6 Different Angles

The blue angles are equal to the sum of the angles in the 3 pink triangles, minus the sum of the angles in the green triangles. Hence, the sum is

$3 \times 180^ \circ - 180^ \circ = 360 ^ \circ$

In all there are 3 big triangles & 1 small triangle at the centre.

Sum of measure of angles of all 3 triangles = 180º + 180º + 180º = 540º

But the small triangle at the centre too measures - 180º

(This small triangles has 3 acute angles of the larger triangles)

Thus, Sum of 6 blue angles = 540º - 180º = 360º

Ans. 360º

3x180 -180=360 Here is how:

BLUE+RED=540
RED=180

BLUE=540-180=360

When traversing the diagram, you turn a total of two revolutions, or $720^\circ$ , before returning at the starting point. This is the sum of the external angles. Since $\text{internal angle} = 180^\circ - \text{external angle},$ and there are six internal angles here, we have $\sum\text{internal angle} = \sum 180^\circ - \sum\text{external angle} = 6\times 180^\circ - 720^\circ = \boxed{360^\circ}.$

I held my pen above the line at the right hand side of the diagram. I then rotated it through the acute angles at the top right, bottom left, bottom right, top left, middle left and middle right of the diagram so that after each rotation the pen lined up with another line of the diagram. After passing through all the angles, the pen came back to its original position, having made one complete rotation of $\boxed{360^\circ}$ .

You can gain some confidence in this method by using it to show that the sum of the angles in a triangle is $180^\circ$ and in a quadrilateral is $360^\circ$ .

3X360=540 (3 large triangles) 540 - 180 (common small triangle) = 360

Outer angles = (180 - x) + (180 - y) + (180 - z)

x + y + z = 180

Therefore,

(3 X 180) - 180 = Answer

540 - 180 = 360

360, perfect triangle in the middle which means, 180/3=60. so 180 degrees per triangle minus 60 = 120. which means per triangle has a remaining angle of 120. 120*3= 360 degrees.

3(big triangles) x 180 = 540

1(small middle triangle) = 180

Deducting the sum of angles from the middle triangle will give you the sum of the 6 angles.

540 - 180 = 360

There are three triangle. The sum of the angles inside a triangle is 180. 3 x 180 = 540. The sum of the non blue angles is 180. Therefore 540 - 180 = 360

the sum of all 3 triangles angles is 540, and you know that 3 of thoes angles are not accounted for, so it must be 540-180=360

totals 360

180=180-ang1'-ang2'+180-ang1''-ang2''+180-ang1'''-ang2''' ----- 3*180-180= sum angles

the sum of all angles in three triangles =3*180, and the 6 angles represent that sum -the sum of angles in the small triangle which is equal 360

Angle 1 = Angle 2 + Angle 3 (Exterior Angles of a triangle equal the sum of the 2 remote interior angles)

Angle 4 = Angle 5 + Angle 6 (Exterior Angles of a triangle equal the sum of the 2 remote interior angles)

Angle 1 + Angle 4 + Angle 7 + Angle 8 = 360° (The sum of the interior angles of any 4 sided polygon equals 360°)

Angle 2 + Angle 3 + Angle 5 + Angle 6 + Angle 7 + Angle 8 = 360° (Substitution)

Imagine walking along the lines. You walk, turn, walk, turn again in the same direction, walk, turn again in the same direction etc. If you end up back where you started then one way or another you’ve gone in a circle. (360deg).

central triangle is an equilateral, thus each angle is 60°. Each central triangle's angle is also an angle of the other 3 triangles. Thus an easy subtraction gives you the answer.

Apply the Exterior Angle Theorem for a triangle twice, then use the fact that the sum of the measures of the angles of a quadrilateral is 360. Easy!

Sum of the three large triangles (3*180) minus sum of the small inner triangle formed (180)

I just know man.

Because the angles encompass the exterior angles- it is effectively the pie pieces cut out of a hypothetical circle. They have to be 360 degrees, a complete boxing of the compass.

The shape is made of 3 triangles - seperate these triangles and the sum of all the angles is 3 x 180 = 540 deg when we overlap the triangles again we get a smaller triangle created - this is made up of 3 of the previous angles. the small triangle angles add up to 180 degrees - so we subtract this from the total so 540 -180 = 360 QED

3 180 = sum of angles in triangles with blue angles, that sum includes sum of angles of their intersections which is 180, therefore (3-1) 180 = 360

Visualize 3 triangles intersecting (two of the blue angles and a third unshaded). Starting with triangle in top left and moving clockwise you have triangle abC, deF, & ghI (lower case are shaded blue, upper case is not). From basic geometry we know the 3 angles of each triangle add to 180.

Start with the triangle in the top left. Label its angles a, b, & C. Angle C is created from two intersecting lines. Either of the adjacent angles is equal to 180 - [C]. Naturally this adjacent angle must be equal to the sum of two shaded angles of the triangle, (that is, a+b).

Notice now that you have a quadrilateral in bottom left: it has angles g & h, as well as the angle a+b that you derived, and finally a fourth angle yet undetermined. This fourth angle can be determined similarly to the 180 - [C] = (a+b) angle we derived and it is found to be d+e.

So we are left with a quadrilateral that has the four angles: g, h, (a+b), & (d+e). Like a triangle's angles add up to 180, a quadrilaterals add to 360. Therefor g + h + (a+b) + (d+e) must equal 360.

Each angle of the three outer triangles are equal to one angle of a triangle next to it. And to one angle of the small interior triangle. So basically it's 180 × 2, because each angle degree appears twice.

Each of the three large triangles has angles adding to 180, of course, so the six blue angles plus the three angles of the center triangle add up to 3x180. But the center triangles' angles add up to 180, so the sum of the blue angles is 2x180, or 360.

Each triangle sums to 180 degrees. So, for each triangle, simply represent the sum of the blue angles with a variable and solve for the third angle: 1) A1 = 180 - A. 2) A2 = 180 - B. 3) A3 = 180 - C.

Keep in mind that the Sum A+B+C represents the sum of all blue angles.

Also notice that these three angles are part of the smaller, center triangle, so their sum total has to be 180 degrees:

A1+A2+A3 = 180,

Giving us:

540 - (A+B+C) = 180,

Which after simplification gives:

A+B+C = 360

Thus the total value represented by the three angles is 360 degrees.

There are three overlapping Triangles 1, 2, and 3 . Consider 1 with marked angles 1 and 2 with Apex angle =x, therefor x=180 - (1+2 ) Hence (1+2) = 180-x . Similarly (3+4)=180-y and (5+6)=180-z Add these three to give (1+2) +(3+4) +(5+6)= 3x180-(x+y+z), But (x+y+z) = 180 therefor (1+2)+(3+4)+(5+6)=(3x180)-180=2x180 =360

Twist the shape to a rectangle. And we all know rectangle is 360

regardless of starting point in order to return to starting point one must go a full circle so to speak

sum of 1 triangle is always 180 degrees, multiplied by 2 is 360

If we call the angle of the internal angles of the small triangle a,b and c, and consider the fact that the internal angles of a triangle are 180 degrees, we then have 180-a+180-b+180-c=180x3 - (a+b+c). a+b+c=180, so the sum of the 6 blue angles is 180x3-180=360

Let a1,b1,c1 be the angles of triangle1, a2,b2,c2 for triangle 2, and a3,b3,c3 of triangle3, where c1,c2,c3 are angles of the smaller intersecting triangle 4.

• Step 1a) c1+c2+c3=180.
• Step 1b) c1=(180-(a1+b1)) c2=(180-(a2+b2)) c3=(180-(a3+b3))
• Step 2) (180-(a1+b1))+(180-(a2+b2))+(180-(a3+b3))=c1+c2+c3=180
• Step 3) 180-a1-b1+180-a2-b2+180-a3-b3=180
• Step 4) 180+180+180-180=a1+b1+a2+b2+a3+b3
• Step 5) 360 = a1+b1+a2+b2+a3+b3

Small triangle sum of angles: $a + b + c = 180$

Upper left triangle: $180 - (\alpha + \beta) = a$

Upper right triangle: $180 - (\gamma + \delta) = b$

Lower triangle: $180 - (\epsilon + \zeta) = c$

Now, $\left( (180 - (\alpha + \beta) \right) + \left( (180 - (\gamma + \delta) \right) + \left( (180 - (\epsilon + \zeta) \right) = a + b+c=180$

Solving for $(\alpha + \beta) + (\gamma + \delta) + (\epsilon + \zeta)$ yields 360.

The sum of each triangle's angle is 180 degrees. The small triangle formed by the intersection of the three triangles is an equilateral triangle where all angles are 60 degrees. (180-60)3 = (120)3 = 360

The sum of the angles of a triangle is 180 degrees The triangle with no blue angles is made of the non blue angle of each of the other three triangles If their sum (180) is subtracted from the sum of the other tree (540) the result is 360

You have two right triangles on the upper left and upper right.The lower angles have to be 90 degrees. The bottom triangle is an equalateral triangle so the two marked angles have to be 45 degree. So far that is 90 X 2 +45 X 2 which is 270 degrees. A simple observation of the legs on the upper right triangle show that the short one is half as long as the long one so the marked angle is 30 degrees.We are now up to 300 degrees worth of marked angles. The left upper triangle's remaining marked angle is twice as big as the one on the right therefore it has to be 60 degrees and the final answer is 360 degrees. Besides if you take all the blue markings out and place them together flat edge to flat edge they make a perfect circle and guess what, voila 360 degrees.

Let's assign labels to each of the angles:

The sum of A, B, and C is 180 degrees. Same goes for angles D, E, and F, as well as angles G, H, and I.

      A + B + C = 180
      D + E + F = 180
      G + H + I = 180

So now we can calculate the sum of all the angles:

      (A + B + C) + (D + E + F) + (G + H + I) = X

But we only want the blue shaded ones. So we have to remove the three angles that make up the smaller, center triangle:

      (A + B + C) + (D + E + F) + (G + H + I) - (A + D + G) = X

We don't know the actual values of A, B, C, or any of the angles, but we do know what they add up to for each triangle - 180 degrees. Each triangle is represented by the grouping right able, so substituting 180 for each group, we get:

      180 + 180 + 180 - 180 = X

And then it's simple addition and subtraction from there:

      180 + 180 + 180 - 180 = 360

Taking any one side, it is effectively turning through 360, albeit displaced and extended.

i saw it as 2 triangles in my head and figured 180 degrees in each triangle times 2

There are three large triangles each at 180 degrees, one angle of each provides an angle to the smaller internal triangle also required to add up to 180 degrees. Therefore 3 x 180 = 540. 540 - 180 = 360. They remaining 6 angle must add up to 360 total degrees.

3*180 -(x+y+z) , x+y+z = 180, 2(180) = 360

middle is an equilateral triangle, each internal angle of middle triangle completes 3rd corner of a triangle with two blue angles, therefore sum of each pair of blue angles = 120

Each angle has a complementary one such that if we move a corner along one of the three 'long lines', its complementary angle changes such that the sum of the two is constant. Thus, moving a corner in such a way, we keep the sum of the angles constant. It is trivial to move points around like this such that the lines form two triangles on top of each other at the centre (the small triangle we see already). Thus, the angles sum to twice the sum of the angles in a triangle: 360 degrees.

Of course, we never appealed to the specific case shown, so in fact one could move any points around however one likes in order to construct a 'double triangle' and the result would still hold.

All of the triangles can be seen as one contigious figure if they are seen to be connected by one continous line, and the sum of the angles in any contigious figure is always 360 degrees.

The sum of the internal angles of a triangle is 180 ((3-2) 180). The smallest triangle contains the three angles missing from each of the "larger" triangles. Therefore the sum is 3 180 (for the three larger triangles) -180 (the sum of the missing angles formed by the smallest triangle. Or 540-180=360 degress

From the diagram, we see that

1 + 2 + C = 180

3 + 4 + A = 180

5 + 6 + B = 180

A + B + C = 180

Now from the above equations, we see that

1 + 2 + 3 + 4 + 5 + 6 = 180 + 180 + 180 - (C + A + B) = 180 * 3 - 180 = 360

Another way:

Rotate line XY clockwise through all the angles in the following sequence: 2 5 6 3 4 1. The line made one complete revolution; so the sum of those angles is 360.

The sum of the angles in a triangle is always 180, so the measure of all the angles of all the triangles would be 180x3=540. BUT! We aren't counting the angles that fall inside the little central triangle, which (since they themselves form a triangle) add up to 180. Thus, the answer is 180 x 2 = 360!

there are three big triangles and one small then small triangle is an Equilateral triangle so all the big triangles having miss an angle of 60...... so 120+120+120=360