Parallelogram proofs

The question

Given PQRS is a parallelogram; The diagonals meet at T

Prove PT ≅ RT

Prove the property

How it works.

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

01
The plan: a diagonal makes two triangles, the parallel sides give equal angles, the diagonal is shared, so the triangles are congruent — then CPCTC gives the part you want. Here: given PQRS is a parallelogram, the diagonals meet at T, prove PT ≅ RT.
02
1. PQRS is a parallelogram. It is given.
03
2. PQ // SR. A parallelogram is a quadrilateral with both pairs of opposite sides parallel — that is the definition.
04
3. PQ ≅ RS. This is a theorem already proved, so it can be used here.
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 (CPCTC): once the triangles match, every matching side and angle is equal.