Adjacent Angles Property: Relationship Between Acute and Obtuse Pairs

Q: What exactly are adjacent angles?

Adjacent angles share a common vertex and side, with no overlap. Think of them as two angles sitting next to each other on a straight line!

Q: Why can't both angles be acute?

If both were acute (less than $90°$), their sum would be less than $180°$. But adjacent angles on a straight line must add to exactly $180°$!

Q: Can both angles be obtuse instead?

No! If both were obtuse (greater than $90°$), their sum would exceed $180°$. This violates the supplementary angle rule for straight lines.

Q: What if one angle is exactly 90°?

The problem states neither angle is a right angle, so we don't consider $90°$ cases. But if we did, the other would also be $90°$!

Q: How do I remember this rule?

Think of a balance: if neither angle is the 'middle' ($90°$), one must tip toward small (acute) and the other toward large (obtuse) to reach $180°$!

Adjacent Angles with Supplementary Constraints

If two adjacent angles are not right angles, then one of them is obtuse and the other is acute.

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Step-by-step video solution

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00:06 Let's figure out if this is true or false.

00:10 We'll draw some adjacent angles.

00:13 Remember, these angles add up to one hundred eighty degrees, which makes a straight line.

00:19 So, if they aren't the same, one angle is acute, and the other is obtuse.

00:26 And that's how we solve this problem!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

If two adjacent angles are not right angles, then one of them is obtuse and the other is acute.

Step-by-step solution

To solve the problem, let’s consider the nature of adjacent angles:

Step 1: Adjacent angles are two angles that share a common side and vertex. If two adjacent angles form a straight line, their measures sum up to $180^\circ$ .
Step 2: According to the problem, neither angle is a right angle, meaning neither is $90^\circ$ .
Step 3: Given this constraint, analyze the possibilities:

If one angle is acute (less than $90^\circ$ ), then the other must be more than $90^\circ$ to make the total $180^\circ$ . Therefore, the other angle is obtuse.
If one angle is obtuse (greater than $90^\circ$ ), then the other must be less than $90^\circ$ to make the total $180^\circ$ . Thus, the other angle is acute.

Since both scenarios involve one angle being acute and the other obtuse, we verify that the statement is correct.

Therefore, the statement is true.

Final Answer

True

Key Points to Remember

Essential concepts to master this topic

Rule: Adjacent angles on a straight line sum to $180°$
Technique: If one angle is $70°$ (acute), the other is $110°$ (obtuse)
Check: Verify acute + obtuse = $180°$ and neither equals $90°$ ✓

Common Mistakes

Avoid these frequent errors

Assuming both angles can be acute or both obtuse
Don't think both angles can be acute like 70° + 80° = 150°! This doesn't add to 180° and violates the straight line rule. Always remember that if neither angle is 90°, one must be less than 90° (acute) and the other greater than 90° (obtuse).

Practice Quiz

Test your knowledge with interactive questions

It is possible for two adjacent angles to be right angles.

FAQ

Everything you need to know about this question

What exactly are adjacent angles?

Adjacent angles share a common vertex and side, with no overlap. Think of them as two angles sitting next to each other on a straight line!

Why can't both angles be acute?

If both were acute (less than $90°$ ), their sum would be less than $180°$ . But adjacent angles on a straight line must add to exactly $180°$ !

Can both angles be obtuse instead?

No! If both were obtuse (greater than $90°$ ), their sum would exceed $180°$ . This violates the supplementary angle rule for straight lines.

What if one angle is exactly 90°?

The problem states neither angle is a right angle, so we don't consider $90°$ cases. But if we did, the other would also be $90°$ !

How do I remember this rule?

Think of a balance: if neither angle is the 'middle' ( $90°$ ), one must tip toward small (acute) and the other toward large (obtuse) to reach $180°$ !

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