In order to determine which, if any, of the statements is not necessarily true, we will analyze each one individually.
Analyzing A
If we begin by analyzing the first statement, we notice that ∠ a and ∠ d are vertical angles. This means they are congruent by the Vertical Angles Theorem.
The first statement is, therefore, true.
Analyzing B
The second statement tells us that ∠ d and ∠ r are supplementary angles. This angle pair is an example of consecutive interior angles and because the two lines cut by the third line are parallel, we know by the Consecutive Interior Angles Theorem that these angles are supplementary.
Therefore, the second statement is also true.
Analyzing C
The third statement tells us that ∠ u and ∠ n have the same measure. To investigate this statement, we first note that ∠ u and ∠ k are consecutive interior angles and since the two lines cut by the third line are parallel, they are supplementary.
We also notice that ∠ k and ∠ n are vertical angles which means they are congruent by the Vertical Angles Theorem.
Since ∠ u and ∠ k are supplementary angles, ∠ u and ∠ n must be so as well. Therefore, the only way u = n is true is if both of these are right angles. Therefore, the third statement is not necessarily true.
Analyzing D
Let's also investigate the fourth statement. Note that ∠ t and ∠ q are corresponding angles and so are ∠ q and ∠ m. Because the two lines cut by the third line are parallel, we know that all of these angles are congruent by the Corresponding Angles Theorem.
The fourth statement is also true and, therefore, the only statement that is not true is C.
