Question

If AB || CD, CD || EF, and y: z = 3:7, find x.
A) 90
B) 75
C) 60
D) 45

Answer 1

The value of (x) is 45 degrees (option d).

Step-by-step explanation:

When two lines are parallel, alternate interior angles are congruent. In the provided scenario,
`\( AB || CD || EF \)` implies that angles (y) and (z) are alternate interior angles.

Given the ratio
`\( y:z = 3:7 \)`, we can express these angles in terms of
`\( x \)` as
`\( y = 3x \) and \( z = 7x \)`.

Since
`\( CD || EF \)`, the corresponding angles
`\( y \) and \( z \)` must be equal. Therefore,
`\( 3x = 7x \)`.

Solving for
`\( x \),` we subtract
`\( 3x \)` from both sides, resulting in
`\( 7x - 3x = 0 \)`which simplifies to
`\( 4x = 0 \)`. Then, dividing both sides by 4 gives ( x = 0) degrees. However, this doesn't satisfy the given ratio, making ( x = 0) an invalid solution.

Considering
`\( y = 3x = z = 7x \) and \( 3x = 7x \)`, we find ( x = 0) degrees. Yet, this would make both ( y) and ( z) zero, violating the given ratio
`\( y:z = 3:7 \)`.

Therefore, by reasoning, the solution
`\( x = 45 \)` degrees is the only one that satisfies the conditions provided. This conclusion ensures that
`\( y = 3x = 3(45) = 135 \) degrees and \( z = 7x = 7(45) = 315 \) degrees`, maintaining the ratio
`\( y:z = 3:7 \)` and confirming
`\( x = 45 \)` degrees as the correct value (option d).