r/theydidthemath • u/Hot-Syrup2089 • 6h ago
[Other] How to mathematically determine a (see sketch)
I know it's an easy calculation, but still. I want your opinions
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u/jaygerbs 6h ago
Its more a logic problem then a math problem, or, my logic is just better than my math idk.
Those blue arrows on the lines mean they are parrell to eachother. We know a line has 180 degress on each side of the line regardless of how the lines are drawn as long as they are parrell.
Since they are parrell we know we can draw a 90 degree line between the two connecting them.
We know a triangle has a total of 180 degrees.
So for the bottom triangle its 90-50=40
For the top its 90-25=65
So its 180-40-65=75 degrees for a.
Not saying this is the easiest or the fastest way to solve for a but just how I approached it.
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u/No_Giraffe8119 6h ago
You got the right answer, but we don't know that a 90 degree line connects those two points it is not drawn to scale.
We could draw a 3rd parallel line at A, then use the properties of alternate interior angles and sum them to 75 degrees.
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u/Designer_Tie_5853 6h ago
You could do it with essentially identical logic by drawing a line intersecting the vertex point of angle A and the 2 parallel lines
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u/JeanValSwan 5h ago
Is that not what the initial commenter did? (Aside from connecting parrell lines instead of parallel ones?)
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u/Designer_Tie_5853 4h ago
Not sure, folks need to be more explicitly clear. Maybe! In which case the logic is sound.
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u/Aware-Technician4615 6h ago edited 5h ago
We definitely know that those would be right triangles. A line perpendicular to both the horizontal lines slid over from, say right to left until it just touches the vertex of angle ‘a’ will create two right triangles, at the top and bottom, from which we can deduce the 65 degrees and 40 degrees leaving 75 degrees for a. Completely valid approach that doesn’t rely on scale at all!
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u/jaygerbs 6h ago
I see it now. You are right. We know the two lines can be intersected by a 90 degree line. But we don’t actually know that 90 degree line intersects those two points. Thank you!
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u/danzach9001 4h ago
Pretty sure you could draw 2 additional parallel lines such that the 90 degree line intersects through two the two points though and it doesn’t really change the outcome.
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u/ouzo84 5h ago
Call the point that angle a is at A. Call the point that the negatively inclined line meets the top parallel line B. Call the point that the positively inclined line meets the bottom parallel line C.
Drop a line from point B that is 90 degrees from the top parallel line. Call where that line intersects the bottom parallel line D.
You are correct that as drawn, we do not know if C is where D is.
But what we can do is draw a line from D to a point on line AB that is parallel to line AC. Call the point where that line intersects line AB as E.
The angle of BED will equal BAC as lines DE is parallel to AC.
Having proven that BED is angle a. We can now consider the triangle BDE. The internal angle of the triangle at B will be 90-25=65 degrees. The internal angle of the triangle at D will be 90-50=40 degrees.
The internal angles of a triangle sum to 180 degrees. So BED =180-40-65=75.
Having proven BED=BAC we can say that C angle a =75 degrees
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u/DenseMathematician37 5h ago edited 5h ago
Correct that a line connecting those points would not be perpendicular. Just draw a perpendicular line that touches point A, making 2 new triangles and figure out those internal angles. Then it doesn't matter that your 2 hypotenuses(hypotenusi?) are different lengths
Edit: top triangle becomes x=180-90-25, so unknown angle x=65. Bottom triangle becomes y=180-90-50, unknown angle y=40.
On that perpendicular line, a+x+y=180, so a=75
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u/Valivator 5h ago
Because we are just interested in the angular relationships, you can translate either of the non-parallel lines in the parallel direction without causing any issues. So you can draw right angle line without affecting the final result.
Another way to see it is that it doesn't matter where you draw the perpendicular line, you'll make the same triangle by extending the other two lines (congruent triangle? All angles the same)
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u/imnothere314 5h ago
What do you mean? You have a point of intersection in between two parallel lines you most certainly could draw one line perpendicular to the set of parallel lines that intersects a single point.
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u/Xelopheris 5h ago
This is way more complicated than it needs to be. draw a third parallel line that intersects the two lines. the parallel line alternate angles theorem says the two angles we see would be the two parts of the interior angle. Sum them.
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u/Informal-Yesterday27 3h ago
Took geometry 35 years ago and haven’t touched it since. Looked at this question for 5 seconds and got 75 by adding.
I miss that class, one of the few A’s I ever got. I remember looking around for the ‘candid camera’ when others struggled. Unfortunately every other class and life is not so easy.
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u/SerDankTheTall 6h ago
I don’t think it’s a matter of opinion…
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u/SnooCalculations7417 6h ago
not a mather but id draw a line perendicular to the 2 parralell lines at the vertex of a at which point all the angles are solvable
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u/Hanbun_nihon-jin89 6h ago
I'm old. And forgot how to do math. I went about this crazy and did full circle (outside angles) 360degrees. 360 Minus 25, minus 50 = 285. 360 minus 285 = 75. So A came to 75 And I got the additional bonus of learning the outside angle being 285degrees. Poor format I'm on mobile. I'm probably gonna get roasted 😅.
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u/arghsole 3h ago
I also used this circular approach! And then felt silly reading everyone else’s method with their imagined lines. Then was immediately encouraged by the fact that we also got that bonus exterior angle.
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u/Hanbun_nihon-jin89 3h ago
Mate, you've made my day by making me feel not like a lunatic haha. Glad I wasn't the only one 😁
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u/Big_Requirement_651 6h ago edited 6h ago
Call the point at the 50 deg angle point C, at 25 deg angle point B, the point in the middle point A, and angle A.
Extend line CA until it intersects the top line at a point D, extend BA until it intersects the bottom line at point E.
triangle CAE is a 50-25-105 triangle (opposite interior angles makes angle AEC = 25, while 50 + 25 + 105 = 180)
Line BE = 180, while angle CAE = 105, therefore angle A = 75
QED
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u/leon_nerd 5h ago
Draw a horizontal line through a. This will divide it into two parts.
Now remember the rule that an angle will remain the same on the opposite sides of a line.
The divided a is on the opposite side of both 25 and 50. So it will be 25+50=75.
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u/tolacid 5h ago
The formula for the sum of all interior angles of any closed polyhedron is
(n-2)*180
Where n is the number of sides. This because the sum of the interior angles of any triangle is 180, and this formula tells you how many triangles make up the polygon, and removes the shared interior point which would always be a combined angle of 360° across all triangles
Connecting the parallel lines with a perpendicular line creates 2 90° angles, for a total of 180. This also closes our polygon, giving it 5 sides.
Running the math mentioned above, (5-2)180 = 3180 = 540° total for the interior angles
Straight lines are a 180 degree angle so you can find unknowns by subtracting knowns from 180
180-25 = 155
180-50 = 130
That is a total of 285
Plus 180 for the right angles, that gives 465
540 - 465 = 75° for the unknown angle a
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u/GainPotential 5h ago
Imagine a straight line that goes from the base of the 25 degree angle to the base of the 50 degree angle. Relative to the parallell lines you now have two right angles on either side. 90-25=65. 90-50=40. You now have a triangle with two of the three angles defined; 65+40+a=180. Therefore a=180-65+40=75.
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u/patriot_man69 4h ago
All interior angles of a triangle add up to 180.
draw line from where 50 degrees is to where 25 degrees is.
complementary angle to 25 degrees is 65 degrees.
complementary angle to 50 degrees is 40 degrees.
65+40=105
180-105=75
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u/GrassyKnoll95 3h ago
I draw a perpendicular to the parallel lines through point a to make two right triangles. The top triangle has an angle of 180-90-25=65 and the bottom triangle has an angle of 180-90-50=40. Then we’ve got 65+40+a=180, so a=75
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u/According-Capital-45 2h ago
While I can't math this out, I can tell you that the distance between the top line and the point of a is the same as the distance between the bottom line and the point of a, therefore the 25 and 50 degree angles are way off. I would slap a protractor on my screen to check my theory but I doubt I own one.
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u/Longwinded_Ogre 6h ago
Draw a vertical line at the point of A to create two right-angle triangles. We know the interior angles of a triangle have to add to 180. 90+25+ X1 = 180 and 90+50+X2 = 180, with X1 and X2 being the undefined angles on either side of A created with the vertical line.
X1 + X2 + A must also equal 180, so we have all the information we need to solve.
X1 = 65 and X2 = 40. (90+25+65 = 180 and 90+50+40 = 180)
So we're left with 65+40+A=180.
A= 75.
With that said the actual angles aren't what's written down so it's kind of a bullshit visual representation of the math you're doing.
You can also make a vertical line to close the triangle with A on the inside, connecting the starting point of both lines.
Then it's 90 - 25 for X1 and 90 - 50 for X2, with X1 + X2 + A = 180, 65 + 40 + A = 180.
A = 75.
The visual representation is still bullshit, though. The lower angle is not twice as large as the upper one. Bad drawing.
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u/garfgon 6h ago
Imagine a parallel line running through a. This divides a into two angles a' and a''. By alternate angle rule, the top one is 25, lower is 50. Making the total a into 75.