道理 is extremely common in daily conversation. It's used to express agreement, disagreement, and to persuade others.
Expressing Agreement: The most common phrase is 有道理 (yǒu dàolǐ), which literally means “has reason.” It's the perfect way to say “That makes sense,” “You've got a point,” or “That's reasonable.”
Expressing Disagreement: The opposite is 没道理 (méi dàolǐ), meaning “has no reason.” Use this to say “That's unreasonable,” “That makes no sense,” or “That's absurd.”
Discussing and Persuading: The verb phrase 讲道理 (jiǎng dàolǐ) means “to reason with someone,” “to talk sense,” or “to be reasonable.” It's often used when you feel someone is being irrational. For example, “我们坐下来讲讲道理” (Let's sit down and talk this through reasonably).
Lecturing (Slightly Negative): The phrase 大道理 (dà dàolǐ), or “big principles,” can refer to profound truths, but it's often used with a slightly negative connotation to describe when someone is lecturing you with abstract, high-sounding platitudes that aren't very practical. For example, “别跟我讲这些大道理” (Don't give me all that preachy talk).