a How does the opposite side of the slope angle compare to its adjacent side?
B
b The angle is less than 45^(∘). What does this tell us about the length of the opposite side relative to the adjacent side?
C
c What do we know about the sum of the angles in a triangle?
A
a We agree, see solution.
B
b Isiah is correct. See solution.
C
c x<9 units
Practice makes perfect
a In a right triangle where the non-right angles are both 45^(∘), the legs are congruent.
Consequently, if the opposite side of an angle is longer than its adjacent side, the angle will be greater than 45^(∘), and vice verse. Let's identify the opposite and adjacent side to the slope angle θ in the given triangle.
Since the opposite side of θ is greater than its adjacent side, the measure of θ must be greater than 45^(∘). Therefore, we agree with Thalia.
b In a right triangle, if the opposite side of an angle is greater than its adjacent side, then the angle must be greater than 45^(∘). Let's use the example from Part A.
In Part B the slope angle is 19^(∘). Since the slope angle is less than 45^(∘), the length of the opposite side must be less than the adjacent side. Let's illustrate this to scale.
Since the angle's opposite side is less than its adjacent side, the ratio of Δ y to Δ x has to be less than 1, and not 2.675. Therefore, we know that Isiah is correct, and Lyra is not.
c When the opposite side of an angle in a right triangle is greater than the adjacent side, the angle must be greater than 45^(∘). The angle in Part C is 76^(∘), which means the opposite side must be greater than the adjacent side. Therefore, we know that x<9 units.
