One angle is eight more than three times another angle

One angle is eight more than three times another angle. If the angles are reflexive, what is the measure of each angle?

The correct answer and explanation is:

We are given the following information:

• One angle is eight more than three times another angle.
• Both angles are reflex angles.

Step 1: Understand reflex angles

A reflex angle is any angle greater than 180° and less than 360°.

Step 2: Let the angles be variables

Let the smaller angle be x.
Then the larger angle is 3x + 8.

Step 3: Apply the reflex angle condition

Both x and 3x + 8 must be greater than 180° and less than 360°.

So we write:

1. 180<x<360180 < x < 360
2. 180<3x+8<360180 < 3x + 8 < 360

Now solve the second inequality:

• 3x+8>180⇒3x>172⇒x>1723≈57.333x + 8 > 180 \Rightarrow 3x > 172 \Rightarrow x > \frac{172}{3} \approx 57.33
• 3x+8<360⇒3x<352⇒x<3523≈117.333x + 8 < 360 \Rightarrow 3x < 352 \Rightarrow x < \frac{352}{3} \approx 117.33

So, combining both inequalities:

• 180<x<360180 < x < 360
• 57.33<x<117.3357.33 < x < 117.33

The common valid range is:
180<x<117.33180 < x < 117.33 — this is a contradiction!

This means x cannot be a reflex angle (since it must be greater than 180°), but it must also be less than 117.33, which is impossible.

So the assumption that x is the smaller reflex angle is incorrect. Instead, let’s reverse it.

Step 4: Let the larger angle be x, and the smaller angle be x−83\frac{x – 8}{3}

Now both angles must still be reflex, so:

• 180<x−83<360180 < \frac{x – 8}{3} < 360
• 180<x<360180 < x < 360

Solve the first inequality:

• x−83>180⇒x−8>540⇒x>548\frac{x – 8}{3} > 180 \Rightarrow x – 8 > 540 \Rightarrow x > 548

But 548 is greater than 360, so again, contradiction.

Conclusion:

The assumption that both angles are reflex cannot be true.

Correct interpretation:

Let’s assume only one of the angles is reflex. Try testing values.

Let’s solve the equation:
Let smaller angle = x
Then larger angle = 3x + 8

Try x = 88
Then larger angle = 3×88+8=264+8=272°3×88 + 8 = 264 + 8 = 272°

Now check:

• x = 88° (not reflex)
• 272° is reflex ✅

So, if only one angle is reflex, and 272° > 180°, this satisfies the condition.

✅ Final Answer:

• One angle is 88°
• The other angle is 272°
• Only 272° is reflex

300-Word Explanation:

To solve this problem, we begin with the relationship: one angle is eight more than three times another. Let’s assign variables: let the smaller angle be xx, then the larger angle is 3x+83x + 8.

We are also told the angles are reflex. A reflex angle is any angle greater than 180° and less than 360°. Initially, we try assuming both angles are reflex, but doing so leads to contradictory conditions. For example, if both xx and 3x+83x + 8 must be greater than 180°, then solving the inequalities leads to a result that suggests x>180x > 180 and x<117.33x < 117.33 at the same time, which is impossible.

We then try reversing the variable assignment, making the larger angle xx, and the smaller one x−83\frac{x – 8}{3}. But solving this again leads to contradictions, such as requiring x>548x > 548, which exceeds the maximum limit for an angle (360°).

This tells us our assumption that both angles are reflex must be incorrect. Instead, we test values where only one of the angles is reflex. Trying x=88°x = 88°, the other angle becomes 3×88+8=272°3×88 + 8 = 272°, which is a valid reflex angle.

Thus, the correct and consistent solution is that one angle is 88°, and the other is 272°, satisfying both the algebraic condition and the reflex angle definition. Only the 272° angle is reflex.