Nathan constructed the angle bisector of ∠ABC by constructing ray BX . Nathan then constructed the angle bisector of ∠XBC by constructing ray BY . If the measurement of ∠ABC is 84 degrees, what is the measurement of ∠YBC ? (1 point) Responses 63 degrees 63 degrees 42 degrees 42 degrees 21 degrees 21 degrees 168 degrees

Answer 1

To find the measurement of angle \( \angle YBC \), we can break the problem down into steps.

1. First, we know that \( \angle ABC = 84^\circ \).
2. Nathan constructs the angle bisector of \( \angle ABC \) by creating ray \( BX \). This means that \( \angle ABX \) will be half of \( \angle ABC \): \[ \angle ABX = \frac{1}{2} \times 84^\circ = 42^\circ \]
3. Now we need to find \( \angle XBC \). Since \( \angle ABC \) consists of \( \angle ABX \) and \( \angle XBC \), we can express it as: \[ \angle ABC = \angle ABX + \angle XBC \] Therefore, \[ \angle XBC = \angle ABC - \angle ABX = 84^\circ - 42^\circ = 42^\circ \]
4. Next, Nathan constructs the angle bisector of \( \angle XBC \) by creating ray \( BY \). This means that \( \angle XBY \) will be half of \( \angle XBC \): \[ \angle XBY = \frac{1}{2} \times 42^\circ = 21^\circ \]
5. Finally, \( \angle YBC \) is part of \( \angle XBC \): \[ \angle YBC + \angle XBY = \angle XBC \] That is, \[ \angle YBC + 21^\circ = 42^\circ \] Thus, \[ \angle YBC = 42^\circ - 21^\circ = 21^\circ \]

Therefore, the measurement of \( \angle YBC \) is \( \boxed{21} \) degrees.