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If ABCD is a parallelogram, and line PQ cut its sides as shown. Which

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Bunuel
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If ABCD is a parallelogram, and line PQ cut its sides as shown. Which [#permalink] New post   01 Jul 2022, 01:30
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If ABCD is a parallelogram, and line PQ cut its sides as shown. Which of the following must be equal to w?

(A) y + z
(B) x + z
(C) x + y
(D) 180 – z
(E) 180 – y


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Re: If ABCD is a parallelogram, and line PQ cut its sides as shown. Which [#permalink] New post   01 Jul 2022, 06:33
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As \(ABCD \) is a parallelogram, we know that \(\angle D = y^{\circ}\).

Let the vertex by \(x^{\circ} = R\). \(PRD\) is a triangle and thus \(x^{\circ} + y^{\circ} + z^{\circ} = 180^{\circ}\).

Let the vertex by \(w^{\circ} = S\). Triangle \(PSC\) is similar to triangle \(PRD\) given that lines \(BC\) and \(AD\) are parallel, and therefore have the same corresponding angles. This means that \(x^{\circ}+w^{\circ}= 180^{\circ}\).

Substituting \(x^{\circ}+w^{\circ}= 180^{\circ}\) into \(x^{\circ} + y^{\circ} + z^{\circ} = 180^{\circ}\):

\(x^{\circ} + y^{\circ} + z^{\circ} = x^{\circ}+w^{\circ}\)

\( y^{\circ} + z^{\circ} = w^{\circ}\)

Answer A
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Re: If ABCD is a parallelogram, and line PQ cut its sides as shown. Which [#permalink] New post   13 Jul 2022, 10:11
Substituting [m]x^{\circ}+w^{\circ}= 180^{\circ}

How did you get this? could you please help me a bit here
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Re: If ABCD is a parallelogram, and line PQ cut its sides as shown. Which [#permalink] New post   26 Jul 2022, 07:33
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Bunuel wrote:

If ABCD is a parallelogram, and line PQ cut its sides as shown. Which of the following must be equal to w?

(A) y + z
(B) x + z
(C) x + y
(D) 180 – z
(E) 180 – y



Dhruva2000

Since ABCD is a parallelogram and opposite angles in a parallelogram are equivalent, D = y.
We have a triangle with angles x, y, z.
So x+y+z=180.

Since BC is parallel to AD, interior angles x and w sum to 180.
So x+w=180.

x+y+z = x+w
y+z = w
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Re: If ABCD is a parallelogram, and line PQ cut its sides as shown. Which [#permalink] New post   26 Jul 2022, 07:54
(1st) adjacent interior angles in a parallelogram are supplementary —- SUM = 180 degrees

Hence, angle <BCD = 180 - y

(2nd) Angle <PCB and Angle <BCD form a straight line angle.

<PCB + (180 - y) = 180

<PCB = y

(3rd). Angle W is an exterior angle of the triangle at the top right corner with vertices P and C.

The SUM of the 2 remote interior angles of this triangle equal = the exterior angle W

i.e.,
(Angle <PCB) + (Angle at vertex P equal to z) = exterior angle w

(y) + (z) = w

Answer *A*

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Re: If ABCD is a parallelogram, and line PQ cut its sides as shown. Which [#permalink] New post   13 Oct 2023, 21:29
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Re: If ABCD is a parallelogram, and line PQ cut its sides as shown. Which [#permalink]
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