Four Triangles which are Congruent to One Another

             Statement

            Reason

1. PN = $\frac{1}{2}$PQ.

1. N is the midpoint of PQ.

2. LM = $\frac{1}{2}$PQ.

2. By the Midpoint Theorem.

3. PN = LM.

3. From statement 1 and 2.

4. Similarly, PM = NL.

4. Proceeding as above.

5. In ∆PMN and ∆LNM,

(i) PN = LM

(ii) PM = NL

(iii) NM = NM.

5.

(i) From 3.

(ii) From 4.

(iv) Common side.

6. Therefore, ∆PMN ≅ LNM.

6. By SSS criterion of congruency.

7. Similarly, ∆NQL ≅ LNM.

7. Proceeding as above.

8. Also, ∆MLR ≅ LNM.

8. Proceeding as above.

9. Therefore, ∆PMN ≅ LNM ≅ NQL ≅ MLR. (Proved)

9. From statements 6, 7 and 8.