2025 AMC 10A Problems/Problem 6: Difference between revisions
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Therefore, we have: <imath>3a+3b=720 \implies a+b=240</imath> (total sum of all angles in a convex hexagon is 720) and also <imath>20+2a+b=360 \implies 2a+b=340</imath> (the rays will form an inner angle of <imath>\frac{60}{3}=20</imath> degrees). Subtracting the two equations yields <imath>a=100</imath> and <imath>b=140</imath>. Hence our smallest angle in this convex hexagon is <imath>\boxed{\textbf{(C) }100}</imath>. ~hxve | Therefore, we have: <imath>3a+3b=720 \implies a+b=240</imath> (total sum of all angles in a convex hexagon is 720) and also <imath>20+2a+b=360 \implies 2a+b=340</imath> (the rays will form an inner angle of <imath>\frac{60}{3}=20</imath> degrees). Subtracting the two equations yields <imath>a=100</imath> and <imath>b=140</imath>. Hence our smallest angle in this convex hexagon is <imath>\boxed{\textbf{(C) }100}</imath>. ~hxve | ||
==Video Solution== | |||
https://youtu.be/gWSZeCKrOfU | |||
~MK | |||
==See Also== | ==See Also== | ||
Revision as of 17:32, 6 November 2025
In an equilateral triangle each interior angle is trisected by a pair of rays. The intersection of the interiors of the middle 20°-angle at each vertex is the interior of a convex hexagon. What is the degree measure of the smallest angle of this hexagon?
Solution 1
Assume you have a diagram in front of you.
Because each angle of the triangle is trisected, we have 9 20° angles. Using a side of the triangle as a base, we have an isosceles triangle with two 20° angles. Using this we can show that the third angle is 140°.
Following that, we use the vertex angles to show that one angle of the hexagon is 140°. And with rotational symmetry, three.
The average of all 6 angles has to be 120°, so the answer is 
- SpectralScholar
Solution 2
It is obvious that of the 6 angles inside the convex hexagon, there are only two different angle measures, 3 of one and 3 of another. A convex quadrilateral formed by the 2 rays of any angle in the equilateral triangle and two sides of the convex hexagon will have a total degree of 360.
Therefore, we have: 
(total sum of all angles in a convex hexagon is 720) and also 
(the rays will form an inner angle of 
degrees). Subtracting the two equations yields 
and 
. Hence our smallest angle in this convex hexagon is . ~hxve
Video Solution
~MK
See Also
| 2025 AMC 10A (Problems • Answer Key • Resources) | ||
| Preceded by Problem 5 |
Followed by Problem 7 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
| All AMC 10 Problems and Solutions | ||
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