**Q) I**n the given figure, ∠ CEF = ∠ CFE. F is the midpoint of DC. Prove that .

**Ans:**

**Step 1: From given conditions:**

- ∵ ∠ CEF = ∠ CFE

∴ CE = CF (sides of opp. Angles)

2. ∵ F is midpoint of CD

∴ FD = CF

**∴ ****CE = CF = FD **……… (i)

**Step 2: Construct to find relation:**

Let’s draw DG ǁ BE

∴ by Basic Proportionality Theorem in Δ ABE: ………… (ii)

**Step 3: establish co-relation:**

∴ by Mid Point Theorem in Δ GDC :

∵ DF = FC ……. from equation (i)

∴ = 1 =

**∴ GE = CE**

∵ CE = CF = FD ….. from equation (i)

**∴ ****CE = CF = ****FD**** = ****GE**** **……. (iv)

**Step 4: Proving the relation:**

∵ ……. from equation (ii)

and GE = FD ………… from equation (iv)

**∴ **

**Hence Proved!**