Geometry Worksheets | Angles Worksheets - Free Printable

Let's solve each problem step by step based on the image you provided. This worksheet focuses on inscribed angles in circles and their relationships with intercepted arcs.

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🔷 Part 1: Decide if each angle is an inscribed angle. If it is, name the angle and intercepted arc.

An inscribed angle is an angle whose vertex lies on the circle, and its sides are chords of the circle. The intercepted arc is the arc that lies between the two points where the angle's sides intersect the circle.

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#### 1)
- Triangle U XV, with angle at V.
- Vertex V is on the circle; sides UV and VX are chords → this is an inscribed angle.
- Angle: ∠UVX
- Intercepted arc: arc UX

✔ Yes, inscribed angle: ∠UVX, intercepted arc: $\overset{\frown}{UX}$

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#### 2)
- Points S, G, R. Angle at G.
- Vertex G is on the circle; sides GS and GR are chords → inscribed angle.
- Angle: ∠SGR
- Intercepted arc: arc SR

✔ Yes, inscribed angle: ∠SGR, intercepted arc: $\overset{\frown}{SR}$

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#### 3)
- Triangle NMO, angle at M.
- Vertex M is on the circle; NM and MO are chords → inscribed angle.
- Angle: ∠NMO
- Intercepted arc: arc NO

✔ Yes, inscribed angle: ∠NMO, intercepted arc: $\overset{\frown}{NO}$

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#### 4)
- Points R, S, T. Angle at S.
- Vertex S is on the circle; RS and ST are chords → inscribed angle.
- Angle: ∠RST
- Intercepted arc: arc RT

✔ Yes, inscribed angle: ∠RST, intercepted arc: $\overset{\frown}{RT}$

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✔ Summary for Part 1:

| # | Inscribed? | Angle | Intercepted Arc |
|---|------------|--------|------------------|
| 1 | Yes | ∠UVX | $\overset{\frown}{UX}$ |
| 2 | Yes | ∠SGR | $\overset{\frown}{SR}$ |
| 3 | Yes | ∠NMO | $\overset{\frown}{NO}$ |
| 4 | Yes | ∠RST | $\overset{\frown}{RT}$ |

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🔷 Part 2: Find the measure of the indicated angle or arc.

Use the Inscribed Angle Theorem:
> The measure of an inscribed angle is half the measure of its intercepted arc.

So:
$$
\text{m}∠ = \frac{1}{2} \times \text{m(arc)}
$$

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#### 5)
Given: $ m∠CDE = 75^\circ $, find $ \overset{\frown}{CE} $

- ∠CDE is an inscribed angle intercepting arc CE.
- So:
$$
m∠CDE = \frac{1}{2} \times m\overset{\frown}{CE}
$$
$$
75^\circ = \frac{1}{2} \times m\overset{\frown}{CE}
$$
$$
m\overset{\frown}{CE} = 2 \times 75^\circ = \boxed{150^\circ}
$$

✔ $ \overset{\frown}{CE} = 150^\circ $

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#### 6)
Given: $ m\overset{\frown}{SU} = 175^\circ $, find $ m∠STU $

- ∠STU is an inscribed angle intercepting arc SU.
- So:
$$
m∠STU = \frac{1}{2} \times m\overset{\frown}{SU} = \frac{1}{2} \times 175^\circ = \boxed{87.5^\circ}
$$

✔ $ m∠STU = 87.5^\circ $

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#### 7)
Given: $ m∠TUX = 89^\circ $, $ m\overset{\frown}{UX} = 127^\circ $, find $ m\overset{\frown}{UT} $

Wait — let’s check consistency.

We are told:
- ∠TUX = 89° → this is an inscribed angle.
- It intercepts arc TX (not UX). But we’re given $ \overset{\frown}{UX} = 127^\circ $.

Wait — look carefully: point U, T, X, V.

∠TUX has vertex at U, so it intercepts arc TX.

But we're given $ \overset{\frown}{UX} = 127^\circ $. That’s not the intercepted arc.

Wait — maybe there's a typo?

Wait — no. Let’s re-analyze.

Points: U, T, V, X on circle. Quadrilateral UTVX? Or just triangle?

Angle ∠TUX → vertex at U, sides UT and UX → so it intercepts arc TX.

But we’re given $ \overset{\frown}{UX} = 127^\circ $? That doesn’t make sense.

Wait — the diagram shows:

- Arc UX is labeled as 127°.
- But ∠TUX is at U, so it should intercept arc TX, not UX.

Unless... is the angle ∠TUX intercepting arc TX?

Yes. So why is $ \overset{\frown}{UX} $ given?

Wait — perhaps I misread.

Look again:

> $ m∠TUX = 89^\circ $, $ \overset{\frown}{UX} = 127^\circ $, find $ \overset{\frown}{UT} = ? $

Ah! Maybe ∠TUX is not an inscribed angle? Wait — vertex at U, which is on the circle → yes, it is.

But ∠TUX intercepts arc TX, so:

$$
m∠TUX = \frac{1}{2} \times m\overset{\frown}{TX}
\Rightarrow 89^\circ = \frac{1}{2} \times m\overset{\frown}{TX}
\Rightarrow m\overset{\frown}{TX} = 178^\circ
$$

Now, total circle is 360°.

We know:
- $ \overset{\frown}{UX} = 127^\circ $
- $ \overset{\frown}{TX} = 178^\circ $

But wait — arc TX includes arc TU and arc UX? Not necessarily.

Let’s think about the order of points.

Assume points are placed in order: U, T, V, X around the circle.

Then arc TX goes from T to X passing through V.

Arc UX goes from U to X — possibly the short way?

But we have both $ \overset{\frown}{UX} = 127^\circ $ and $ \overset{\frown}{TX} = 178^\circ $

But arc TX and arc UX may overlap or be adjacent.

Wait — better approach:

From above:
- $ m\overset{\frown}{TX} = 178^\circ $
- $ m\overset{\frown}{UX} = 127^\circ $

But these two arcs share point X, but not necessarily overlapping.

Wait — unless the points are arranged as U, T, X, V or something.

Alternatively, perhaps the arc $ \overset{\frown}{UX} $ is not the same as the one involved.

Wait — maybe the angle ∠TUX is at U, so it intercepts arc TX, so:

$$
m∠TUX = \frac{1}{2} \times m\overset{\frown}{TX} = 89^\circ \Rightarrow m\overset{\frown}{TX} = 178^\circ
$$

Now, we are also given $ m\overset{\frown}{UX} = 127^\circ $

We need to find $ m\overset{\frown}{UT} $

If we assume the points go around the circle in order: U → T → V → X → U

Then:
- Arc UX can be split into: $ \overset{\frown}{UT} + \overset{\frown}{TX} $?

No — that would only work if going from U to X via T.

But arc UX could be the minor arc directly from U to X.

But we are told $ \overset{\frown}{UX} = 127^\circ $, and $ \overset{\frown}{TX} = 178^\circ $

But if $ \overset{\frown}{TX} = 178^\circ $, and $ \overset{\frown}{UX} = 127^\circ $, they can't both be minor arcs unless the circle is large.

Wait — contradiction?

Wait — maybe $ \overset{\frown}{UX} $ is not the arc from U to X directly, but rather a different arc?

Wait — perhaps the notation means the arc from U to X not passing through T?

But then how do we relate?

Alternative idea:

Maybe the angle ∠TUX is not intercepting arc TX?

Wait — let’s draw it mentally.

- Point U: vertex of angle
- Rays: UT and UX
- So the angle "opens" toward the interior of the circle, intercepting arc TX (the arc opposite to U)

Yes — so intercepted arc is TX.

So $ m\overset{\frown}{TX} = 2 \times 89^\circ = 178^\circ $

Now, we are given $ \overset{\frown}{UX} = 127^\circ $

But arc UX is from U to X — which might be the other way?

So the full circle is:

Total circumference = 360°

Suppose arc TX = 178°, arc UX = 127° — but these arcs likely overlap or are parts of the same path?

Wait — perhaps arc UX is not the same as arc from U to X via T.

Let’s suppose the points are ordered: U, T, V, X, back to U.

Then:
- Arc UX (minor) = 127° — direct path
- Arc TX = 178° — from T to X via V

But arc UX and arc TX share point X, but not necessarily connected.

But we need to find arc UT.

Let’s suppose:

From U to T: arc UT = ?

From T to X: arc TX = 178°

From X to U: arc XU = 127°? But that would make total circle: UT + TX + XU = ?

But that’s not right — unless the points are in order U, T, X, U.

Wait — if arc TX = 178°, and arc UX = 127°, but both start at X?

Wait — no. Let’s clarify:

We have:
- $ m\overset{\frown}{UX} = 127^\circ $ — arc from U to X
- $ m\overset{\frown}{TX} = 178^\circ $ — arc from T to X

But we don’t know the relative positions.

Wait — perhaps the arc $ \overset{\frown}{UX} $ is part of the arc from U to X via T?

Wait — no.

Another possibility: maybe the angle ∠TUX is not intercepting arc TX?

Wait — standard rule: inscribed angle at U with rays to T and X intercepts arc TX (the arc not containing U).

So yes, $ m\overset{\frown}{TX} = 2 \times 89^\circ = 178^\circ $

Now, we are asked to find $ m\overset{\frown}{UT} $

We are also given $ m\overset{\frown}{UX} = 127^\circ $

But arc UX is from U to X — which might be the other arc.

Wait — perhaps the arc $ \overset{\frown}{UX} $ is the minor arc, and $ \overset{\frown}{TX} $ is another arc.

But unless we know how the points are arranged, we can’t proceed.

Wait — perhaps there's a mistake in interpretation.

Let me re-read:

> $ m∠TUX = 89^\circ $, $ \overset{\frown}{UX} = 127^\circ $, find $ \overset{\frown}{UT} $

Wait — could $ \overset{\frown}{UX} $ be the arc from U to X passing through T?

Then $ \overset{\frown}{UX} = \overset{\frown}{UT} + \overset{\frown}{TX} $

But we already found $ \overset{\frown}{TX} = 178^\circ $, and $ \overset{\frown}{UX} = 127^\circ $ — but 178 > 127, impossible.

So $ \overset{\frown}{UX} $ cannot include $ \overset{\frown}{TX} $.

Therefore, arc UX must be the other arc from U to X, not passing through T.

So the total circle is:

- Arc UX (minor) = 127°
- Arc TX = 178°
- But arc TX and arc UX may be on different sides.

Wait — but we need to find arc UT.

Perhaps the points are in order: U, T, X, V, U.

Then arc from U to T: $ \overset{\frown}{UT} $

Arc from T to X: $ \overset{\frown}{TX} = 178^\circ $

Arc from X to U: $ \overset{\frown}{XU} = 127^\circ $

Then total circle:
$$
\overset{\frown}{UT} + \overset{\frown}{TX} + \overset{\frown}{XU} = 360^\circ
\Rightarrow \overset{\frown}{UT} + 178^\circ + 127^\circ = 360^\circ
\Rightarrow \overset{\frown}{UT} = 360 - 305 = \boxed{55^\circ}
$$

✔ So $ m\overset{\frown}{UT} = 55^\circ $

This makes sense.

But is this valid? Only if arc UX is from X to U directly, and arc TX is from T to X via V.

But in that case, arc UX (from U to X) would be the long way around, but we’re told $ \overset{\frown}{UX} = 127^\circ $, which is less than 180°, so it must be the minor arc.

But if arc TX = 178°, and arc UX = 127°, and they are on different paths, then the only way is:

- From U to T: ?
- From T to X: 178°
- From X to U: 127°

But then arc from U to X via T would be $ \overset{\frown}{UT} + 178^\circ $, and arc from U to X directly is 127°.

But the total circle is $ \overset{\frown}{UT} + 178^\circ + 127^\circ = 360^\circ \Rightarrow \overset{\frown}{UT} = 55^\circ $

And arc UX (direct) = 127°

So yes, consistent.

Thus:

✔ $ \overset{\frown}{UT} = \boxed{55^\circ} $

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🔷 Part 3: Solve for x

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#### 8)
Given: $ m∠GHK = 52.5^\circ $, $ \overset{\frown}{GK} = 10x + 5^\circ $

∠GHK is inscribed, intercepting arc GK.

So:
$$
m∠GHK = \frac{1}{2} \times m\overset{\frown}{GK}
\Rightarrow 52.5^\circ = \frac{1}{2} \times (10x + 5)
$$

Multiply both sides by 2:
$$
105 = 10x + 5
\Rightarrow 10x = 100
\Rightarrow x = \boxed{10}
$$

✔ $ x = 10 $

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#### 9)
Given: $ m∠FGH = 70^\circ $, $ \overset{\frown}{GF} = 135^\circ $, $ \overset{\frown}{GH} = 10x + 5^\circ $

∠FGH is inscribed, vertex at G, sides GF and GH → intercepts arc FH

Wait — arc FH?

But we are given $ \overset{\frown}{GF} $ and $ \overset{\frown}{GH} $

Wait — the angle at G, between F and H → so it intercepts arc FH

So:
$$
m∠FGH = \frac{1}{2} \times m\overset{\frown}{FH}
\Rightarrow 70^\circ = \frac{1}{2} \times m\overset{\frown}{FH}
\Rightarrow m\overset{\frown}{FH} = 140^\circ
$$

Now, arc FH is composed of arc FG and arc GH?

Wait — depending on order.

Assume points: F, G, H on circle.

Then arc FH = arc FG + arc GH

But we are given:
- $ \overset{\frown}{GF} = 135^\circ $ — that’s from G to F
- $ \overset{\frown}{GH} = 10x + 5^\circ $

But arc GF = 135° — so arc FG = 135° (same measure)

Then arc FH = arc FG + arc GH = 135° + (10x + 5) = 140°

Wait — but earlier we found $ m\overset{\frown}{FH} = 140^\circ $

So:
$$
135 + (10x + 5) = 140
\Rightarrow 140 + 10x = 140
\Rightarrow 10x = 0
\Rightarrow x = \boxed{0}
$$

But let’s double-check.

Wait — arc FH = arc FG + arc GH?

Only if G is between F and H.

But if $ \overset{\frown}{GF} = 135^\circ $, that’s from G to F — so arc from F to G is 135°.

Then arc from F to H via G would be arc FG + arc GH = 135° + (10x + 5)

But we want arc FH, which is the arc from F to H not passing through G?

Wait — the inscribed angle ∠FGH intercepts arc FH, which is the arc opposite to G — so the arc from F to H not passing through G.

So if arc FG = 135°, and arc GH = 10x + 5, then the arc from F to H via G is $ 135 + (10x + 5) $

Then the arc from F to H not via G is the rest of the circle.

But the inscribed angle intercepts the arc not containing G, so it’s the minor arc FH?

Wait — no — the intercepted arc is the one between F and H, not containing G.

So if G is on the circle, and angle is at G, then intercepted arc is FH, the arc not containing G.

So the measure of arc FH (not containing G) is what matters.

But we are given $ \overset{\frown}{GF} = 135^\circ $, $ \overset{\frown}{GH} = 10x + 5^\circ $

These are arcs from G to F and G to H.

So the arc from F to H via G is $ \overset{\frown}{FG} + \overset{\frown}{GH} = 135^\circ + (10x + 5^\circ) $

Then the arc from F to H not via G is the rest of the circle:
$$
m\overset{\frown}{FH} = 360^\circ - [135 + (10x + 5)] = 360 - 140 - 10x = 220 - 10x
$$

But from the inscribed angle:
$$
m∠FGH = \frac{1}{2} \times m\overset{\frown}{FH} = 70^\circ
\Rightarrow m\overset{\frown}{FH} = 140^\circ
$$

So:
$$
220 - 10x = 140
\Rightarrow 10x = 80
\Rightarrow x = \boxed{8}
$$

✔ $ x = 8 $

Wait — earlier I assumed arc FH was via G, but actually, the intercepted arc is the one not containing G, so we must subtract.

Yes — so correct solution is:

- Arc from F to H not passing through G: $ m\overset{\frown}{FH} = 140^\circ $
- Total circle = 360°
- Arc from F to G to H = $ 135^\circ + (10x + 5^\circ) = 140 + 10x $
- Then:
$$
(140 + 10x) + 140 = 360
\Rightarrow 280 + 10x = 360
\Rightarrow 10x = 80
\Rightarrow x = \boxed{8}
$$

✔ $ x = 8 $

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✔ Final Answers:

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Part 1: Inscribed Angles

| # | Inscribed? | Angle | Intercepted Arc |
|---|------------|--------|------------------|
| 1 | Yes | ∠UVX | $\overset{\frown}{UX}$ |
| 2 | Yes | ∠SGR | $\overset{\frown}{SR}$ |
| 3 | Yes | ∠NMO | $\overset{\frown}{NO}$ |
| 4 | Yes | ∠RST | $\overset{\frown}{RT}$ |

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Part 2: Find Measures

5) $ \overset{\frown}{CE} = \boxed{150^\circ} $
6) $ m∠STU = \boxed{87.5^\circ} $
7) $ \overset{\frown}{UT} = \boxed{55^\circ} $

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Part 3: Solve for x

8) $ x = \boxed{10} $
9) $ x = \boxed{8} $

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