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In the diagram, BC = CD = BD, angle FDE = 40°, angle EFD = 75°, and li

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Bunuel
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In the diagram, BC = CD = BD, angle FDE = 40°, angle EFD = 75°, and li [#permalink] New post   18 Jun 2019, 00:49
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In the diagram, BC = CD = BD, angle FDE = 40°, angle EFD = 75°, and line BG is parallel to line DF. What is the measure of angle A in degrees?

A. 25
B. 35
C. 45
D. 55
E. 65


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D
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Re: In the diagram, BC = CD = BD, angle FDE = 40°, angle EFD = 75°, and li [#permalink] New post   18 Jun 2019, 01:30
Given info:
1) BC = CD = BD
2) angle FED = 40
3) angle EFD = 75
4) BG || DF


Using (1), angles cbd=bcd=bdc=60
Using (2) and (4), and applying concept of Vertically opposite angles:
angle D made by line segment CD is 40.
Angle between extended DF and BD is 60+40 = 100

Applying sum of interior angles on same transverse is 180
Therefore, upper angle B and D sum to 180
angle B + D = 180
=> B= 180-100 = 80

Now, angle B (made by extended line segment BG and CB):
= 80-60 = 20

Again, applying vertically opposite angles:
angle ABG = 20

From (3) and (4);
angle DFE = BGF = 75

angle BGA = 180 - 75 = 105

therefore, angle BAG = 180 - 105 - 20 = 55

Option D as correct answer
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Re: In the diagram, BC = CD = BD, angle FDE = 40°, angle EFD = 75°, and li [#permalink] New post   18 Jun 2019, 01:44
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Bunuel wrote:

In the diagram, BC = CD = BD, angle FDE = 40°, angle EFD = 75°, and line BG is parallel to line DF. What is the measure of angle A in degrees?

A. 25
B. 35
C. 45
D. 55
E. 65


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Capture.PNG


Find Angle E using sum of angles in a triangle rule = E=65.
angle C=60 as CBD is equilateral triangle (from stem: BC = CD = BD)
So using sum of angles in a triangle rule ACE arrive at angle A 60+65+A=180; A=55 IMO D
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Re: In the diagram, BC = CD = BD, angle FDE = 40°, angle EFD = 75°, and li [#permalink] New post   18 Jun 2019, 02:19
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In this Question, the only property that is to be used is Sum of triangles.

Triangle BCD is an equilateral triangle, hence all the angles are 60. then in triangle, DFE, angle E = 180-75-40, ie., 65
so, triangle AEC, A = 180-65-60
180-125 = 55 ie. D


:) +KUDOS, if you like the answer.
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Re: In the diagram, BC = CD = BD, angle FDE = 40°, angle EFD = 75°, and li [#permalink] New post   18 Jun 2019, 02:58
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Bunuel wrote:

In the diagram, BC = CD = BD, angle FDE = 40°, angle EFD = 75°, and line BG is parallel to line DF. What is the measure of angle A in degrees?

A. 25
B. 35
C. 45
D. 55
E. 65


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Capture.PNG


In Triangle DEF, LE +LF + LD = 180
--> LE + 75 + 40 = 180
--> LE = 65

Triangle BCD is Equilateral --> LC = 60

In Triangle ACE, LA + LC + LE = 180
--> LA + 60 + 65 = 180
--> L = 180 - 125 = 55

IMO Option D

Pls Hit Kudos if you like the solution
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Re: In the diagram, BC = CD = BD, angle FDE = 40°, angle EFD = 75°, and li [#permalink] New post   18 Jun 2019, 10:53
Bunuel wrote:

In the diagram, BC = CD = BD, angle FDE = 40°, angle EFD = 75°, and line BG is parallel to line DF. What is the measure of angle A in degrees?

A. 25
B. 35
C. 45
D. 55
E. 65


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other side angle is 65 and ∆ BCD is equilateral so 60
180-60+65; 55
angle A is 55
IMO D
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Re: In the diagram, BC = CD = BD, angle FDE = 40°, angle EFD = 75°, and li [#permalink] New post   18 Jun 2019, 12:32
Bunuel wrote:

In the diagram, BC = CD = BD, angle FDE = 40°, angle EFD = 75°, and line BG is parallel to line DF. What is the measure of angle A in degrees?

A. 25
B. 35
C. 45
D. 55
E. 65


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Capture.PNG


1. Since in △BCD , BC = CD = BD, its an equilateral Triangle and ∠CBD = ∠BDC = ∠BCD = 60°
2. In △DEF , ∠DEF= 180° - (40° + 75°) => 65°
3. In △ACE, ∠CAE = 180° - (60° + 65°) => 55°

Hence, Answer must be (D)
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Re: In the diagram, BC = CD = BD, angle FDE = 40°, angle EFD = 75°, and li [#permalink] New post   12 Sep 2019, 20:48
Great approach Arvind42
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Re: In the diagram, BC = CD = BD, angle FDE = 40°, angle EFD = 75°, and li [#permalink] New post   31 Aug 2023, 10:20
Since BC = CD = BD, Triangle BCD is an equilateral triangle. Imagine line DF is extended upwards till X and BG is extended upwards till Y.

Since line BG is parallel to line DF,
By using angles in parallel lines property,
Angle BGF = 75=> Angle BGF + Angle BGA = 180 => Angle BGA = 105
Angle XBD + Angle YDB = 180
Angle XBC + 60 + Angle YDC + 60 = 180
Angle XBC + Angle YDC = 60
By using vertically opposite angles, Angle YDC = 40
Therefore, Angle XBC = 60 - 40 = 20
By using vertically opposite angles, Angle ABG = 20

By using angle sum property of a triangle,
Angle ABG + Angle BAG + Angle BGA = 180
20 + Angle BAG + 105 = 180
Angle BAG = 55

Hence D
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Re: In the diagram, BC = CD = BD, angle FDE = 40°, angle EFD = 75°, and li [#permalink]
31 Aug 2023, 10:20
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