Answer:

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Step-by-step explanation:

To find the values of x, y, and z in the figure, let's analyze the given information and use the properties of parallel lines and transversals.

Given information:

1. AB || CD || EF

2. PQ and RS are two transversals intersecting each other at a point on line CD.

3. Angle POD = 38°

We need to find the values of x, y, and z.

First, let's identify the angles formed by the parallel lines and transversals:

Angle POD = 38° (Given)

Angle AOP = x° (Vertical angles with angle POD)

Angle BOQ = y° (Vertical angles with angle AOP)

Angle ROQ = z° (Vertical angles with angle BOQ)

Since AB || CD || EF, we can use the properties of alternate interior angles and corresponding angles:

1. Alternate interior angles: When a transversal intersects two parallel lines, the alternate interior angles are equal.

2. Corresponding angles: When a transversal intersects two parallel lines, the corresponding angles are equal.

Using these properties, we can establish the following relationships:

1. Angle POQ = Angle AOP + Angle BOQ = x° + y°

2. Angle QOR = Angle BOQ + Angle ROQ = y° + z°

Now, since PQ and RS are transversals, the angles POQ and QOR add up to 180° (supplementary angles):

POQ + QOR = x° + y° + y° + z° = 180°

Simplifying the equation:

2y + x + z = 180

Also, as the angles on a straight line add up to 180°, we have:

POD + DOC = 180°

Given that Angle POD = 38°, we can find Angle DOC:

38° + Angle DOC = 180°

Angle DOC = 180° - 38°

Angle DOC = 142°

Now, using the properties of corresponding angles, we can find the values of x, y, and z:

1. Angle DOC = Angle BOQ (corresponding angles)

So, 142° = y°

2. Angle BOQ = Angle AOP (corresponding angles)

So, y° = x°

3. Using the equation 2y + x + z = 180:

2(142) + x + z = 180

284 + x + z = 180

x + z = 180 - 284

x + z = -104

Now, since we have two variables (x and z) in one equation, we need additional information to find their specific values. The question does not provide further details, so we cannot determine the exact values of x and z with the given information.

In summary:

x = y = 142°

z is unknown (z = 180 - x - y)

x + z = -104 (without additional information, we cannot determine the exact values of x and z)