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Here we will prove converse of basic proportionality theorem.
The line dividing two sides of a triangle proportionally is parallel to the third side.
Given: In ∆XYZ, P and Q are points on XY and XZ respectively, such that XPPY = XQQZ.
To prove: PQ ∥ YZ
Proof:
Statement |
Reason |
1. XPPY = XQQZ. |
1. Given |
2. PYXP = QZXQ |
2. Taking reciprocals of both sides in statement 1. |
3. PYXP + 1 = QZXQ + 1 ⟹ PY+XPXP = QZ+XQXQ ⟹ XYXP = XZXQ |
3. By adding 1 on both sides of statement 2. |
4. In ∆XYZ and ∆XPQ, (i) XYXP = XZXQ (ii) ∠YXZ = ∠PXQ |
4. (i) From statement 3. (ii) Common angle |
5. Therefore, ∆XYZ ∼ ∆XPQ |
5. By SAS criterion of similarity. |
6. Therefore, ∠XYZ = ∠XPQ |
6. Corresponding angles of similar triangles are equal. |
7. YZ ∥ PQ |
7. Corresponding angles are equal. |
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