SOLUTION: Geometry Inscribed Angles (Page 2 of 2) - Studypool - Free Printable

Let’s solve problem 14 step by step.

We are told:

- The measure of angle STU is (x² - 6x)°
- The measure of arc SU is (7x + 10)°
- We need to find the measure of angle STU.

First, recall a key circle theorem:
The measure of an inscribed angle is half the measure of its intercepted arc.

In this case, angle STU is an inscribed angle that intercepts arc SU. So:

> m∠STU = (1/2) × m(arc SU)

Now plug in the expressions given:

> x² - 6x = (1/2)(7x + 10)

Multiply both sides by 2 to eliminate the fraction:

> 2(x² - 6x) = 7x + 10
> 2x² - 12x = 7x + 10

Bring all terms to one side to form a quadratic equation:

> 2x² - 12x - 7x - 10 = 0
> 2x² - 19x - 10 = 0

Now solve this quadratic equation.

We can factor it or use the quadratic formula. Let’s try factoring first.

We need two numbers that multiply to (2 × -10) = -20 and add to -19.

Those numbers are -20 and +1.

So rewrite the middle term:

> 2x² - 20x + x - 10 = 0

Group:

> (2x² - 20x) + (x - 10) = 0
> 2x(x - 10) + 1(x - 10) = 0
> (2x + 1)(x - 10) = 0

Set each factor equal to zero:

> 2x + 1 = 0 → x = -1/2
> x - 10 = 0 → x = 10

Now we have two possible values for x: -1/2 and 10.

But we must check which one makes sense in context — because angles and arcs in geometry problems like this are usually positive measures.

Check x = -1/2:

- m∠STU = (-1/2)² - 6(-1/2) = 1/4 + 3 = 3.25° → positive, okay
- m(arc SU) = 7(-1/2) + 10 = -3.5 + 10 = 6.5° → also positive

Wait — but let’s verify if the relationship holds:

Is 3.25 = (1/2)(6.5)? Yes, 3.25 = 3.25 → so mathematically it works.

But now check x = 10:

- m∠STU = (10)² - 6(10) = 100 - 60 = 40°
- m(arc SU) = 7(10) + 10 = 80°
- Is 40 = (1/2)(80)? Yes → 40 = 40 → also works.

So both values satisfy the equation.

However, in most school-level geometry problems, unless otherwise specified, we assume positive integer solutions when possible — especially since arc and angle measures are typically written as whole numbers in diagrams.

Also, looking at the handwritten note on the worksheet, it says “x = -1/2 or x = 10” — so both are accepted mathematically.

But the question asks: find m∠STU

So we should compute the actual angle measure for both cases.

Case 1: x = -1/2 → m∠STU = 3.25°
Case 2: x = 10 → m∠STU = 40°

Which one is intended?

Looking back at the diagram (even though we’re not describing it), in typical textbook problems, they expect the positive integer solution unless stated otherwise.

Moreover, if you look at other problems on the page (like #9–13), all answers are integers.

Also, in problem 14, the arc is labeled as (7x+10)° — if x=10, arc is 80°, angle is 40° — clean numbers.

If x=-1/2, arc is 6.5°, angle is 3.25° — decimals, less likely for this level.

Therefore, although both are mathematically valid, the expected answer is probably 40°.

But wait — let’s double-check the original problem statement.

It says: “find m∠STU”

And gives two expressions. It doesn’t specify to find x — it wants the angle measure.

Since both x values give valid angle measures, but only one is likely intended, and since 40° is a nice number and matches the pattern of other problems, we go with that.

Alternatively, perhaps the problem expects us to list both? But the format suggests a single answer.

Looking again at the student’s work on the page — they wrote “x = -1/2 or x = 10” — meaning they found both roots.

But then didn’t compute the final angle.

The question is to find m∠STU, not x.

So we must plug back in.

But which one?

Actually — there’s another consideration: in circle geometry, an inscribed angle intercepting an arc must be less than 180°, and the arc must be less than 360° — both are satisfied here.

But more importantly — if x = -1/2, then the expression for the angle is (x² - 6x) = 0.25 + 3 = 3.25°, which is fine.

But let’s see what happens if we consider the diagram — even though we’re not supposed to describe it — in standard position, point T is on the circle, S and U are endpoints of the arc — so angle at T should be facing the arc SU.

No issue with either value.

However, in many textbooks, when a quadratic yields two positive solutions, both may be acceptable — but here one is negative x, leading to small decimal angle.

Given that, and since the problem likely expects a whole number, and 40° is clean, I think the intended answer is 40°.

But to be thorough — let’s see if there’s any restriction we missed.

The problem says: “m∠STU = (x² - 6x)°” — if x = -1/2, then x² - 6x = 0.25 + 3 = 3.25 — still positive.

Arc SU = 7*(-0.5)+10 = 6.5 — also positive.

So no violation.

But perhaps in the context of the diagram, the angle looks larger — but we can’t rely on that.

Wait — another thought: sometimes in these problems, if the angle is expressed as a quadratic, and the arc as linear, they expect the solution where x is positive and large enough to make sense visually.

But strictly speaking, both are correct.

However, looking at the student’s own work — they stopped at finding x, didn’t compute the angle.

The question is to find m∠STU.

So we must provide the numerical value.

Perhaps the problem allows two answers? But the format suggests one.

Let me check online or standard practice — actually, in such cases, if both are mathematically valid, and no diagram constraints, both should be accepted.

But since this is a homework sheet, and other answers are integers, and 40 is integer, while 3.25 is not, likely 40 is expected.

Moreover, if you plug x=10 into the arc: 7*10+10=80, half is 40 — perfect.

For x=-0.5: arc=6.5, half is 3.25 — also correct, but messy.

I think for educational purposes, and matching the style of the other problems, we’ll go with 40°.

But to be accurate — the problem might accept both, but since it says "find m∠STU", implying one value, and given the context, 40° is the intended answer.

Final decision: m∠STU = 40°

Let me confirm calculation once more:

x = 10

Angle: x² - 6x = 100 - 60 = 40

Arc: 7x + 10 = 70 + 10 = 80

Half of 80 is 40 — yes.

Perfect.

Final Answer:
40