Question

In a triangle ABC,AB=AC .Point D and E are on the sides BC and AC respectively such that AD=AE .If angle BAD=30 then measure of angle EDC is
a) 100 b) 150 c) 200 d) 250

Answer 1

In ABC
<A =. < C = <1
In ADE
< ADE = <AED [ AD =. AE] = <2
<EDC = <3
<1 +<BAD =<ADC[ ext. angle]
<1+30°=<2+<3
<1=<2+<3-30°....….......................................(1)
< DEC = 180°-<2
in DEC
<1+180°-<2+<3=180 °
<1=<2-<3…......................... (2)
equating (1) and (2)
<2+<3 +30°= <2-<3
2<3=30°
<3=15°
<EDC =15 °

Answer 2

The measure of ∠EDC is 15°.

Option B is correct.

Correct question:

In a triangle ABC,AB = AC . Point D and E are on the sides BC and AC respectively such that AD = AE .If angle BAD = 30° then measure of angle EDC is

a) 10° b) 15° c) 20° d) 25°

Concept used:

Isosceles triangle Property:

If two sides are equal in an isosceles ∆, then the angles opposite to the two sides are also equal.

Angle Sum Property of a Triangle:

If we measure the three angles of a triangle, we find that their sum is equal to two right angles or 180° i.e., for ∆ABC, ∠A+ ∠B + ∠C = 180°.

Exterior Angle of a Triangle:

An exterior angle of a triangle is formed when a side of the triangle is extended.

An exterior angle of a triangle is equal to the sum of the two interior opposite angles.

Given: In ∆ ABC,AB = AC .

Point D and E are on the sides BC and AC such that AD = AE.

∠ BAD = 30°

To find: The measure of ∠EDC.

Solution:

Step 1: Apply Isosceles triangle Property in ∆ABC and in ∆ADE:

AB = AC

⇒∠B = ∠C = x° (Isosceles triangle Property) ……(1)

AD = AE

⇒∠D = ∠E = y° (Isosceles triangle Property) …….(2)

In ∆ADE,

Sum of angles of a ∆ = 180°

∠DAE + ∠ADE + ∠AED = 180°

∠DAC + y° + y° = 180°

[From eq. 2]

∠DAC + 2y° = 180°

∠DAC = 180° − 2y° ………(3)

Step 2: Apply Exterior angle property :

Let ∠EDC  = a°

In ΔDEC, ∠y is an exterior angle

∠DEA = ∠EDC + ∠ECD (Exterior angle property)

y° = a° + x°

a° = y° − x° …….(4)

Step 3: Find ∠EDC:

In ΔABC,

Sum of angles of a triangle = 180°

∠ABC + ∠ACB + ∠BAC = 180°

∠ABC + ∠ACB + (∠BAD + ∠DAC) = 180°

x° + x° + (30° + 180° − 2y°) = 180°

[Given- ∠BAD = 30° and From eq.1 and 3]

2x° − 2y° = 180° - 180°- 30°

2x° − 2y° = - 30°

2(x° - y°) = - 30°

2(y° − x°) = 30°

y° − x° = $\frac{30 \°}{2}$

y° − x° = 15° = a°

[From eq.4]

∠EDC  = a° = 15°

∠EDC = 15°

Hence ∠EDC is 15°. Option B is correct.

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