Parallelogram proofs, step by step

The parallelogram theorems are where a geometry student first strings the earlier pieces together.

The question

Given PQRS is a parallelogram; The diagonals meet at T

Prove PT ≅ RT

Prove the property

ProvePT ≅ RT

PQRS is a parallelogram—Given

PQ // SR—Definition of Parallelogram

PQ ≅ RS—Opposite sides of a parallelogram are congruent

∠QPT ≅ ∠SRT, ∠PQT ≅ ∠RST—Alternate Interior Angles

△PQT ≅ △RST—ASA

PT ≅ RT—CPCTC

Every proof of a parallelogram property runs the same way: draw a diagonal to make two triangles, use the parallel sides to find equal angles, notice the diagonal is shared, close with a congruence criterion, then read off the part with CPCTC. Here: given PQRS is a parallelogram, The diagonals meet at T, prove PT ≅ RT.
1. PQRS is a parallelogram. This fact is handed to you in the Given — nothing has to justify it.
2. PQ // SR. A parallelogram is DEFINED as a quadrilateral with both pairs of opposite sides parallel — so naming it parallelogram is the same as saying those sides are parallel.
3. PQ ≅ RS. This is the theorem proved on the first sheet of this set, so it can be used here as a known fact rather than re-proved.
4. ∠QPT ≅ ∠SRT, ∠PQT ≅ ∠RST. The two lines are parallel, so a transversal makes alternate interior angles congruent.
5. △PQT ≅ △RST. Two pairs of angles and the side between them are equal: ASA.
6. PT ≅ RT. Corresponding parts of congruent triangles are congruent — once the two triangles are congruent, every matching pair of sides and angles is equal, including pairs nobody gave you.

How it works.

01
Every proof of a parallelogram property runs the same way: draw a diagonal to make two triangles, use the parallel sides to find equal angles, notice the diagonal is shared, close with a congruence criterion, then read off the part with CPCTC. Here: given PQRS is a parallelogram, The diagonals meet at T, prove PT ≅ RT.
02
1. PQRS is a parallelogram. This fact is handed to you in the Given — nothing has to justify it.
03
2. PQ // SR. A parallelogram is DEFINED as a quadrilateral with both pairs of opposite sides parallel — so naming it parallelogram is the same as saying those sides are parallel.
04
3. PQ ≅ RS. This is the theorem proved on the first sheet of this set, so it can be used here as a known fact rather than re-proved.
05
4. ∠QPT ≅ ∠SRT, ∠PQT ≅ ∠RST. The two lines are parallel, so a transversal makes alternate interior angles congruent.
06
5. △PQT ≅ △RST. Two pairs of angles and the side between them are equal: ASA.
07
6. PT ≅ RT. Corresponding parts of congruent triangles are congruent — once the two triangles are congruent, every matching pair of sides and angles is equal, including pairs nobody gave you.