Given the 𝐴𝐵𝐶𝐷𝐸 is a regular pentagon, find the measure of angle 𝐴𝐵𝑋.
Firstly, let’s mark the angle that we’re looking to find on the diagram. 𝐴𝐵𝑋 is the angle formed when you travel from 𝐴 to 𝐵 to 𝑋. So it’s this angle here, marked in orange. We can see that the diagram consists of a triangle, triangle 𝐵𝑋𝑌, and then the pentagon 𝐴𝐵𝐶𝐷𝐸 which we’re told is regular. Let’s think about how we’re going to approach this problem.
The angle that we’re looking for, angle 𝐴𝐵𝑋, sits on a straight line with two other angles: angle 𝑋𝐵𝑌 inside the triangle and angle 𝐴𝐵𝐶 inside the pentagon. If we can work out these two other angles, then we can calculate angle 𝐴𝐵𝑋 using the fact that angles on a straight line sum to 180 degrees.
Let’s think about the angle in the triangle first of all. Remember, the angle sum in a triangle is always 180 degrees. And as we’ve been given the measures of the other two angles, we can calculate the third. So angle 𝑋𝐵𝑌 is 180 degrees minus 79 degrees minus 64 degrees which is 37 degrees.
Next, let’s think about the angle in the pentagon, angle 𝐴𝐵𝐶. A key fact about polygons is that the sum of their interior angles can be calculated by multiplying 180 by 𝑛 minus two, where 𝑛 represents the number of sides in the polygon. Our polygon is a pentagon which has five sides. Therefore, the sum of its interior angles is found by multiplying 180 by three which is 540.
Now this is the sum of all of the interior angles in the pentagon, not the size of each individual angle. The key piece of information given in the question is that 𝐴𝐵𝐶𝐷𝐸 is a regular pentagon. Which means that all of its interior angles are the same size. Therefore, each interior angle can be found by dividing the sum by five. 540 divided by five which is 108 degrees.
So now we know the size of angle 𝐴𝐵𝐶 and the size of angle 𝑋𝐵𝑌. Remember, these two angles sit on a straight line with angle 𝐴𝐵𝑋, which we’re looking to calculate. So angle 𝐴𝐵𝑋 is equal to 180 degrees minus 108 degrees minus 37 degrees. The measure of angle 𝐴𝐵𝑋 is 35 degrees.