Question Title Here

[sandyRoe]

A large square divided into 9 equal smaller squares. The bottom left square is bisected, lower left to upper right, and the angle at the bottom corner is 'a'.
A line runs from the left corner of the middle lower square to the top right corner of the central square. The angle where it meets the Base line is 'b'.
A line from the left corner of the square , bottom right, runs to the top right corner of the square to the right on the top line.
The angle it makes at the bottom is 'c'.
The question asks what is the sum of the 3 angles?
I think it's 90 degrees but can't see why.

[sandyRoe]

Angle 'a' is obviously 45 degrees

[Jonathan-Joe]

Using trigonometry, tangent (angle) = opposite / adjacent
tan(a) = 1/1; a = 45 deg
tan(b) = 2/1; b = 63.43 deg
tan(c) = 3/1; c = 71.57 deg
a + b + c = 45 + 63.43 + 71.57 = 180 deg
But I have seen a more elegant solution using geometry - unfortunately can't recall where - will try a search for it.

[Jonathan-Joe]

Found what is essentially the same problem - the diagram here superimposes the three rectangles.
https://www.theguardian.com/science/alexs-adventures-in-numberland/2014/oct/21/martin-gardner-mathematical-puzzles-birthday
Scroll down to "Three Squares" then to "UPDATE: I’ve posted the answers here."
Here we prove angle a = (90 - b) + (90 - c).
so a = 180 - b - c
then a + b + c = 180

[Old_Geezer]

A picture is worth a thousand words, so I have been told.

[Jonathan-Joe]

Sorry Sandy, the Martin Gardner problem and my post reverse your order of a, b and c. Hope you can translate from one to the other.

[sandyRoe]

If I could have managed the link I'd have saved myself a lot of typing on this phone.

[Togo]

We know that the internal angles of a triangle=180degrees
All three triangles described ha a right angle leaving two angles that have a sum of 90 degrees
The square that has angle A in it has equal sides at right angles making a ratio of 1 to 1 ie 45 to 45 degrees in the adjacent angles.
The triangle with angle b has a ratio of 2 to 1 in its sides of the right angle. ie 60 degrees and 30 degrees angle b being 60 degrees
The triangle with angle c has a ratio of 3 to 1 (the remaining 90 degrees divided by 4 =  22.5) 22.5 X 3 = 67.5degtees for angle c.
45+60+67.5=172.5 degrees for angles a,b,c.

[Togo]

Brief synopsis. Think 3 right angled triangles have opposite and adjacent sides having ratios of 1/1 2/1 3/1. The remaining angles must have the same ratios of the 90 degrees that are left when the right angle is deducted from the internal angles of a triangle.

[Togo]

JoJo was correct of course. ( Just did the maths using. Tom On A Ship Of His Caught All Herring)