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Triangles
EXERCISE 6.2
CBSE_NCERT_TEXTBOOK
Grade 10
Question:
In Fig. 6.20, DE || OQ and DF || OR. Show that EF || QR.
Step-by-Step Solution
Key Concept: Use the concept of similarity of triangles formed by parallel lines. Since DE is parallel to OQ and DF is parallel to OR, the angles of triangle DEF are equal to the corresponding angles of triangle OQR (AA similarity). Hence triangle DEF ∼ triangle OQR, which implies the third pair of corresponding sides are parallel, i.e., EF ∥ QR.
1. Identify the given parallels:\
- DE ∥ OQ \
- DF ∥ OR
2. Compare the angles:\
- Because DE ∥ OQ, the angle formed by DE with DF equals the angle formed by OQ with OR. Hence \[ \angle EDF = \angle QOR \] (corresponding angles).\
- Because DF ∥ OR, the angle formed by DF with DE equals the angle formed by OR with OQ. Hence \[ \angle E D F = \angle R O Q \] (again, corresponding angles).
3. Establish similarity:\
- In triangle OQR, the two angles \(\angle QOR\) and \(\angle ROQ\) are known.\
- In triangle DEF, we have shown that \(\angle EDF = \angle QOR\) and \(\angle E D F = \angle R O Q\).\
- Therefore, by the AA (Angle‑Angle) criterion, \[ \triangle DEF \sim \triangle OQR \].
4. Correspondence of sides:\
- From the similarity, the side opposite \(\angle QOR\) in \(\triangle OQR\) is \(QR\).\
- The side opposite \(\angle EDF\) in \(\triangle DEF\) is \(EF\).\
- Hence the pair of corresponding sides are \(EF\) and \(QR\).
5. Conclude parallelism:\
- Corresponding sides of similar triangles are parallel. Therefore, \[ EF \parallel QR. \]
Thus, the required result is proved.
- DE ∥ OQ \
- DF ∥ OR
2. Compare the angles:\
- Because DE ∥ OQ, the angle formed by DE with DF equals the angle formed by OQ with OR. Hence \[ \angle EDF = \angle QOR \] (corresponding angles).\
- Because DF ∥ OR, the angle formed by DF with DE equals the angle formed by OR with OQ. Hence \[ \angle E D F = \angle R O Q \] (again, corresponding angles).
3. Establish similarity:\
- In triangle OQR, the two angles \(\angle QOR\) and \(\angle ROQ\) are known.\
- In triangle DEF, we have shown that \(\angle EDF = \angle QOR\) and \(\angle E D F = \angle R O Q\).\
- Therefore, by the AA (Angle‑Angle) criterion, \[ \triangle DEF \sim \triangle OQR \].
4. Correspondence of sides:\
- From the similarity, the side opposite \(\angle QOR\) in \(\triangle OQR\) is \(QR\).\
- The side opposite \(\angle EDF\) in \(\triangle DEF\) is \(EF\).\
- Hence the pair of corresponding sides are \(EF\) and \(QR\).
5. Conclude parallelism:\
- Corresponding sides of similar triangles are parallel. Therefore, \[ EF \parallel QR. \]
Thus, the required result is proved.
Correct Answer: EF ∥ QR
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