Question

In the adjoining figure AB = AC and def is an equilateral triangle then ,

Answer 1

Triangle ABC is an equilateral triangle. ⇒ AB = BC = AC DE = 1/2 AB EF = 1/2 BC ⇒ EF = 1/2 AB [Since AB = BC = AC] DF = 1/2 AC ⇒ DF = 1/2 AB [Since AB = BC = AC] DE = EF = DF ∴ ΔDEF is an equilateral triangle.

Answer 2

Given:

AB = AC

DEF is an equilateral triangle

To Find:

The relation between angles a, b, and c

Solution:

(A) a + b + c = 180° is correct.

According to the Mid Point Theorem, the third side of a triangle is parallel to the line segment that joins the mid-points of the other two sides.

Applying it in ΔABC,

DE ║ BC     - 1

DF ║ AC     - 2

EF ║ AB      - 3

From 1, since DE ║ BC and DF acts as the transversal,

∠EDF = ∠DFB = a     (Alternate interior angles)      - 4

Similarly from 2 and 3,

∠DEF = ∠ADE = b                     - 5

∠DFE = ∠FEC = c                      - 6

Adding equations 4, 5, and 6,

∠DEF + EDF + ∠DFE = a + b + c

But DEF is a triangle and according to the angle sum property of a triangle, the sum of all its angles should be 180°

⇒ ∠DEF + EDF + ∠DFE = 180°

or a + b + c = 180°