Bisectors of the Angles of a Parallelogram form a Rectangle

Here we will prove that the bisectors of the angles of a parallelogram form a rectangle.

Given: PQRS is a parallelogram in which PQ ∥ SR and SP ∥ RQ. The bisectors of ∠P, ∠Q, ∠R and ∠S are PJ, QK, RL and SM respectively which enclose the quadrilateral JKLM.

To prove: JKLM is a rectangle.

Proof:

Statement	Reason
1. ∠QPS + ∠PSR = 180° Therefore, $\frac{1}{2}$∠QPS + $\frac{1}{2}$∠PSR = 90°	1. PQ ∥ SR.
2. ∠SPM + ∠PSM = 90°	2. PJ and SM are bisectors of ∠QPS and ∠PSR respectively.
3. ∠PMS = 90° ⟹ JM ⊥ ML.	3. Sum of the three angles of ∆PSM is 180°.
4. Taking bisectors of ∠S and ∠R, ML ⊥ LK; Taking bisectors of ∠R and ∠Q, KL ⊥ JK; Taking bisectors of ∠Q and ∠P, JK ⊥ JM.	4. Similarly.
5. JK ∥ ML, JM ∥ KL.	5. Two lines perpendicular to the same line are parallel.
6. JKLM is a parallelogram. (Proved).	6. By statement 5 and one angle, say ∠JML = 90°.

9th Grade Math

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Statement	Reason
1. ∠QPS + ∠PSR = 180° Therefore, \(\frac{1}{2}\)∠QPS + \(\frac{1}{2}\)∠PSR = 90°	1. PQ ∥ SR.
2. ∠SPM + ∠PSM = 90°	2. PJ and SM are bisectors of ∠QPS and ∠PSR respectively.
3. ∠PMS = 90° ⟹ JM ⊥ ML.	3. Sum of the three angles of ∆PSM is 180°.
4. Taking bisectors of ∠S and ∠R, ML ⊥ LK; Taking bisectors of ∠R and ∠Q, KL ⊥ JK; Taking bisectors of ∠Q and ∠P, JK ⊥ JM.	4. Similarly.
5. JK ∥ ML, JM ∥ KL.	5. Two lines perpendicular to the same line are parallel.
6. JKLM is a parallelogram. (Proved).	6. By statement 5 and one angle, say ∠JML = 90°.

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