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If AB = BC, DA = AC, and angles DCB and ABC are right angles as shown

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If AB = BC, DA = AC, and angles DCB and ABC are right angles as shown  [#permalink]

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New post 08 Jul 2020, 03:18
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If AB = BC, DA = AC, and angles DCB and ABC are right angles as shown above, then what is the value of xº?

A. 30º
B. 45º
C. 60º
D. 90º
E. 105º



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Re: If AB = BC, DA = AC, and angles DCB and ABC are right angles as shown  [#permalink]

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New post 08 Jul 2020, 03:23
1
AB = BC and B = 90°
ABC is a 90°,45°,45° right angle triangle.
Similarly Angle C is a 90° angle, consisting of two angles BCA and ACD. 45° and 45° each

AC=AD so ACD = ADC
Hence angle A is 90° (180-45-45)

Answer is D

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Re: If AB = BC, DA = AC, and angles DCB and ABC are right angles as shown  [#permalink]

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New post 08 Jul 2020, 03:28
Bunuel wrote:
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If AB = BC, DA = AC, and angles DCB and ABC are right angles as shown above, then what is the value of xº?

A. 30º
B. 45º
C. 60º
D. 90º
E. 105º


BAC = BCA (Angles opp to equal sides are equal)
BAC + BCA + 90 = 180
=> BCA = 45º = BAC

ACD = 90º - BCA = 45º
ACD = ADC = 45º (Angles opp to equal sides are equal)
x + ACD + ADC = 180º
=> X = 180º - 90º = 90º

Hence, OA is (D)
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Re: If AB = BC, DA = AC, and angles DCB and ABC are right angles as shown  [#permalink]

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New post 08 Jul 2020, 03:33
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Bunuel wrote:
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If AB = BC, DA = AC, and angles DCB and ABC are right angles as shown above, then what is the value of xº?

A. 30º
B. 45º
C. 60º
D. 90º
E. 105º



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Angle D = Angle ACD \(= \frac{(180-x)}{2}\)

Angle DAB = x+45º

Sum of all angles of Trapezium ABCD = 360º

i.e. Angle D + Angle DAB = 360-(90+90) = 180

i.e. \(\frac{(180-x)}{2}\) + \(x+45º\) \(= 180\)

i.e. 180-x+2x = 2*135º

i.e. x = 90º

Answer: Option D
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Re: If AB = BC, DA = AC, and angles DCB and ABC are right angles as shown  [#permalink]

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New post 08 Jul 2020, 03:39
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IMO D.

AC bisects 90 angle .
Therefore in triangle ACD angle C and angle D are same (45) (because of equal sides)

x + 45 + 45 =180.

x is 90.

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Re: If AB = BC, DA = AC, and angles DCB and ABC are right angles as shown  [#permalink]

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New post 08 Jul 2020, 03:57
If AB = BC, DA = AC, and angles DCB and ABC are right angles as shown above, then what is the value of xº?

A. 30º
B. 45º
C. 60º
D. 90º
E. 105º

ABC is a isosceles right angle triangle so Angle B = 90. Angle BCA and BAC= 45.
Angle DCB is 90 (given)
But DCB = DCA + BCA = 90= 45+ DCA….
so DCA = 45 and hence angle ADC = 45.
Using the sum of the angles of a triangle we get
ADC + DAC + ACD = 180
180 = 2 (45) = DAC = 180-90= 90

OA D
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Re: If AB = BC, DA = AC, and angles DCB and ABC are right angles as shown  [#permalink]

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New post 08 Jul 2020, 19:27
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Bunuel wrote:
Image
If AB = BC, DA = AC, and angles DCB and ABC are right angles as shown above, then what is the value of xº?

A. 30º
B. 45º
C. 60º
D. 90º
E. 105º



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As Triangle ABC is an isosceles right angle triangle - Angle ACB = 45 degrees.
And given that Angle C - 90 degrees
Angle ACD will be 90 - 45 = 45 degrees

And as Triangle ACD is also an isosceles triangle and AC = AD
Angle ACD = Angle ADC = 45 degrees.

Hence Angle x = 180-45-45 = 90 degrees.

OA: D
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Re: If AB = BC, DA = AC, and angles DCB and ABC are right angles as shown   [#permalink] 08 Jul 2020, 19:27

If AB = BC, DA = AC, and angles DCB and ABC are right angles as shown

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