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Question

In the diagram below, points A, E, and F lie on the same line. If ABCDE is a regular pentagon, and angle EFD=90 degrees, then how many degrees are in the measure of angle FDE?

http://latex.artofproblemsolving.com/3/8/d/38d4959c0ccf347f48c0a45878e116382300d594.png

Answer 1

Divide the pentagon by 5=360/5=72 degrees

extend lines to edges to make a triangle =180 degrees

180 degrees subtract 72 degrees=108 degrees

the triangle formed is isosceles triangle meaning the base angles are equal=108/2=54 degrees

at AED, THE ANGLE THERE IS: 54 degrees X 2=108 degrees. to complete a straight line 180-108=72 degrees 

therefore, FDE=72+90+Y=180, y=angleD=12Degrees

Answer 2

Greetings Round Rock, TX! You clearly have four great answers provided below. I'm responding to provide more benefit -- something that will help you with future problems as well (i.e., teach you to fish!). 

 

With regular polygons, the external angles will always add to 360 degrees.

 

What are external angles? Well, they're formed by extending the side of the polygon. As you extend the side, you'll notice you're forming an additional angle...much like DEF. [Cool factoid - for any polygon with five or more sides, the external angle will always be acute (i.e., less than 90 degrees)]. 

 

How does this help? Well, if you have five sides of a regular pentagon, then the 360 degrees are equallly divided amongst the five external angles...meaning each external angle is 72 degrees (360/5). You can then quickly calculate the internal angle to be 108 (180 - 72). 

 

This approach is very simple for all regular polygons. Best wishes with your future math studies!

Answer 3

We are given that figure ABCDE is a regular pentagon.

 

From the properties of regular pentagons, we know that the interior angles are all 108^o.

 

Thus, ∠AEF = 108^o. As a result, ∠DEF will equal 180^o - 108^o, or 72^o.

 

We know that the sum of the internal angles of a right triangle is 180^o.

Since ∠EFD = 90^o, then 180^o - 90^o - 72^o =  ∠EDF = 18^o

Answer 4

Great question. So one thing to remember here is that the sum of the interior angles of any polygon can me calculated. A great website that explains this is http://www.mathsisfun.com/geometry/interior-angles-polygons.html. So we can see that a pentagon has a total of 540°. Because it is a regular pentagon (meaning all the angles have the same measure) we can calculate the measure of each angle by dividing 540° by 5 (540°/5 = 108°). This means the measure of angle AED is 108°.

 

We also know that points A, E and F are linear, meaning the measure of angle AEF is 180° (it's a straight line). Using this information we can subtract ∠AED from ∠AEF to get ∠FED. 180° - 108° = 72°.

 

If ∠FED is 72° then we can calculate the m∠FDE because we know that the sum of the interior angles of a triangle is 180°. ∠EFD is 90°, so 72° + 90° = 162°. So then ∠FDE = 180° - 162°.

 

I'll leave you to finish this one on your own.

 

I'm available for additional tutoring help online if you'd like.

Answer 5

It is important to note that in a regular pentagon the interior angles are 108 degrees.

Thus AED is 108 degrees.

FED is supplementary to this angle and is therefore 72 degrees.

since EFD is 90 degrees and the sum of angles in a triangle is 180,

FDE must therefore be 18 degrees.

 

If you need further help, I am available for tutoring.

Answer 6

Because ABCDE is a regular pentagon, this means that each side is the same length and also that the interior angles are all equal.  We use the formula for the sum of interior angles of a polygon, 180(n-2), to determine that the sum of these angles must be 540.  Now, since we know that the interior angles of the pentagon must all be equal, we know that each of these 5 angles = 540/5, or 108 degrees.

 

So, we know that angle AED is 108 degrees and that any straight line is 180 degrees.  This tells us that angle DEF must equal 180 - 108, or 72 degrees.  Finally, we know that the sum of the interior angles of a triangle must equal 180 degrees.  Now we can use this information to find the number of degrees in FDE: 180 - 90 - 72 = 18 degrees.