To obtain a we have to add 3 to b. What does this tell us about the relationship between a and b?
E
e
Use the sine and cosine ratio.
A
a
a is greater than b.
B
b
Not enough information.
C
c
b is greater than a.
D
d
a is greater than b.
E
e
a is equal to b.
Practice makes perfect
a
To calculate the measure of an interior angle of a regular n-gon we can use the formula 180^(∘)(n-2)n, where n is the number of sides in the polygon. A pentagon is a polygon with 5 sides.
The sum of the central angles in any regular polygon equals 360^(∘). Since an equilateral triangle has 3 congruent central angles, we can determine the measure of a by dividing 360^(∘) by 3.
360^(∘)/3= 120^(∘)
As we can see, a=120^(∘) and b=108^(∘). Therefore a is greater than b.
From our calculations, we see that b is greater than a.
d
The equation tells us that to obtain a we have to add 3 to b. This must mean that a is greater than b.
a = b + 3
a is 3 greater than b
e
Like in Part A, we can use trigonometric ratios to write equations containing a and b. For the triangle on the left we need to use the cosine, and in the triangle on the right we need to use the sine ratio.
Notice that both triangles have the same hypotenuse. It will be easier to compare a and b if we express their values using the same trigonometric function. To do so, let's find the measure of the missing angle in the right triangle. Ratio b 7 could be then expressed using the sine ratio.
In a right triangle two acute angles are complementary. Therefore, the sum of their measures will add up to 90^(∘). This allows us to write an equation that we can solve for x.
x + 67^(∘) = 90^(∘) ⇔ x = 23^(∘)
The measure x is the same as the measure of the acute angle in the left triangle. Therefore, we can equate the sine ratios.
a/7 = sin 23^(∘) = b/7
We can multiply the equation between the outward terms by 7.
a/7 = b/7 ⇔ a = b
We see that a is equal to b.
