# In Fig. 6.38, altitudes AD and CE of Δ ABC intersect each other at the point P. Show that:

(i) ΔAEP ~ ΔCDP

(ii) ΔABD ~ ΔCBE

(iii) ΔAEP ~ ΔADB

(iv) ΔPDC ~ ΔBEC

**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.

In ΔAEP and ΔCDP

∠AEP = ∠CDP = 90

[∵ CE ⊥ AB and AD ⊥ BC; altitudes]

∠APE = ∠CPD (Vertically opposite angles)

⇒ ΔAEP ~ ΔCPD (AA criterion)

(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 ΔABD and ΔCBE

∠ADB = ∠CEB = 90

∠ABD = ∠CBE (Common angle)

⇒ ΔABD ~ ΔCBE (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 ΔAEP and ΔADB

∠AEP = ∠ADB = 90

∠PAE = ∠BAD (Common angle)

⇒ ΔAEP ~ ΔADB

(iv) 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.

In ΔPDC and ΔBEC

∠PDC = ∠BEC = 90

∠PCD = ∠BCE (Common angle)

⇒ ΔPDC ~ ΔBEC

**Video Solution:**

## In Fig. 6.38, altitudes AD and CE of Δ ABC intersect each other at the point P. Show that: (i) ΔAEP ~ ΔCDP (ii) ΔABD ~ ΔCBE (iii) ΔAEP ~ ΔADB (iv) ΔPDC ~ ΔBEC

### NCERT Class 10 Maths Solutions - Chapter 6 Exercise 6.3 Question 7:

In Fig. 6.38, altitudes AD and CE of Δ ABC intersect each other at the point P. Show that: (i) ΔAEP ~ ΔCDP (ii) ΔABD ~ ΔCBE (iii) ΔAEP ~ ΔADB (iv) ΔPDC ~ ΔBEC

In the above figure altitudes, AD and CE of D ABC intersect each other at the point P. Hence proved that triangle AEP is similar to triangle CDP and triangle ABD is similar to triangle CBE and triangle CDP and triangle AEP is similar to triangle ABD and triangle CDP and triangle PDC is similar to triangle BEC.