And yes, I missed a few of my geometry classes and I am trying to catch up. This was one of the assignments from the class I missed. After looking at this I realized what you are saying. I know the interior angles of a triangle add up to 180. And, the pentagon on the picture add up to 540(each side is 108), and the angles of the hexagon add up to 720(each angle is 120 because they are equal). Also, I know all the sides are congruent that are labeled and I know both triangles are 90 degree triangles. But I still don’t understand how to find the angle of BAC. I believe the answer is 30 degrees because It looks like a 30-60-90 triangle to me., but I have to explain how I came up with thirty degrees and I’m not positive that is correct?Are you saying you have not been taught how to calculate
the inside angles of a regular polygon?
If so, go here:
Interior Angles of Polygons
Another example ... The Interior Angles of a Triangle add up to 180www.mathsisfun.com
Right! Angles in pentagon = 108, in hexagon = 120.
Surrounding point A are 4 angles: see that?
The 2 angles above the triangle are 108 and 120; ok?
The 4 angles total 360; ok?
So the other 2 angles total 360 - 120 - 108 = 132; ok?
Since these 2 angles are equal, then they each equal 66; got that?
And since angle BAC is one of them, then....?
Yes, thank you so much!Suppose we decompose a regular polygon having \(n\) sides into \(n\) isosceles triangles. If we look at just one of these triangles, we know the sum of the two equal angles (which will be equal to an interior angle \(\theta_n\)) and the other angle is \(180^{\circ}\):
[MATH]\theta_n+\frac{360^{\circ}}{n}=180^{\circ}[/MATH]
Hence:
[MATH]\theta_n=180^{\circ}-\frac{360^{\circ}}{n}=\frac{180^{\circ}(n-2)}{n}[/MATH]
Let \(\beta=\measuredangle BAC\)...then we know:
[MATH]2\beta+\theta_5+\theta_6=360^{\circ}[/MATH]
[MATH]2\beta+\frac{180^{\circ}(5-2)}{5}+\frac{180^{\circ}(6-2)}{6}=360^{\circ}[/MATH]
Can you finish?
Angle BAC equals 66. Thanks a lot!Right! Angles in pentagon = 108, in hexagon = 120.
Surrounding point A are 4 angles: see that?
The 2 angles above the triangle are 108 and 120; ok?
The 4 angles total 360; ok?
So the other 2 angles total 360 - 120 - 108 = 132; ok?
Since these 2 angles are equal, then they each equal 66; got that?
And since angle BAC is one of them, then....?