Angle Bisectors Between Parallels — Meet at 90°
November 4, 2024 2025-11-04 0:22Angle Bisectors Between Parallels — Meet at 90°
Angle Bisectors Between Parallels ⇒ They Meet at \(90^\circ\)
If two lines are parallel and a transversal cuts them, the **bisectors** of the adjacent interior angles are **perpendicular**. Here’s the 1-minute why + a quick practice.
Concept & Proof (one look)
Let \( \ell \parallel m \) and \(Q\) be the intersection of the transversal with \( \ell \), \(O\) with \( m \). Suppose \(SQ\) bisects \( \angle PQO\) and \(SO\) bisects \( \angle QOR\). Denote \( \angle PQS=x \) and \( \angle ROS=y \).
- Adjacent interior angles on parallels are supplementary: \[ \angle PQR+\angle QRO=180^\circ. \]
- Since each is bisected, \[ 2x+2y=180^\circ \;\Rightarrow\; x+y=90^\circ. \]
- In triangle \( \triangle QOS \), \[ x+y+\theta=180^\circ \ \Rightarrow\ \theta=90^\circ. \]
Cheat code: Between parallels, **sum of the two bisected adjacent angles is a right angle** ⇒ the bisectors are perpendicular.
Show figure
CAT-style Quick Check
Question. In the configuration above with \( \ell \parallel m \), \(SQ\) bisects \( \angle PQR\) and \(SO\) bisects \( \angle QRO\). What is \( \angle QOS \)?
- (A) \(60^\circ\)
- (B) \(75^\circ\)
- (C) \(90^\circ\)
- (D) \(120^\circ\)
Show answer & explanation (figure inside)
Adjacent interior angles are supplementary: \( \angle PQR+\angle QRO=180^\circ \).
After bisection: \( 2x+2y=180^\circ \Rightarrow x+y=90^\circ \).
In \( \triangle QOS\): \( x+y+\theta=180^\circ \Rightarrow \theta=90^\circ \).
Correct option: (C)
© AzuCATion — “Think CAT, Think AzuCATion.”
