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In the figure shown, AB=AC and CE=CF. What is the value of x?

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In the figure shown, AB=AC and CE=CF. What is the value of x?  [#permalink]

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New post 28 Sep 2018, 16:14
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Question Stats:

83% (01:56) correct 17% (01:43) wrong based on 43 sessions

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In the figure shown, AB=AC and CE=CF. What is the value of x?

(A) 90
(B) 110
(C) 120
(D) 130
(E) 140


Source: http://www.GMATH.net
(Exercise 12 of the Quant Class #02 included in our free test drive!)

Attachment:
GMATH_figure028.gif
GMATH_figure028.gif [ 2.66 KiB | Viewed 494 times ]

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Re: In the figure shown, AB=AC and CE=CF. What is the value of x?  [#permalink]

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New post 28 Sep 2018, 19:05
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fskilnik wrote:
In the figure shown, AB=AC and CE=CF. What is the value of x?

(A) 90
(B) 110
(C) 120
(D) 130
(E) 140



Source: http://www.GMATH.net
(Exercise 12 of the Quant Class #02 included in our free test drive!)

Attachment:
GMATH_figure028EDIT (2).jpg
GMATH_figure028EDIT (2).jpg [ 19.86 KiB | Viewed 507 times ]

We need only four properties:
• Angles opposite congruent sides are congruent
• Vertical angles are congruent
• Angles that are colinear sum to 180°
• The sum of interior angles of a triangle = 180°

Given: ∠ F = 40°
∠ F = ∠ CEF = 40° (angles opposite congruent sides are congruent)

Sum of the two congruent angles
(40° + 40°) = 80°

In ∆ CEF, the third angle,
∠ECF must = (180°- 80°) = 100°

∠ ECF and ∠ ACB, are colinear
∠ ACB must = (180° - 100°) = 80°

∠ACB = ∠ABC = 80° (opposite sides AB = AC)

Vertical angles: ∠CEF = ∠BED
From above ∠CEF = 40° = ∠BED

The two known angles of ∆BDE sum to (40° + 80°) = 120°
Third angle BDE = (180° - 120°) = 60°

Finally, because the angles are colinear: \((x+∠BDE) = 180°\)
\((x+60°)=180°\)

\(x=120°\)

Answer C
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Re: In the figure shown, AB=AC and CE=CF. What is the value of x?  [#permalink]

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New post 30 Sep 2018, 08:49
fskilnik wrote:
In the figure shown, AB=AC and CE=CF. What is the value of x?

(A) 90
(B) 110
(C) 120
(D) 130
(E) 140

Dear generis,

Thank you for your excellent (and very detailed) contribution. Kudos!

The "alternate" way, presented in the figures below, has (I believe) one merit:

It takes into account the "exterior angle property", avoiding some intermediate calculations.


Image


This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.
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Re: In the figure shown, AB=AC and CE=CF. What is the value of x?   [#permalink] 30 Sep 2018, 08:49
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