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Chapter 3: Q13. (page 113)

Given: AB¯∥CD¯; BF¯bisects ∠ABE; DG¯bisects ∠CDB.

Prove: BF¯∥DG¯.

Short Answer

Expert verified

It is proved that BF¯∥DG¯.

Step by step solution

01

Step 1. Draw a diagram.

02

Step 2. Description of step.

As per the given information, BF¯bisects ∠ABEand DG¯bisects ∠CDBwhich implies that,

∠ABF=∠FBE∠CDG=∠GDB

03

Step 3. Description of step.

If two parallel lines are bisected by a transversal then the corresponding angles are congruent.

Here,AB¯∥CD¯ andDE¯ is transversal then ∠CDX≅∠ABD.

04

Step 4. Description of step.

Now, ∠CDX+∠CDG+∠GDB=180and ∠ABD+∠ABF+∠FBE=180then from these two equations we get,

∠CDX+∠CDG+∠GDB=∠ABD+∠ABF+∠FBE∠CDX+∠CDG+∠CDG=∠ABD+∠ABF+∠ABF∠CDX+2∠CDG=∠ABD+2∠ABF2∠CDG=2∠ABF∠CDG=∠ABF

05

Step 5. Description of step.

If two parallel lines are bisected by a transversal then the corresponding angles are congruent.

Here, AB¯∥CD¯andGD¯ is transversal then ∠CDG≅∠AYG.

06

Step 6. Description of step.

As ∠CDG≅∠ABFand ∠CDG≅∠AYGimplies ∠ABF≅∠AYG, which are corresponding angles. Therefore, BF¯∥DG¯.

Hence it is proved that BF¯∥DG¯.

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