89 of 100: Pentahexacross
Two vertices of a regular pentagon, in red, intersect with two vertices of a regular hexagon, in blue. What is the measure of the yellow angle?
There are several shapes and angle relationships in this diagram to work with.
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8 solutions
We share a common solution to this problem
Didn't thought to do it like that good solution.
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Yes it needs nothing more than 108 and 120.
And a little suggestion for you too A = 120 -180 + B so you would need just to know the specific angles of the regular pentagon and hexagon for your solution.
I like this solution! If you want to get more upvotes or help more people, you will need to elaborate more on the diagram and actually write some words so people will understand more.
This one is my solution although a little simplification, the whole hexagon angle (the yellowish one and the 72 one both have to be together 120, so you just needed to know what angle has the 72 one witch you can get from the quadrilateral witch has to have the sum of it's angles = 360 from witch you know that the 72 angle = (360 - 108)/2). So you didn't need to calculate the 36 angle. (btw if it helps you can take parts of my text for your description).
My little advice write bigger text, except for that good solution.
Nice, neat solution
Thanks for ur suggestions and comments !!
CAN'T READ IT MAN
∠ A B E is an interior angle of a hexagon, so its measure is ( ( 6 − 2 ) × 1 8 0 ) / 6 = 1 2 0 , and ∠ B D F is an interior angle of a pentagon, so its measure is ( ( 5 − 2 ) × 1 8 0 ) / 5 = 1 0 8 .
The measure of ∠ A B C is m ∠ A B E − 9 0 = 1 2 0 − 9 0 = 3 0 .
The measure of ∠ B D E is m ∠ B D F − 9 0 = 1 0 8 − 9 0 = 1 8 .
C B and D E are parallel because they are perpendicular to the same line, so ∠ C B D is congruent to ∠ B D E because they are opposite interior angles and their measures are equal.
Finally, m ∠ A B D = m ∠ A B C + m ∠ C B D = 3 0 + 1 8 = 4 8 ∘ .
This is nice.
We know that the interior angles of a regular hexagon are 1 2 0 ∘ each and the interior angles of a regular pentagon are 1 0 8 ∘ each.
Since a regular pentagon is inscribable in a circle, the trapezoid formed inside the hexagon must also be inscribable in a circle and is therefore a cyclic quadrilateral. Since the opposite angles of cyclic quadrilaterals are supplementary, we can find the measure of the angles in the trapezoid as 1 0 8 ∘ and 7 2 ∘ . We then find that the measure of the yellow angle is 1 2 0 ∘ − 7 2 ∘ = 4 8 ∘ .
I actually used logic to solve this one. For this, I actually looked at the yellow angle, then noticed the 4 choices; the yellow angle looked like a 4 5 ∘ angle, which means that A and D are already eliminated. Now I am left with only 2 choices 4 8 ∘ and 5 4 ∘ ; since I said the yellow degree looks like a 4 5 ∘ and 4 8 ∘ is the closer to 4 5 ∘ than 5 4 ∘ , which makes 4 8 ∘ the most logical answer.
Not really logic though. It's called instinct and it's risky to only rely on it.
Because of the 2 paralel lines angle A is equal with angle A' and the specific angles for a regular hexagon and pentagon are P=180
3/5=108 and H=180
4/6=120, so the final formula for the yellow angle is ? = H - A' = H - A = H - 180 + P = 120 - 180 + 108 = 48
Interior angle of regular hexagon is ( 4 × 1 8 0 ) / 6 = 1 2 0 ∘ and its external angle will be 6 0 ∘ . On the other hand B is an interior angle of a regular pentagon, so its ( 3 × 1 8 0 ) / 5 = 1 0 8 ∘ .
By parallel lines B = A + 6 0 and A = B − 6 0 = 1 0 8 − 6 0 = 4 8 ∘ .