Geometry worksheet focusing on arcs and sectors, including angle and arc measure calculations and arc length problems with diagrams.

Let’s solve each problem step by step. We’ll go one at a time, carefully checking our work.

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Problem 1:

We are given circle with center H. Angle JHK = 21°. Points J, K, L, M on circle.

- mJK̂ = measure of arc JK → same as central angle JHK = 21°
- m∠JHM = ? Since JM is diameter (straight line through center), ∠JHM = 180° - ∠JHK? Wait — actually, looking at diagram: points J, H, M are colinear? Yes — JM is diameter. So ∠JHM is straight angle? But wait — point K is between J and M? Actually, from diagram: rays HJ, HK, HL, HM. Given ∠JHK = 21°, and assuming symmetry or other info? Wait — no other angles given. But we see that KL and LM might be symmetric? Not necessarily.

Wait — let’s look again. The diagram shows:

- Center H.
- Points J, K, L, M on circle.
- ∠JHK = 21°
- Also, it appears that JL and KM are diameters? Or maybe not. Actually, in many such problems, if two lines cross at center, they form vertical angles.

But here, only one angle given: 21° at ∠JHK.

Actually, looking at standard worksheet patterns — often when you have intersecting chords at center, opposite angles are equal.

Assume that JL and KM are both diameters? Then they intersect at H, forming vertical angles.

So if ∠JHK = 21°, then vertically opposite angle ∠LHM = 21°.

Then adjacent angles would be 180° - 21° = 159°.

So:

- mJK̂ = JHK = 21°
- m∠JHM = angle from J to M passing through... if M is opposite J, then ∠JHM = 180°? But that can’t be right because K is in between.

Wait — perhaps better to assume that the four points divide the circle into arcs based on central angles.

Given ∠JHK = 21°, and assuming that the figure has symmetry where ∠KHL = ∠LHM = etc.? No, not stated.

Actually, re-examining typical problems: often in these diagrams, if two lines intersect at center, they create two pairs of vertical angles.

Suppose lines JL and KM intersect at H. Then:

∠JHK and ∠LHM are vertical → both 21°

Then ∠JHM and ∠KHL are the other pair → each 180° - 21° = 159°

Yes, that makes sense.

So:

- mJK̂ = central angle ∠JHK = 21°
- m∠JHM = angle from J to M — which goes across the larger part? If M is opposite J, but K is between, then ∠JHM = JHK + ∠KHM? But we don’t know ∠KHM.

Wait — if KM is a straight line (diameter), then ∠JHK + ∠JHM = 180°? Only if K, H, M are colinear — which they are if KM is diameter.

Looking at diagram description: likely KM is diameter, so points K, H, M are colinear → so ∠JHK + ∠JHM = 180°

Thus:

m∠JHM = 180° - 21° = 159°

Similarly, since JL may also be diameter? Not sure. But for arcs:

Arc KL: what is its measure? If we assume that after K comes L, and then M, and back to J.

If ∠JHK = 21°, and if we assume that the next angle KHL is something else — but not given.

Wait — perhaps all we need is to use vertical angles and linear pairs.

Standard approach:

When two diameters intersect at center, they form two pairs of vertical angles.

Assume JL and KM are both diameters → intersect at H.

Then:

∠JHK = 21° → vertical angle ∠LHM = 21°

Then the other two angles: ∠JHM and ∠KHL are each 180° - 21° = 159°

Now, arcs correspond to central angles.

So:

- mJK̂ = ∠JHK = 21°
- mKL̂ = ∠KHL = 159° (since K to L passes through the big angle)
Wait — no: arc KL corresponds to central angle ∠KHL.

If ∠KHL = 159°, then arc KL = 159°

But let's list:

Points around circle: say J, K, L, M in order.

Central angles:

∠JHK = 21° → arc JK = 21°

KHL = ? If total around point is 360°, and we have two 21° and two 159°, sum: 21+21+159+159=360 — yes.

So:

- arc JK = 21°
- arc KL = 159° (angle at H between K and L)
- arc LM = 21° (vertical to JK)
- arc MJ = 159° (vertical to KL)

But the question asks for:

mJK̂ = 21°

m∠JHM = this is angle from J to M — which could be minor or major? Usually, unless specified, it's the smaller one, but 159° is less than 180, so ok.

From J to M: if going via K and L, it's 21+159+21=201°, too big. Directly? If M is adjacent to J, then angle JHM should be the angle at H between J and M.

In the diagram, if JL and KM are diameters, then J and L are opposite, K and M are opposite.

So vector HJ and HM: angle between them.

Since K and M are opposite, and J to K is 21°, then J to M is 180° - 21° = 159°? Let's think coordinates.

Place H at origin.

Put point J at (1,0). Since ∠JHK=21°, put K at 21° from J, so at angle 21°.

If KM is diameter, then M is opposite K, so at 21° + 180° = 201°.

Then angle between J (0°) and M (201°) is min(201, 360-201)=min(201,159)=159° — yes.

So m∠JHM = 159°

Now arcs:

mKL̂: from K to L. Where is L? If JL is diameter, and J is at 0°, then L is at 180°.

K is at 21°, L at 180°, so arc KL = 180° - 21° = 159°

mJM̂: from J to M. J at 0°, M at 201°, so minor arc is min(201, 159) — wait, 201 - 0 = 201, but the other way is 159° (from 0 to 201 clockwise is 159°? From 0 to 201 counterclockwise is 201°, clockwise is 159° — so minor arc is 159°.

But typically, arc JM means the arc not containing others? In context, probably the minor arc.

But let's see the notation: mJM̂ — likely the arc from J to M not passing through K and L? Or passing? Ambiguous.

In many textbooks, when they write mAB without specification, it's the minor arc.

Here, from J to M: shortest path is 159° (as calculated).

But earlier we have arc MJ = 159° in our division.

To avoid confusion, let's use the central angles we have.

We have four arcs:

- JK: 21°
- KL: 159° (from K at 21° to L at 180°)
- LM: from L at 180° to M at 201° → 21°
- MJ: from M at 201° to J at 0°/360° → 159° (360-201=159)

So:

mJK̂ = 21°

m∠JHM = angle at H between J and M. Vectors HJ and HM: J at 0°, M at 201°, difference 201°, but the smaller angle is min(201, 360-201)=159° — so 159°

mKL̂ = arc from K to L = 180° - 21° = 159°

mJM̂ = arc from J to M. Minor arc is 159° (going directly, not via K,L) — but in the circle, from J to M, there are two ways: one is J-K-L-M = 21+159+21=201°, other is J-M directly = 159°. So minor arc is 159°.

But sometimes they mean the arc corresponding to the angle. I think it's safe to say mJM = 159°

mMKL̂ — this is arc from M to K to L? That would be M to K to L.

From M to K: if M at 201°, K at 21°, going backwards: from 201 to 360/0 to 21 = (360-201)+21=159+21=180°? Or forward: 201 to 21 is not direct.

Arc MKL means starting at M, going through K, to L.

So positions: M at 201°, K at 21°, L at 180°.

From M (201°) to K (21°): if going increasing angle, 201 to 360 is 159°, then 0 to 21 is 21°, total 180° to reach K.

Then from K (21°) to L (180°) is 159°.

So total arc MKL = 180° + 159° = 339°? That seems large.

Perhaps it's the arc from M to L passing through K.

From M to K to L: M to K is 180° (as above), K to L is 159°, but that's overlapping or something.

Better to calculate the central angle for arc MKL.

Points M, K, L. Arc from M to L via K.

The central angle is the angle at H from M to L passing through K.

Vector HM to HL via HK.

Angle from M to K: as above, 180° (since M and K are not opposite; distance is |201-21|=180°? 201-21=180 — oh! 201 - 21 = 180, so M and K are diametrically opposite? Earlier I assumed KM is diameter, so yes, M and K are opposite, so arc MK is 180°.

Then from K to L is 159°, but if we go from M to K to L, and M to K is 180°, K to L is 159°, but that would be more than half circle.

The arc MKL is the arc from M to L that contains K.

Since M and K are opposite, and L is at 180°, K at 21°, M at 201°.

From M (201°) to L (180°) via K (21°): this would be from 201 down to 180 is 21° the short way, but via K means going the long way: from 201 to 360/0 to 21 to 180.

From 201 to 360: 159°, 0 to 21: 21°, 21 to 180: 159°, total 159+21+159=339°.

The other way from M to L directly is |201-180|=21°, so minor arc ML is 21°, but arc MKL is the major arc, 339°.

In some contexts, they might want the measure of the arc named by three points, which is the arc from first to last passing through the middle point.

So mMKL̂ = arc from M to L via K = 339°.

But let's confirm with the values.

Perhaps in the diagram, it's different. To save time, let's move to other problems and come back if needed.

For now, let's assume:

1. mJK̂ = 21°
2. m∠JHM = 159°
3. mKL̂ = 159°
4. mJM̂ = 159° (minor arc)
5. mMKL̂ = 339° (major arc from M to L via K)

But this seems messy. Perhaps mJM̂ is the arc not containing K and L, which is 159°, and mMKL̂ is the rest.

Another thought: in some worksheets, when they ask for mMKL, it might be the arc from M to K to L, which is the sum of arc MK and arc KL.

Arc MK: since M and K are opposite, arc MK = 180° (semicircle)

Arc KL = 159° as before

But arc MK and arc KL overlap at K, so adding them double-counts K, but for arc length, it's the path, so from M to K is 180°, then K to L is 159°, so total arc length for M to L via K is 180° + 159° = 339°, yes.

And the whole circle is 360°, so the other arc from M to L directly is 21°, which matches arc ML = 21° as we had.

So for problem 1:

- mJK̂ = 21°
- m∠JHM = 159°
- mKL̂ = 159°
- mJM̂ = 159° (assuming minor arc, but 159° is correct for the arc not containing K and L? From J to M, the minor arc is 159°, which does not contain K and L? J at 0°, M at 201°, minor arc is from 0 to 201 clockwise: 0 to 360 is 360, minus 201 is 159, so from 0 down to 201, which passes through 359,358,...,201, so it does not pass through K at 21° or L at 180°? 21° and 180° are not on that arc if we go from 0 to 201 clockwise: from 0 to 360/0, then to 201, so angles from 0 down to 201, which includes 359,358,...,201, so 21° is not included (since 21 > 201 in numerical value but in circle, 21 is between 0 and 180, while 201 to 360 is the other side).

Actually, from J at 0° to M at 201°, the minor arc is the shorter one, which is 159°, and it goes from 0° to 201° in the direction of decreasing angle or increasing? Standard position: usually counter-clockwise is positive.

So from 0° to 201° counter-clockwise is 201°, clockwise is 159° (since 360-201=159).

The clockwise arc from 0 to 201 goes through 359,358,...,201, so it includes angles from 201 to 360/0, so 21° is not on it (21 is between 0 and 180), 180 is not on it. So yes, minor arc JM = 159° does not contain K or L.

Similarly, arc MKL from M to L via K: M at 201°, K at 21°, L at 180°. From 201 to 21 counter-clockwise: 201 to 360 is 159°, 0 to 21 is 21°, total 180° to reach K, then from K at 21° to L at 180° is 159° counter-clockwise, so total 180° + 159° = 339°.

So I think it's correct.

But let's box the answers for problem 1 as:

mJK̂ = 21°
m∠JHM = 159°
mKL̂ = 159°
mJM̂ = 159°
mMKL̂ = 339°

Now Problem 2:

Circle with center X. Angles given: SXT = 72°, ∠TXU = 87°. Points S,T,U,V,W on circle.

First, find m∠UXV.

Note that SV and TW are likely diameters or something. Looking at diagram description: probably SV and TW are lines through center.

Assume that SV and TW are diameters intersecting at X.

Then, we have angles at center.

Given ∠SXT = 72°, ∠TXU = 87°.

Point U is between T and V? Probably.

So from S to T is 72°, T to U is 87°, so S to U is 72+87=159°.

Then, since SV is diameter, angle from S to V is 180°, so from U to V is 180° - 159° = 21°.

Similarly, TW is diameter, so from T to W is 180°.

From T to U is 87°, so from U to W is 180° - 87° = 93°.

Now, m∠UXV: angle at X between U and V.

As above, if S to V is 180°, S to U is 159°, so U to V is 21°, so ∠UXV = 21°.

Is that it? But let's see the points.

Also, we need arcs.

mST̂ = arc ST = central angle ∠SXT = 72°

mWT̂ = arc WT. W to T. If TW is diameter, arc WT could be 180°, but usually minor arc.

From W to T: if we go directly, and since TW is diameter, minor arc is 180°? No, minor arc is less than or equal to 180, so if it's diameter, arc WT is 180°.

But in the context, probably they want the arc not containing other points.

Let's define positions.

Set point S at 0°.

Then ∠SXT = 72°, so T is at 72°.

TXU = 87°, so U is at 72° + 87° = 159°.

SV is diameter, so V is at 180° (since S at 0°, opposite is 180°).

TW is diameter, T at 72°, so W is at 72° + 180° = 252°.

Now, check: from T to W should be 180°, 252 - 72 = 180, yes.

Now, m∠UXV: U at 159°, V at 180°, so angle at X between U and V is |180 - 159| = 21°.

So m∠UXV = 21°

mST̂ = arc from S to T = 72° - 0° = 72°

mWT̂ = arc from W to T. W at 252°, T at 72°. Minor arc: min(|252-72|, 360-|252-72|) = min(180, 180) = 180°? |252-72|=180, so exactly 180°, so arc WT = 180°

But is that what they want? Probably yes.

mTV̂ = arc from T to V. T at 72°, V at 180°, so 180 - 72 = 108°

mTVŴ = arc from T to W via V. T at 72°, V at 180°, W at 252°.

From T to V: 108°, V to W: 252 - 180 = 72°, so total 108 + 72 = 180°? But that's the same as direct T to W.

Arc TVW means from T to W passing through V.

From T (72°) to V (180°) to W (252°).

From 72 to 180 is 108°, 180 to 252 is 72°, total 180°.

The other way from T to W directly is also 180° (since diameter), so it's the same.

But typically, arc TVW might imply the arc containing V, which is this one, 180°.

Since it's a semicircle, it's fine.

So for problem 2:

m∠UXV = 21°
mST̂ = 72°
mWT̂ = 180°
mTV̂ = 108°
mTVW = 180°

Now Problem 3:

Circle with center D. Diameter GB? Points C,D,B,F,G on circle. Angle at B is 34°, but it's an inscribed angle? The diagram shows angle at B is 34°, and it's angle FBG or something.

Looking: "34°" near point B, and it's likely angle FBD or something.

The text says: "34°" and it's at point B, and there's a triangle or something.

Probably, angle at B is an inscribed angle subtending arc FD or something.

Standard: if angle at circumference is 34°, then arc it subtends is twice that.

But let's see the diagram description: points C,G,B on a line? Probably GB is diameter.

Assume GB is diameter, so G-D-B colinear.

Point F on circle, and angle at B is 34°, which is angle FBG or angle FBD.

Likely, angle FBG = 34°, and since GB is diameter, angle in semicircle is right angle, but here it's at B.

Angle at B is formed by points F, B, G.

So angle FBG = 34°.

This is an inscribed angle that subtends arc FG.

Inscribed angle subtends the arc between the two points.

So angle at B, formed by chords BF and BG, subtends arc FG.

So measure of inscribed angle is half the arc it subtends.

So angle FBG = (1/2) * arc FG

So 34° = (1/2) * mFĜ

Thus mFĜ = 68°

But the questions are for arcs CD, FD, DCF, GDP.

Points: C, D, B, F, G.

D is center.

GB is diameter, so G and B are ends of diameter.

C is another point, probably on the circle.

Angle at B is 34°, which is angle between FB and GB.

Since GB is diameter, and F on circle, then angle at F in triangle GFB should be 90°, but here angle at B is given.

So in triangle GFB, angle at B is 34°, angle at F is 90° (since GB diameter), so angle at G is 180-90-34=56°.

But we need arcs.

Arc FG: as above, subtended by angle at B, so mFĜ = 2 * 34° = 68°

Now, since GB is diameter, arc GFB is semicircle, 180°.

Arc GF + arc FB = 180°? Arc from G to B via F.

Arc G to F is 68°, so arc F to B is 180° - 68° = 112°

But we need arc CD, etc.

Where is C? Probably C is on the other side.

The diagram might have point C such that GC is something.

Perhaps C is symmetric or something.

Another possibility: the 34° is angle at B for triangle CBD or something.

Let's read the problem: "34°" is written near B, and there's a line from B to F and B to G, and G to D to B is diameter.

Also, there is point C, and D is center.

Probably, arc CD is asked, so C is another point.

Perhaps angle at B is angle CBD or angle FBD.

I recall that in some problems, if you have diameter GB, and point F on circle, angle FBG = 34°, then arc FG = 68°, as above.

Then, if C is the other end or something.

Perhaps C is such that DC is perpendicular or something.

Another thought: the 34° might be the central angle, but it's at B, not at D.

The label "34°" is at vertex B, so it's an angle at B, so inscribed angle.

So likely, angle FBG = 34°, subtending arc FG, so arc FG = 68°.

Then, since GB is diameter, arc G to B is 180°, so arc FB = 180° - arc FG = 180° - 68° = 112°? Arc from F to B along the circle not containing G.

In triangle GFB, with GB diameter, angle at F is 90°, angle at B is 34°, so angle at G is 56°.

Angle at G is angle FGB, which is an inscribed angle subtending arc FB.

So angle at G = (1/2) * arc FB

So 56° = (1/2) * mFB̂, so mFB̂ = 112°

Similarly, angle at B = 34° = (1/2) * mFG, so mFĜ = 68°

Now, what about point C? The diagram has point C, and we need arc CD, etc.

Probably, C is on the circle, and perhaps GC is a radius or something.

Perhaps C is the point such that DC is drawn, and maybe angle at D or something.

Another idea: perhaps the 34° is angle CBD, and D is center, so if B is on circle, D center, then DB is radius, and if C is on circle, then angle CBD is at B, between C, B, D.

But D is center, so BD is radius, BC is chord.

Angle at B between points C, B, D.

This is an angle in triangle CBD.

But we need more information.

Perhaps in the diagram, there is a right angle or something.

Let's look at the names: arcs CD, FD, DCF, GDP.

GDP suggests points G,D,P, but P is not mentioned. In problem 3, points are C,D,B,F,G, no P.

In the user's image description, for problem 3, it's "3." with circle, points C,G,B on a line? And F, and angle 34° at B.

And arcs: mCD, mFD̂, mDCF, mGDP̂ — but GDP has P, which is not in the diagram. Probably typo, or perhaps it's GDB or something.

In the text: "mGDP̂" but in problem 3, no P. Perhaps it's mGDB or mGBD.

Maybe P is a mistake, and it's mGDF or something.

Another possibility: in some versions, it's mGDF.

Let's assume that "P" is a typo, and it's "F" or "B".

Perhaps for mGDP, it's arc from G to D to P, but no P.

I think it's likely a typo, and it's mGDF or mGBD.

Perhaps D is center, so arc GDP doesn't make sense because D is center, not on circle.

Arcs are between points on the circle, so G, D, P — but D is center, so probably not.

Unless it's arc GP or something.

I think there might be a mistake in my assumption.

Let's search for standard problems.

Perhaps the 34° is the central angle for arc FB or something.

Another idea: in the diagram, the 34° is at the center? But the label is at B, and B is on the circle, so likely not.

Let's read the problem again: "3. [diagram] mCD̂ = ___ mFD̂ = ___ mDCF̂ = ___ mGDP̂ = ___" and "34°" near B.

Perhaps angle at B is 34°, and it's angle of the triangle, and D is center, so we can find other angles.

Assume that GB is diameter, D center, so GD = DB = radius.

Point F on circle, so DF = radius.

Angle at B is 34°, which is angle FBD or angle FBG.

Suppose it's angle FBG = 34°, as before.

Then in triangle FBD, we have points F,B,D.

DB is radius, DF is radius, so triangle FBD is isosceles with DF = DB.

Angle at B is 34°, so in triangle FBD, angle at B is 34°, sides DF = DB, so base angles equal.

Vertices F,B,D.

Sides: DF and DB are both radii, so equal, so triangle FBD is isosceles with DF = DB, so base is FB, so angles at F and B are equal? No.

Sides from D: DF and DB are equal, so the base is FB, so the base angles are at F and B.

So angle at F and angle at B are equal.

But angle at B is given as 34°, so angle at F is also 34°, then angle at D is 180-34-34=112°.

So central angle FDB = 112°.

Then arc FB = central angle = 112°.

Then, since GB is diameter, arc G to B is 180°, so arc G to F = arc GB - arc FB = 180° - 112° = 68°, same as before.

Now, where is C? Probably C is another point on the circle.

Perhaps C is such that GC is drawn, and maybe angle at G or something.

Perhaps the diagram has point C on the extension or something.

Another common configuration: perhaps C is the point diametrically opposite or something.

Perhaps arc CD is from C to D, but D is center, so not on circle.

I think there's a mistake; arcs are between points on the circle, so mCD̂ means arc from C to D, but D is center, so impossible.

Unless D is on the circle, but in the diagram, D is labeled as center.

In the user's description, for problem 3, "D" is likely the center, as in other problems.

For example, in problem 1, H is center, in 2, X is center, in 3, D is probably center.

So arc CD doesn't make sense if D is center.

Unless "D" in mCD̂ is a point on the circle, but in the diagram, D is center.

Perhaps in some notations, but unlikely.

Let's look at the arc names: mCD, mFD̂, mDCF̂, mGDP̂.

mFD̂: F to D, again D center.

This is confusing.

Perhaps "D" in the arc name is a typo, and it's supposed to be another letter.

For example, in mCD̂, perhaps it's mCB̂ or mCĜ.

Perhaps for mFD̂, it's mFB̂ or mFĜ.

Another idea: in some worksheets, they use the center in the arc name to indicate the arc, but usually not.

Perhaps for mDCF̂, it's arc from D to C to F, but D is center.

I think there might be a mislabeling.

Perhaps in problem 3, "D" is not the center; but in the diagram, it's likely is.

Let's assume that for arc measures, when they say mCD̂, they mean the arc from C to D, but since D is center, it must be that D is on the circle, but that contradicts.

Unless in this diagram, D is on the circle, but typically not.

Let's check problem 4: in problem 4, U is center, and arcs like mPQ, mSR̂, etc., so U is center, not on circle.

So for problem 3, D is center, so arcs like mCD̂ don't make sense.

Perhaps "mCD̂" means the arc from C to the point diametrically opposite or something, but unlikely.

Another possibility: "D" in mCD̂ is a typo, and it's "B" or "G".

For example, mCB̂ or mCĜ.

Perhaps it's mCF̂ or something.

Let's look at the angle: 34° at B, and we have points C,G,B on a line, so G-D-B colinear, diameter.

Point F on circle, and perhaps C is on the circle on the other side.

Perhaps angle at B is 34°, and it's angle between CB and FB or something.

Assume that the 34° is angle CBD, and D is center, so in triangle CBD, with D center, B on circle, C on circle, so DB and DC are radii, so triangle CBD is isosceles with DB = DC.

Angle at B is 34°, so if DB = DC, then angles at B and C are equal, so angle at C is 34°, angle at D is 112°.

Then arc BC = central angle ∠BDC = 112°.

Then, since G-D-B is diameter, arc G to B is 180°, so arc G to C = arc GB - arc CB = 180° - 112° = 68°, if C is on the same side.

Then we can find other arcs.

But we need arc CD, which would be from C to D, still problem.

Perhaps for mCD̂, they mean the arc from C to the point, but D is not on circle.

I think the only logical explanation is that "D" in the arc names is a typo, and it's supposed to be "B" or "G" or "F".

For example, in many similar problems, they ask for arc CF, arc FB, etc.

Perhaps mCD̂ means arc from C to D, but since D is center, it's not defined, so likely it's mCB̂ or mCG.

Let's assume that "D" is a typo for "B" in some cases.

For instance, mCD̂ might be mCB̂, mFD̂ might be mFB̂, etc.

Perhaps for mGDP̂, "P" is "B", so mGDB, but D is center.

Another idea: in some notations, mGDP might mean the arc from G to P passing through D, but D is center, so not on arc.

I recall that in some old texts, they might use the center to denote the arc, but it's rare.

Perhaps for mDCF̂, it's the arc from D to C to F, but again D not on circle.

Let's try to interpret mDCF̂ as the arc from D to F via C, but D not on circle.

I think the best guess is that "D" in the arc names is a mistake, and it's supposed to be the points on the circle.

Perhaps in the diagram, there is a point D on the circle, but the center is labeled differently, but in the problem, it's "D" for center.

Let's look at the user's input: for problem 3, "3. [diagram] mCD̂ = ___ mFD̂ = ___ mDCF̂ = ___ mGDP̂ = ___" and "34°" at B.

And in the diagram, likely D is center.

Perhaps for mCD, they mean the arc from C to the point diametrically opposite to D, but that doesn't make sense.

Another thought: in some contexts, mCD might mean the arc whose central angle is at D, but usually it's specified.

Perhaps mCD means the arc subtended by angle at D, but that's not standard.

Let's calculate what we can.

From earlier, if we assume that angle at B is 34°, and it's angle FBG, then arc FG = 68°, arc FB = 112°, as before.

Then, if we assume that C is the point such that GC is drawn, and perhaps angle at G is given or something, but not.

Perhaps the 34° is for angle at B in triangle CBD, and C is on the circle, so as above, if DB = DC, then arc BC = 112°.

Then, since G-D-B diameter, arc G to B = 180°, so if C is on the arc not containing F, then arc G to C = 180° - 112° = 68°.

Then arc C to B = 112°.

Then for arc CD: if D is center, perhaps they mean the arc from C to the point, but let's say for mCD̂, they might mean the arc from C to D, but since D is center, perhaps it's a mistake, and it's mCB̂ = 112°.

Similarly, mFD̂ might be mFB̂ = 112° or mFĜ = 68°.

mDCF̂: arc from D to C to F — perhaps from C to F via D, but D not on circle.

Perhaps mDCF means the arc from D to F, but again.

Another idea: in some worksheets, mDCF might mean the arc from D to F passing through C, but D not on circle.

I think for the sake of time, let's assume that "D" in the arc names is a typo for "B", so mCD̂ = mCB̂, mFD̂ = mFB, etc.

So from above, if angle at B is 34° for angle CBD, and DB = DC, then arc BC = 112°.

Then mCB̂ = 112°.

Then mFB̂: if F is another point, and we have arc FB = 112° from earlier calculation, but that was for different assumption.

Let's stick with one assumption.

Assume that the 34° is angle FBG = 34°, so arc FG = 68°, arc FB = 112°.

Then for point C, perhaps C is G or something, but not.

Perhaps C is the point such that DC is perpendicular to GB or something.

Perhaps in the diagram, there is a right angle at D or something.

Let's look for the answer.

Perhaps the 34° is the central angle for arc FB.

But the label is at B, not at D.

In the diagram, the 34° is written at vertex B, so it's an angle at B.

So likely inscribed angle.

So arc it subtends is 68°.

Then, if we assume that C is the point diametrically opposite to F or something.

Perhaps for mCD̂, they mean the arc from C to D, but since D is center, it's not possible, so perhaps in this context, "D" is a point on the circle, but the center is labeled as D, so conflict.

I recall that in some problems, the center is labeled, and arcs are between other points.

Perhaps for mCD̂, "D" is a typo, and it's "G", so mCĜ.

Let's assume that.

Suppose that the 34° is angle at B for angle FBG = 34°, so arc FG = 68°.

Then, since GB is diameter, arc G to B = 180°, so arc F to B = 112°.

Now, if C is on the circle, and perhaps GC is a radius, but G is on circle, D center, so GD is radius.

Perhaps C is such that DC is drawn, and angle at D is given, but not.

Another common thing: perhaps the 34° is angle at B for the triangle, and D is center, so we can find angle at D.

In triangle FBD, if DB = DF (radii), and angle at B is 34°, then as before, angle at D is 112°, so arc FB = 112°.

Then, if we assume that C is the point on the circle such that GC is the other radius, but G is already there.

Perhaps C is the intersection or something.

Let's notice that in the arc mGDP̂, "P" might be "F", so mGDF.

And mDCF might be mDCF, but D not on circle.

Perhaps for mDCF, it's the arc from D to C to F, but again