a The resulting angles are half as big as the original angle.
B
b The resulting angles are half as big as the original angle.
C
c The resulting angles are half as big as the original angle.
D
d The resulting angles are half as big as the original angle.
A
a Acute
B
b Acute
C
c Acute
D
d Right
Practice makes perfect
a If you bisect an angle you draw a ray which splits the angle into two smaller angles each with a measure half that of the original angle. If ∠ A is acute, its measure is in the following interval.
0^(∘) < m∠ A < 90^(∘)
After we bisect A, the resulting angles are identical and these angles will have a measure which is half as big as m∠ A. If ∠ B is one of the smaller angles, then the following is true.
0^(∘)/2 < m∠ B < 90^(∘)/2 ⇒ 0^(∘) < m∠ B < 45^(∘)
Thus the two smaller angles will be acute.
b If you have a right angle it is 90^(∘). The angles that are created when you bisect it will be half as big, that is they will have the measure 45^(∘). An angle with the measure 45^(∘) is acute.
c When an angle is bisected, two angles half as big as the original angle are created. If we have an obtuse angle A, it will have a measure in the following interval.
90^(∘) < m∠ A < 180^(∘).
If we bisect A, the angle is split into two identical angles that both have a measure half of m∠ A. If we call one of the smaller angles ∠ B, then its measure will be in the following range.
90^(∘)/2 < m∠ B < 180^(∘)/2 ⇒ 45^(∘) < m∠ B < 90^(∘)
An angle in that interval is acute.
d An angle is said to be straight when its measure is 180^(∘). If you bisect such an angle you will get two angles, each with the size half that of the straight angle. The resulting angles will get the measure
180^(∘)/2=90^(∘).
An angle with the measure 90^(∘) is called a right angle.
