Start with using the Polygon Angle-Sum Theorem to find the sum of the interior angle measures in a pentagon. Then think what would happen if we had a particular number of right angles.
C
Practice makes perfect
In this exercise we want to determine the maximum number of right angles possible in a pentagon. A pentagon is a polygon with 5 sides, so it has 5 angles in total. Let's start with calculating the sum of all interior angle measures in a pentagon. To do so we will use the Polygon Angle-Sum Theorem.
(n-2)180
Here n is the number of sides, so in our case n= 5. Let's calculate the sum of the interior angle measures in a pentagon.
The sum of all interior angle measures in a pentagon is 540^(∘). Let's now think if we could have a pentagon with 5 right angles. If it was possible this pentagon would be regular. Recall the formula for the interior angle measure of a regular n-gon.
(n-2)180/n
Now we can substitute 5 for n in this formula and simplify. Note that we already know that (5-2)180= 540.
The measure of an interior angle of a regular pentagon is 108^(∘), which is not a right angle. Therefore, there is no possibility of having a pentagon with 5 right angles. Let's now check what would happen if we had 4 right angles. We will mark the remaining unknown angle as x and find its measure.
For such situation to happen the fifth angle would have to measure 180, which is not possible, since it is a half of a full angle. Let's now think about a pentagon with 3 right angles. We will mark the missing angle measures as x and y, and try to find their sum.
The sum of two remaining angles needs to be 270^(∘) when we have 3 right triangles, which is possible to obtain. Let's try plotting a pentagon satisfying these demands.
We have obtained a pentagon with 3 right angles. However, we previously stated that it is not possible to get a pentagon with 4 or 5 right angles. Therefore, the maximum number of right angles possible in a pentagon is 3, which corresponds to option C.
