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If Angle 8 is congruent to angle 10 and Angle 1 is congruent to angle 7, which describes all the lines that must be parallel?

Lines r and s are crossed by lines t and u to form 16 angles. Clockwise from top left, at the intersection of r and t, the angles are 1, 2, 3, 4; at the intersection of s and t, 5, 6, 7, 8; at the intersection of u and s, 9, 10, 11, 12; at the intersection of u and r, 13, 14, 15, 16.
Only lines r and s must be parallel.
Only lines t and u must be parallel.
Lines r and s and lines t and u must be parallel.
Neither lines r and s nor lines t and u must be parallel.

1 answer

To determine which lines must be parallel, we need to analyze the given congruences of the angles.

From the problem statement, we know that:

  • Angle 8 is congruent to angle 10.
  • Angle 1 is congruent to angle 7.

In the configuration provided, we see that:

  • Angle 8 and angle 10 are formed by transversal line u with lines r and s.
  • Angle 1 and angle 7 are formed by transversal line t with lines r and s.

For two lines to be parallel, certain angle relationships must hold:

  • Corresponding angles are congruent.
  • Alternate interior angles are congruent.
  • Alternate exterior angles are congruent.

Given that angle 8 (which is an interior angle at the intersection of line s and line t) is congruent to angle 10 (which is an interior angle at the intersection of line r and line t), this establishes that lines r and s are parallel (because of the alternate interior angles theorem).

Similarly, the congruence of angle 1 (formed by line r and line t) and angle 7 (formed by line s and line t) suggests something about the relationship between the lines r and s as well.

Since we have established the relationships needed:

  • Angles 1 and 7 being congruent implies that lines r and s must be parallel.
  • Angles 8 and 10 being congruent implies that lines r and s must be parallel as well.

Thus, we can deduce that:

Only lines r and s must be parallel.

Lines r and s must be parallel due to the congruence of their respective angles formed by the transversals. The lines t and u do not necessarily have to be parallel based on the information provided. Therefore the final answer is:

Only lines r and s must be parallel.