# CD and GH are respectively the bisectors of ∠ACB and ∠EGF such that D and H lie on sides AB and FE of ∆ABC and ∆EFG respectively. If ∆ABC ~ ∆FEG, show that:

(i) CD/GH = AC/FG

(ii) ∆DCB ~ ∆HGE

(iii) ∆DCA ~ ∆HGF

**Solution:**

(i) If two angles of one triangle are respectively equal to two angles of another triangle, then the two triangles are similar.

This is referred to as the AA criterion for two triangles.

∠ACB = ∠FGE

⇒ ∠ACB/2 = ∠FGE/2

⇒ ∠ACD = ∠FGH (CD and GH are bisectors of ∠C and ∠G respectively)

In ∆ADC and ∆FHG

∠DAC = ∠HFG [∆ADC ~ ∆FEG]

∠ACD = ∠FGH

⇒ ∆ADC ~ ∆FHG (AA criterion)

[If two triangles are similar, then their corresponding sides are in the same ratio]

⇒ CD/GH = AG/FG

(ii) If two angles of one triangle are respectively equal to two angles of another triangle, then the two triangles are similar.

This is referred as AA criterion for two triangles.

In ∆DCB and ∆HGE

∠DBC = ∠HEG [∆ABC ~ ∆FEG]

∠DCB = ∠HGE [∵ ∠ACB/2 = ∠FGE/2]

⇒ ∆DCB ~ ∆EHG (AA criterion)

(iii) If two angles of one triangle are respectively equal to two angles of another triangle, then the two triangles are similar.

This is referred as AA criterion for two triangles.

In ∆DCA, ∆HGF

∠DAC = ∠HFG [∆ABC ~ ∆FEG]

∠ACD = ∠FGH [∵ ∠ACB = ∠FGE]

⇒ ∆DCA ~ ∆HGF (AA criterion)

**Video Solution:**

## CD and GH are respectively the bisectors of ∠ACB and ∠EGF such that D and H lie on sides AB and FE of ∆ABC and ∆EFG respectively. If ∆ABC ~ ∆FEG, show that: (i) CD/GH = AC/FG (ii) ∆DCB ~ ∆HGE (iii) ∆DCA ~ ∆HGF

### NCERT Class 10 Maths Solutions - Chapter 6 Exercise 6.3 Question 10:

CD and GH are respectively the bisectors of ∠ACB and ∠EGF such that D and H lie on sides AB and FE of ∆ABC and ∆EFG respectively. If ∆ABC ~ ∆FEG, show that: (i) CD/GH = AC/FG (ii) ∆DCB ~ ∆HGE (iii) ∆DCA ~ ∆HGF

CD and GH are respectively the bisectors of ∠ACB and ∠EGF such that D and H lie on sides AB and FE of ∆ABC and ∆EFG respectively. If ∆ABC ~ ∆FEG. Hence proved that CD/GH = AC/FG and ∆DCB ~ ∆HGE and ∆DCA ~ ∆HGF