In this word pair analogy, you must select the odd pair that does not show the same kind of relationship among the given options.

Introduction / Context:
This question comes from the odd one out section using word pairs and analogies. Three of the pairs show one common type of relationship, while the remaining pair differs in meaning. Your task is to identify that different pair. These questions test your vocabulary, sense of synonyms and antonyms, and your ability to compare subtle relationships between words.

Given Data / Assumptions:

• Each option consists of two related words separated by a colon.
• Most of the pairs express a synonym or closely related relationship.
• Exactly one pair has a different type of relationship, such as an antonym relationship.
• We must compare the meaning of each word pair carefully.

Concept / Approach:
To solve odd pair questions, we first classify the relation in each pair as synonym, antonym, degree of intensity, part whole or some other semantic relation. Then we see which one does not fit the pattern followed by the others. Here, we will check whether pairs show similar meaning or opposite meaning. The pair that expresses opposite meaning while others show similarity will be the odd one.

Step-by-Step Solution:
Step 1: Consider option B, Happy : Ecstatic. Happy and ecstatic both refer to strong positive emotions, with ecstatic being a more intense form. This is a synonym or degree relationship.Step 2: Consider option C, High : Giant. Both words convey the idea of something large or tall, so they also express similarity, even if they are not perfect synonyms.Step 3: Consider option D, Trail : Pathway. A trail is essentially a small pathway, so these two words are close synonyms as well.Step 4: Consider option A, Sad : Hopeful. Sad expresses a negative emotion, whereas hopeful expresses a positive, optimistic feeling. These two words are contrasting in nature and are closer to antonyms.Step 5: Therefore, while options B, C and D show broadly similar or synonymous relationships, option A shows an opposite or contrasting relationship.

Verification / Alternative check:
We can verify by trying to substitute the words in simple sentences. Saying "He felt sad and hopeful" indicates two opposing emotional states, while sentences like "He was happy and ecstatic," "The mountain was high and giant," or "We followed a trail, a narrow pathway" show similarity or intensification. This confirms that Sad : Hopeful is structurally different from the others.

Why Other Options Are Wrong:
Option B is wrong as an answer because it clearly shows a similar emotional state with different intensity. Option C is wrong because both words are associated with largeness or greatness, making the pair broadly synonymous. Option D is wrong because trail and pathway are nearly interchangeable in everyday language. None of these pairs show the kind of direct contrast that we see with Sad : Hopeful.

Common Pitfalls:
One common mistake is to think that because High and Giant are not perfect synonyms, their pair must be the odd one. However, in analogy questions, we focus on the general type of relationship, not exact dictionary synonyms. Another pitfall is to overlook that Sad and Hopeful describe opposite emotional directions. Always check whether each pair shows similarity or contrast before deciding the odd one out.

Final Answer:
The odd pair is Sad : Hopeful, so option A is correct.