a Consider the Triangle Longer Side Theorem. What's the size of x given the measure of ∠ 2 when compared to ∠ 1 and ∠ 3?
B
b Which angle is the smallest in the triangle given the inequalities?
A
a x>565km and x> 489 km
B
b x< 489 km
Practice makes perfect
a We can box in the distance from Granite Peak to Fort Peck Lake by examining the triangle's sides and making assumption about the size of ∠ 2.
By the Triangle Longer Side Theorem we can already say that m∠ 3>m ∠ 1. This is because the distance between Fort Peck Lake and Glacier National Park is greater than the distance between Glacier National Park and Granite Peak. Now we have to make assumptions about ∠ 2.
m∠ 2 >m ∠ 3>m∠ 1
If ∠ 2 is greater than both ∠ 3 and ∠ 1, the distance between Granite Peak and Fort Peck Lake will be the longest of the triangle's sides. Since 565 km is the longest of the known sides, all we can say is that x will be greater than this side.
x>565km
m∠ 3 > m ∠ 2>m∠ 1
If ∠ 2 is less than ∠ 3 but greater than ∠ 1, the distance between Granite Peak and Fort Peck Lake will fall in between 489 km and 565 km.
489 km< x < 565 km
m ∠ 3>m∠ 1> m∠ 2
If ∠ 2 is less than both ∠ 3 and ∠ 1, the distance between Granite Peak and Fort Peck Lake will be the shortest of the triangle's sides. Since 489 km is the shortest of the two known sides, all we can say is that x will be less than this value.
x< 489 km
Therefore, two inequalities that represents the possible distances from Granite Peak are
x>565km and x< 489 km
b If m∠ 2Triangle Larger Angle Theorem we know that x is the smallest side in the triangle. Thus, means we can narrow down our answer to one inequality.
