In the figure shown, lines k and m are parallel to each other. Is x = z

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Target question: Is x = z ?
Before we examine the statements, let's first recognize that, since k||m, we can conclude that angles w and z are EQUAL


Okay, now onto the statements

Statement 1: x = w
We already learned that w = z
Statement 1 tells us that w = x
Since x and z are both equal to w, it must also be true that x = z
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: y = 180 - w
We can also REPHRASE this as y + w = 180
Since w = z, we can also replace w with z in the above equation to get: y + z = 180
In the diagram, we can see that since angles y and x are on the same line, then y + x = 180
Since both equations are set equal to 180, we can also write: y + z = y + x
From here, we can subtract y from both sides to get z = x
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Answer: D

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1) statement one tells us that X=W..
but W=Z since both are alternate angles between two llel lines.. suff

2) y=180-w... also y=180-x as y and x are supplementary angles.. so w=x , same info as statement 1... suff
D Ans
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I made a mistake by discarding second statement since it did not provide any new information. Duh.
Answer is D

Guys can you tell me where can learn the angles stuff?

Given : Figure in which k || m
DS : x = z ?
From figure, y = 180 -x

Option 1 : x = w
SO sum of all angles of a quadrilateral = 360
-> y + w +(180 -z) + x(alternate angle) = 360
-> 180 -x + x + 180 - z + x = 360
-> x = z
SUFFICIENT

Option 2 : Y = 180 - w = 180 - x
-> x = w which is smae as option 1
SUFFICIENT

Answer D

The Diagram really confused me, whether "W" was alternate angle of X or Z.
How do I determine this in exams, in such cases ? do i have to see the degree symbol, which was on right side
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alternate angles.

Just derive algebraically that x= w in either statement to determine that each is sufficient.

(1) x = w

From the figure we can conclude w = z, as we're told lines k and m are parallel to each other (alternate angles are equal). From statement 1 we're told x = w. Thus x = w = z. SUFFICIENT.

(2) y = 180 - w

This statement tells us x = w. Since w = z, z = x. SUFFICIENT.

Answer is D.
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(1) As the figure is a parallelogram; \(w=z\); as they are alternate angels, Given \(x=w, \ so \ x=z\); Sufficient.

(2) As \(y=180-w\), \(x=180-180+w=w\), now for the condition described in option 1 \(x=z\); Sufficient.

The answer is \(D\)

St1 - x = w
w = z (alternating angles)

Sufficient.

St2 - y = 180 - w

180 - w = 180 - x (by rule of supplementary angles)

x = w

Sufficient.

D.
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Solution:

Question Stem Analysis:

We need to determine whether x = z. We see that x + y = 180. Furthermore, we are given that lines k and m are parallel, so w = z because they are alternate interior angles.

Statement One Alone:

Since x = w and w = z, then x = z. Statement one alone is sufficient.

Statement Two Alone:

Since y = 180 - w and w = z, then y = 180 - z or z + y = 180. Since x + y = 180 and z + y = 180, we see that x = z. Statement two alone is sufficient.

Answer: D
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When a line transverses two parallel line we end up with a couple of special properties. One of which is that the corresponding acute and obtuse angles are equal to each other. So from analyzing the picture in the stem we can conclude right off the bat that W=Z.

Statement 1:
If X = W, then X = W = Z, so X must be equal to Z

Statement 2:
Given that the angles X and Y are supplementary, we can conclude that X+Y = 180, and furthermore with the information provided in statement two we can say that X + 180 - W = 180, so X = W. Since in the stem we concluded that W=Z then we can now conclude that X = W = Z, so it must be the case that X = Z.

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