What is the sum of the 6 blue angles?

$180^\circ$
$360^\circ$
$450^\circ$
$540^\circ$

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I am not convinced that the pink triangle is an equilateral triangle. since there are no parallel lines here. But since it is the SUM of its angles which is important, and since this will always equal 180 degrees, it doesn't matter whether it is or not. John Oakes

John Oakes
- 4 years, 4 months ago

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Right, we do not need the assumption that the pink triangle is equilateral :)

Chung Kevin
- 4 years, 4 months ago

I didn't understand anything of your solution.

Anik Saha
- 4 years, 6 months ago

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Do you understand what the sum of angles of a pink triangle are? Do you see why it gives us 2 of the blue angles and 1 of the angles in the green triangle?

Chung Kevin
- 4 years, 6 months ago

You took the words right out of my mouth! :)

Hannah Park
- 4 years, 6 months ago

It doesn't matter if the inner triangle is equilateral. It could have been 40, 60, 80, or whatever It would still be one of each of the triangle's angles for them to each add up to 180 and so the inner one does as well.

Kristian Valle
- 2 weeks, 3 days ago

The sum of all the angles within a triangle (any type) is 180°. The triangle created by the intersection of the three triangles is an equilateral triangle (each angle is the same), so 60°. So 180°+180°+180°-60°-60°-60°= 360°

Makes sense?

Sam Fis
- 4 years, 4 months ago

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It need not be true that the small pink triangle is equilateral. The result is still true, as John mentioned.

Chung Kevin
- 4 years, 4 months ago

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

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Nice! That feels so quick :)

Chung Kevin
- 4 years, 5 months ago

Great solution!

Kerushan Govender
- 4 years, 4 months ago

3x180 -180=360 Here is how:

BLUE+RED=540

RED=180

BLUE=540-180=360

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*
external
*
angles. Since
$\text{internal angle} = 180^\circ - \text{external angle},$
and there are six internal angles here, we have
$\sum\text{internal angle} = \sum 180^\circ - \sum\text{external angle} = 6\times 180^\circ - 720^\circ = \boxed{360^\circ}.$

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I like this too! So many nice ways to view this problem :)

Chung Kevin
- 4 years, 6 months ago

?????? not understood

Anik Saha
- 4 years, 6 months ago

Turtle Geometry by Abelson and diSessa is a wonderful book - where taught now I wonder?

Chris Wallace
- 4 years, 4 months ago

With a physical straightedge or mental one you can just traverse/pivot on the outer angles and get 360 without even thinking about the inner angles. In all honesty I think it's more fun/easier to just pivot on the relevant angles.

Brian Bohan
- 1 year, 11 months ago

I think you could also consider the angle a vector moving around the path would rotate through to get back to the start.

Tom Mason
- 1 year, 8 months ago

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 $360^\circ$ .

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I like the "rotate a pen" idea!

Chung Kevin
- 4 years, 6 months ago

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

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

x + y + z = 180

Therefore,

(3 X 180) - 180 = Answer

540 - 180 = 360

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Well done :)

Chung Kevin
- 4 years, 4 months ago

Just because it 'looks like' an equilateral triangle does not mean that it is one absolutely. Therefore, none of the angles in the diagram is absolutely determined. Regardless, the sum of the angles of the central triangle is still 180 degrees.

Patrick Conoley
- 4 years, 4 months ago

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

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Well done :)

Chung Kevin
- 4 years, 4 months ago

That's how I did it too (but without the first calc) - err, (3-1) x 180 = 360.

Brandon Ashworth
- 4 years, 4 months ago

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Although the diagram 'looks' like a solution, do we really know that the sides will make a quadrilateral after flipping?

Patrick Conoley
- 4 years, 4 months ago

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

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Oh, that's a nice equation to put together!

Chung Kevin
- 4 years, 5 months ago

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Nice! That's what I did too :)

Chung Kevin
- 4 years, 5 months ago

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)

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Nice way of presenting it!

Chung Kevin
- 4 years, 5 months ago

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

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*
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

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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.

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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.

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*
- (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
*

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

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

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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

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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.

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The small triangle need not be equilateral. We're still subtracting off all of the angles, so we have $3 \times 180 - 180 = 360$ .

Chung Kevin
- 4 years, 4 months ago

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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
```

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

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

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

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What makes you believe that the middle triangle is an equilateral triangle? It isn't forced to be.

Charles Gaskell
- 4 years, 4 months ago

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It actually doesn't matter what type of triangle it is, since it still has the same sum of internal angles, each of which completes one of the larger triangles. So it will always be (180*3) - 180 = 360 regardless of how the internal angles of the middle triangle are distributed.

James Palmer
- 3 years, 1 month ago

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Agree with that. The bit about the sum of each pair of blue angles being 120 degrees is (I think) incorrect

Charles Gaskell
- 2 years, 11 months ago

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.

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Nice interpretation :)

Chung Kevin
- 4 years, 4 months ago

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*
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

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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.

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We are asked to be creative. Unfold the drawing and you see a rectangle. In doing so, six angles "become" four angles....four corners angles of a rectangle: 90 + 90 + 90 +90 = 360. Just sayin'.

Jeff Kadas
- 4 years, 4 months ago

I "unfolded" the diagram and made a rectangle. a rectangle has a total of 360 degrees

Tobias Nash
- 4 years, 4 months ago

Is the middle triangle an equilateral triangle? How do you know?

Charles Gaskell
- 4 years, 4 months ago

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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

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