a Choose the equator and two lines of longitude as the three great circles.
B
b Repeat the steps in Part A.
C
c What is the sum of the measures?
A
aExample Solution: 90 ^(∘), 90^(∘), and 90^(∘).
B
b See solution.
C
cConjecture: The sum of the measures of a triangle in spherical geometry is greater than 180^(∘).
Practice makes perfect
a We want to mark three great circles on a ball and estimate the measures of angles in the triangle the great circles will form. Let's choose the equator and two perpendicular lines of longitude as the three great circles. This is just one possible triangle that we can make, and there are many other possible answers.
Let's first use a protractor to measure the angle at point A. To do this, we will rotate the ball so that we are looking straight at point A.
Now let's use a protractor.
The line of longitude that goes through the point A is perpendicular to the equator. Now we will measure the angle of our triangle at point B by rotating the ball so that we are looking straight at point B and then using a protractor.
We can now use a protractor again.
The line of longitude that goes through point B is also perpendicular to the equator. To estimate the measure of the last angle, let's look at the ball from the top and use a protractor again.
The third angle in the triangle also has a measure of 90^(∘). This tells us that △ ABC has three right angles.
b The measures of the angles of triangle from Part A are 90^(∘), 90^(∘), and 90^(∘). Now, we will create a second triangle and measure its angles. Like before, let's choose the equator and two lines of longitude.
This time we choose lines of longitude that are not perpendicular. When we measure ∠ ACB we get that m∠ ACB= 60^(∘). Since the lines of longitude are perpendicular to the equator, we get that m∠ CAB=m∠ ABC=90^(∘). Let's create a triangle like below.
When we measure the angles of △ ABC, we get that m∠ ABC=75^(∘), m∠ BCA=45^(∘), and m∠ CAB=80^(∘). Now, let's tabulate the measures and record the sum of the measures of each triangle.
Triangle
Angles
Sum
1
90^(∘), 90^(∘), 90^(∘)
270^(∘)
2
90^(∘), 90^(∘), 60^(∘)
240^(∘)
3
75^(∘), 45^(∘), 80^(∘)
200^(∘)
c Note that the sum of the measures of each triangle in Part B is always greater than 180^(∘). If you make your own triangles, you will always get the same conclusion. Therefore, we can write the following conjecture.
Conjecture
The sum of the measures of a triangle in spherical geometry is greater than 180^(∘).
