it is 360

looking at the triangles, I just figured this is an equilateral triangle. so this means 60 degrees per angle. 60*6 = 360.

There are 6 angles So it forms 2 triangle So 1 triangle=180˚ So 2 triangle=180*2=360˚

observed the three unshaded angle, it is equal to 180 degrees. so just multiple 3*180 -180 which is equal to 360.

The sum of interior angles of all triangles equal 180. So 180° * 3 = 540°. Now because the 3 triangles meet at one point, their corners (unshaded) form a straight line which equals 180. With that, 540° - 180° = 360°.

6 angles are sum of 360. Middle angle is a straight line means 180 which contains 2 angles. Other 4 angles is the sum of 180. Total 180+180=360

sum of internal angles of triangle =180

let the three joint angles to x, y, z together they form a straight angle

so x+y+z= 180

$3*180-180=2*180=\boxed{360}$

Sum of all the angles in a triangle will be 180. There are totally 3 triangles and the total angle will be 540.

To find the sum of gray angles, we can subtract the not shaded angles from the total. The not gray angles from a straight line. The angle for a straight line is 180.

So, subtracting 180 from 540 will be 360, the sum of all gray angles.

Sum of angles of three triangles is 540, minus the supplementary angle made by the three not shaded angles which is 180 is 360 degrees. :)

there are 3 triangles each having 180 deg sum considering all their angles. so, total sum of all the angles is 3*180=540

but, here don't consider those three angles which are not greyed....as they are forming straight line sum of angles is 180

thus all the angles which are greyed effectively gives addition of 360 deg

I think this problem could be made a little less obvious if the top middle line was not parallel to the bottom one, but at some angle.

First we have to remember that the sum of a triangle's angles is 180 degrees. Seeing that the shapes in the figure are all triangles, we can count how many shaded angles there are, divide by 3 and multiply by 180. So: 6 shaded angles/3 = 2 triangles' worth of angles. 2*180 = 360 degrees

At first it looks difficult. But as you could see the sum of three triangles is 540 . and we could minus 180 (as the the sum of three angles at the same line is =180) so the result will be 360

The total sum of angles in the three triangles is= 180 degrees*3=540 degrees. The sum of non-gray angles is=180 degrees as they lie on a straight line.Therefore sum of gray-angles is = (540-180 degrees)=360 degrees.

The three triangles have 3*180=540 degree. The three unshaded angles make a straight line that is 180 degree. So the shaded angles have 540-180=360 degree

all of the triangles contain 180 degrees so multiply it by 3 then subtract the angle formed by the line which is joined by the 3 non shaded angles of the triangles which is also 180

Sum of angles of three triangles is 540, minus the supplementary

angle made by the three not shaded angles which is 180 is 360

degrees. :)

There are 3 triangles and the unmarked ones form a straight line. All angles of straight line=180 Angle of three triangle=180×3=540 Therefore, dark angles=540-180=360

3(no. of triangles)x180(sum of angles in a triangle)-180(sum of non gray angles) = 360

consider the 3 angles in linear pair which are not sheded as x and y and z and the sum of all angles in a triangle is 180 as they are in linear pair. therefore 180 - x + 180 - y + 180 - z = sum of the required angles. 540 - ( x + y + z ) = Sum of grey angles. 540 - 180 . therefore its 360 :-)

Since there are three triangles, therefore the total sum of angles = No. of triangles × 180 = 3 × 180 = 540 and as we put some logic in the problem we observe that we don't need to find the three angles from each triangle and that all these three angles lie on a straight line so we will subtract the sum of the three angles i.e. 180 from the total sum of the angles.Therefore, 540 - 180 = 360 , which is the answer.

shaded angle = 180 + 180 + 180 - ( the sum of unshaded angle)

180 + 180 + 180 - ([a+ b+ c]=180) = sum of required angles = 360.

sum of all angle of triangles 180 + 180 +180 = 540

now minus 180 for all angles in straight line

540 - 180 = 360 degree :P

The simple solution is to consider the triangles joint at the base and to subtract 180 from the sum of the angles of 3 triangles, so 540-180=360

as there is three triangles ... so calculate the total angle of all the triangle (180 times 3 ) = 540 ... the unshaded angle is 180 .. so 540 - 180 = 360

x+y+z = 180 ; A = B = C = 180 ; (A-x)+(B-y)+(C-z) = 3A - (x+y+z) ; 3A - A = 2A = 360

any outside angle of a triangle is equal to opposite two angle inside the triangle. And sum of three angle in Triangle is 180. Thus got the result. But due to the fact that there are no labeling in the figure I am unable to explain it to the point.

3*180 Degree=540 then,540-180=360,so answer is 360.

Let sum of grey angles in first triangle be A, in second be B and in third be C In Triangle first, unknown angle = 180-A Similarly, 180-B and 180-C in second and third triangle. Notice that, (180-A) + (180-B) + (180-C) = 180, which gives, A+B+C=360!

sum of 3 triangle's angles=540,then sum of the non-gray angles is 180 and finally sum is 540-180=360.

this works out quite nicely. first, take a look at the right most triangle (you can do this with the others but i looked at the right most one). set the unmarked value equal to x, and the marked angles become 180-x. notice that the two unmarked angles of the other triangles are on a line with x, so they are also equal to 180-x. this means the sum of the marked angles is the sum of the angles of two triangles or 180 + 180 = 360

These are 3 Different triangles in shape and size.Considering the basic law that the sum of internal angles of any triangle is equal to 180.

Hence these are three triangles then there total sum sum would be, t1<180> + t2<180> + t3<180> = 540 degrees

Now as we can see the there a single vertice of all three of them where they all are meeting each other. We can see that the total angle of those three angles is eventually equal to 180.

As a straight line has an angle of 180.

For getting the sum of the grey angles subtract total sum of angles from the white angles from it,

G<> = S<540> - W<180> G<> = 360

Now that is your actual answer. :)

Alternate angles made between parallel lines are equal and sum of the angles of a triangle is 180. There are two triangles so after adding the angles it would be 360.

If you were to add up all the angles in the three triangles, you would get 540 degrees (3 x 180). Take away the unshaded ones, which happen to lie on a straight line (so = 180), and you get 360 degree.

الثلاثة زوايا الغير محدده تمثل زاوية مستقيمه 180 درجه إذن مجموع درجات الزوايا الملونة يساوى مجموع زوايا الثلاثة مثلثات مطروحا منه الزاوية المستقيمة (3*180) - 180 = 360

Let say angles from right hand side a,b,c,d,e and f. Now, a+b=180-c and second equation , e+f=180-d. Add both equations

There we see 3 triangles. we know sum of interior angles of a triangle is 180 degree. so sum of angles of 3 triangle is 540 degree. also sum of angles made in a straight line on single point is 180 degree. so sum of gray angles is 540-180=360 degree.

The sum of angles in a triangle is 180. The sum of angles of three triangles is 540 (3 x 180). The three unshaded angles lie on a straight line, which always add to 180. 540 - 180 = 360.

(180+180+180)-180=360 as sum of angles of 3 triangles is 540 minus sum of angles on a straight line 180 equals 360

For the 3 triangles the sum of angles is 3* 180 degres but the white agles form a line, so 180 degrees. That means that the the sum of gray angles is 2*180= 360 degrees.

First understand the two basic rules of geometry… 1.The sum of all angles of a triangle is always equal to 180 degree. 2.Vertically opposite angles are equal.

Now, just consider right and left most triangles. Both have one unshaded angle and they have their vertically opposites angles (i.e shaded angles of middle triangle) which are equal to unshaded angles of right and left most triangles.

Hence all shaded angles constitute (are equivalent of) two triangles... that is 180 + 180 = 360 degree.

all triangle are equilateral 60*60

SUM OF ALL ANGLES IN A TRIANGLE IS 180 SUM OF TOTAL ANGLES IN 3 TRIANGLES IS 3X180=540 UN COLORED 3 ANGLES FORMED STRAIGHT LINE SO SUM OF UNCOLORED ANGLES IS 180 SO GREY COLORED ANGLES TOTAL = 540-180=360

Each triangle has 180 degrees, a straight line is also 180 degrees, hence 180*3(triangles) - 180 (straight line or unshaded angles) = 360 degrees.

Its All rational ,assume,that every angle in three angles of triangle is 60, so ,we have here 6 angles ,6 60=360 ....... since they are connected by point that join angles ,so what increases in angle decreases in another,and the sum remain "CONSTANT"

construction: just extend the straight lines and they act like two parallel lines cut by traversal. sole accordingly.

Just notice that the two angles on the two sides of upper horizontal line are equal to the remaining (not shaded) angles of two triangles (the left and the right one) respectively. So the sum of shaded angles equals to double sum of a triangle. That's it. It was very simple.

sum of the angles of the 3 triangles =540 the sum of the supplementary angles at the base =180 540-180=360=the gray angles

60+45+120+90+45:360

Since smaller and bigger lines( sorry ,names not given in the problem)and intesecting lines are making corressponding angles to complete the triangles.Each triangle will have total 180 degrees and so both will have toital 360 degrees Ans. K.K.GARG,India

3 triangles times 180 = 540 - 180(the 3 angles in straight line = 360