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

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GMATH Teacher
Status: GMATH founder
Joined: 12 Oct 2010
Posts: 935
In the figure shown, AB=AC and CE=CF. What is the value of x?  [#permalink]

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28 Sep 2018, 16:14
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Difficulty:

15% (low)

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 [ 2.66 KiB | Viewed 494 times ]

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Fabio Skilnik :: GMATH method creator (Math for the GMAT)
Our high-level "quant" preparation starts here: https://gmath.net
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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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28 Sep 2018, 19:05
1
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 [ 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: $$(x+∠BDE) = 180°$$
$$(x+60°)=180°$$

$$x=120°$$

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GMATH Teacher
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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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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.

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

Regards,
Fabio.
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Fabio Skilnik :: GMATH method creator (Math for the GMAT)
Our high-level "quant" preparation starts here: https://gmath.net
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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