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In the figure shown, AC = 2 and BD = DC = 1. What is the measure of

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In the figure shown, AC = 2 and BD = DC = 1. What is the measure of [#permalink]

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2018.OG.05.026.q.png
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In the figure shown, AC = 2 and BD = DC = 1. What is the measure of angle ABD?

A. 15°
B. 20°
C. 30°
D. 40°
E. 45°
[Reveal] Spoiler: OA

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Last edited by Bunuel on 17 Jun 2017, 13:07, edited 1 time in total.
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Re: In the figure shown, AC = 2 and BD = DC = 1. What is the measure of [#permalink]

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AbdurRakib wrote:
Attachment:
2018.OG.05.026.q.png
A. 15°
B. 20°
C. 30°
D. 40°
E. 45°


Given BD = DC = 1

AC = 2.

AC = AD + DC => 2 = AD - 1

Therefore AD = 1

\(\triangle\) ABD and \(\triangle\) BDC are isosceles triangles.

\(\triangle\) ABD is isosceles, sides AD = BD = 1. Therefore \(\angle\) BAD = \(\angle\) ABD. Let this angle be x.

Given \(\angle\) ADB is \(120^{\circ}\)

\(\angle\) ADB + \(\angle\) BAD + \(\angle\) ABD = 180

\(120 + x + x = 180\)

\(2x = 180 - 120 = 60\)

\(x = \frac{60}{2} = 30^{\circ}\). Answer (C)...

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Re: In the figure shown, AC = 2 and BD = DC = 1. What is the measure of [#permalink]

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AbdurRakib wrote:
Attachment:
2018.OG.05.026.q.png

In the figure shown, AC = 2 and BD = DC = 1. What is the measure of angle ABD?

A. 15°
B. 20°
C. 30°
D. 40°
E. 45°


If AC = 2 and DC = 1, then we can conclude that AD = 1
Add this information to the diagram...
Image

If AD = 1 and BD = 1, then ∆ABD is an ISOSCELES triangle, which means ∠DAB = ∠ABD are EQUAL...
Image

If we let x = BOTH ∠DAB and ∠ABD, then we can use the fact that angles in a triangle add to 180º
We can write: x + x + 120 = 180
Simplify: 2x + 120 = 180
Solve: x = 30

In other words, ∠ABD = 30º

Answer: C

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Re: In the figure shown, AC = 2 and BD = DC = 1. What is the measure of [#permalink]

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Re: In the figure shown, AC = 2 and BD = DC = 1. What is the measure of [#permalink]

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New post 04 Jul 2017, 01:55
Angle BDC is 60 degree(because the angle in a straight line is equal to 180 degree)
Since BD = DC(as given in the question stem) the triangle must be equilateral.
Therefere DC=1.

Since AC=2, AD = AC - DC = 1
Angle ABD = Angle DAB = x(Angles opposite equal sides are equal)


We know that the sum of angles in a triangle is 180 degree,
2x + 120 = 180
Angle ABD(x) = 30 degree(Option C)
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Re: In the figure shown, AC = 2 and BD = DC = 1. What is the measure of [#permalink]

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New post 04 Jul 2017, 09:22
AC-DC=AD
AD = 1, this makes triangle ADC an isosceles triangle

Sum of interior opposite angles = exterior angle.
\(\angle BAD + \angle ABD = \angle BDC\)
Here these two angles are equal, lets say x.

Since AC is a straight line, \(\angle BDC = 180^ {\circ} - 120^ {\circ} = 60^ {\circ}\)

Hence \(\angle BAD + \angle ABD = 60^ {\circ}\)
x+x=60
x=30

Answer is C
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Re: In the figure shown, AC = 2 and BD = DC = 1. What is the measure of [#permalink]

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New post 15 Nov 2017, 15:46
AbdurRakib wrote:
Attachment:
2018.OG.05.026.q.png

In the figure shown, AC = 2 and BD = DC = 1. What is the measure of angle ABD?

A. 15°
B. 20°
C. 30°
D. 40°
E. 45°


Since AC = 2 and DC = 1, AD must be 1. Since BD = 1, this makes triangle ABD an isosceles triangle with angle D as the vertex angle and angles A and ABD as the base angles. We know that angles A and ABD are equal because the sides opposite them are equal. If we let each of the base angles = x, we can create the following equation:

x + x + 120 = 180

2x = 60

x = 30

Answer: C
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Re: In the figure shown, AC = 2 and BD = DC = 1. What is the measure of   [#permalink] 15 Nov 2017, 15:46
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