Geometry worksheet for calculating angle measurements in a circle with tangent lines and labeled arcs.

Here is the step-by-step solution to find the measurement of all the numbered angles and the arc.

Step 1: Find Angle 2
Angle 2 is an inscribed angle that intercepts the arc measuring $85^\circ$. The measure of an inscribed angle is half the measure of its intercepted arc.
$$ \text{Angle } 2 = \frac{85}{2} = 42.5^\circ $$

Step 2: Find Angle 3
Angles on a straight line add up to $180^\circ$. Angle 2 and Angle 3 form a linear pair along the chord RM extended to L? No, looking closely at vertex M, the line segment is a chord RM and a tangent LM. Wait, let's look at the triangle LMR.
Actually, let's look at the straight line passing through R and M? No, R-M is a chord. L-M-N is a tangent line.
Let's look at Angle 1 first using the Tangent-Secant theorem or Triangle sum.
Let's find the arcs first to make it easier.
We know Arc $RM = 85^\circ$ and Arc $MQ = 150^\circ$.
The total circle is $360^\circ$. We need Arc $SQ$ and Arc $RS$.
Let's look at Angle 11 ($40^\circ$). This is an external angle formed by secants intersecting outside the circle.
Formula: $\text{Angle} = \frac{1}{2} (\text{Far Arc} - \text{Near Arc})$.
The secants are from P. They intersect the circle at S, Q and R, M.
So, $\angle P = \frac{1}{2} (\text{Arc } MQ - \text{Arc } RS)$.
Wait, the secant lines are P-S-Q and P-R-M.
So the intercepted arcs are Arc $MQ$ (far) and Arc $RS$ (near).
$$ 40 = \frac{1}{2} (150 - \text{Arc } RS) $$
$$ 80 = 150 - \text{Arc } RS $$
$$ \text{Arc } RS = 150 - 80 = 70^\circ $$

Now we have Arc $RS = 70^\circ$, Arc $RM = 85^\circ$, Arc $MQ = 150^\circ$.
The remaining arc is Arc $SQ$.
$$ \text{Arc } SQ = 360 - (70 + 85 + 150) = 360 - 305 = 55^\circ $$
So, Answer 20 (m arc SQ) = $55^\circ$.

Step 3: Calculate Inscribed Angles
* Angle 3: Intercepts Arc $RS$ ($70^\circ$).
$$ \text{Angle } 3 = \frac{70}{2} = 35^\circ $$
* Angle 4: Intercepts Arc $SQ$ ($55^\circ$).
$$ \text{Angle } 4 = \frac{55}{2} = 27.5^\circ $$
* Angle 6: Intercepts Arc $RS$ ($70^\circ$). Note: Angle 6 is $\angle MQR$? No, Angle 6 is $\angle OQM$? No, it's $\angle MQS$? Let's trace the lines. Angle 6 is $\angle MQS$? No, the vertex is Q. The sides go to M and S? No, the sides go to M and R?
Let's look at Vertex Q. The chords are MQ, SQ, RQ.
Angle 6 is bounded by chord MQ and chord RQ? Or MQ and SQ?
Looking at the diagram, Angle 6 is inside triangle MRQ? No, it's $\angle MQR$?
Let's check Angle 7. Angle 7 is $\angle RQS$?
Let's check Angle 8. Angle 8 is $\angle PQS$? No, it's $\angle SQR$?
Let's re-examine the vertices.
Vertex Q has angles 6, 7, 8.
Angle 6 is $\angle MQR$? It intercepts Arc $MR$? No, Arc $MSR$?
Actually, let's look at the chords connected to Q: M, R, S.
Angle 6 is between chord QM and QR. It intercepts Arc $MR$? No, Arc $M...R$?
The arc opposite Angle 6 (if it is $\angle MQR$) is Arc $MSR$? No, just Arc $MR$ if it doesn't cross the center?
Wait, Arc $RM$ is given as $85^\circ$. But that's the minor arc.
Does Angle 6 intercept Arc $RM$? Yes, if the angle is $\angle MQR$.
Let's assume standard labeling where adjacent numbers are adjacent angles.
Angle 6 is $\angle MQR$. Intercepted Arc is Arc $MR$? No, the points on the circle are M, R, S, Q.
Order on circle: M -> R -> S -> Q -> M.
Arc $MR = 85$. Arc $RS = 70$. Arc $SQ = 55$. Arc $QM = 150$.

* Angle 6: Vertex Q. Rays QM and QR. Intercepts Arc $MR$? No, it intercepts Arc $M...R$ which is Arc $MSR$? Or Arc $MR$ directly?
Since R and M are adjacent in the sequence M-R-S-Q? No, looking at the drawing:
Top right is M. Left is R. Bottom is S. Right is Q.
So the order is M, R, S, Q clockwise?
Arc $MR = 85$. Arc $RS = 70$. Arc $SQ = 55$. Arc $QM = 150$.

Angle 6 is $\angle MQR$. The rays cut off Arc $MR$? No, the interior of the angle contains the center?
Let's look at Angle 3 ($\angle RMS$). It intercepts Arc $RS$ ($70$). So Angle 3 = 35. Correct.
Angle 4 ($\angle SMQ$). It intercepts Arc $SQ$ ($55$). So Angle 4 = 27.5. Correct.
Angle 2 ($\angle LMR$? No, $\angle RMS$? No).
Let's re-read Angle 2. Vertex M. Between Tangent LM and Chord MR?
If Angle 2 is between Tangent LM and Chord MR, it intercepts Arc $MR$.
Measure = $\frac{1}{2} \text{Arc } MR = \frac{85}{2} = 42.5^\circ$.

Let's check Angle 5. Vertex M. Between Chord MQ and Tangent MN.
Intercepted Arc is Arc $MQ$? No, Arc $MSQ$?
The angle between tangent and chord equals half the intercepted arc.
Angle 5 intercepts Arc $MQ$? No, it's adjacent to Angle 4.
Angle 4 is $\angle SMQ$. Angle 5 is $\angle QMN$.
So Angle 5 intercepts Arc $MQ$? No, Arc $MSQ$?
The arc "inside" the angle is Arc $MQ$? No, the chord is MQ. The arc is the one not containing the other points?
Arc $MQ = 150^\circ$.
So Angle 5 = $\frac{150}{2} = 75^\circ$.

Let's verify Line LMN is a straight line (tangent).
Sum of angles at M on the straight line: Angle 2 + Angle 3 + Angle 4 + Angle 5?
Wait, Angle 2 is outside the circle?
Diagram shows:
Line L-M-N is tangent.
Chords from M are MR, MS, MQ.
Angle 2 is $\angle LMR$. Intercepted Arc $MR = 85$. Angle 2 = 42.5.
Angle 3 is $\angle RMS$. Intercepted Arc $RS = 70$. Angle 3 = 35.
Angle 4 is $\angle SMQ$. Intercepted Arc $SQ = 55$. Angle 4 = 27.5.
Angle 5 is $\angle QMN$. Intercepted Arc $MQ$? No, Arc $Q...M$?
The arc corresponding to chord MQ is 150.
Angle 5 = $\frac{150}{2} = 75$.
Check sum: $42.5 + 35 + 27.5 + 75 = 180$. Perfect.

Now for Vertex Q:
Chords are QM, QR, QS.
Angle 6 is $\angle MQR$. Intercepts Arc $MR$? No, Arc $MSR$?
The angle subtends Arc $MR$? No, the points are M, R, S, Q.
Angle $\angle MQR$ subtends Arc $MR$? No, it subtends Arc $M...R$ which goes through nothing?
Wait, the arc from M to R is 85.
So Angle $\angle MQR$ (which is Angle 6 + Angle 7?) No.
Let's identify the specific rays for each number.
At Q:
Ray QM, Ray QR, Ray QS, Ray QP (secant).
Angle 6 is between QM and QR. Intercepts Arc $MR$?
Arc $MR = 85$. So Angle $\angle MQR = \frac{85}{2} = 42.5^\circ$.
Is Angle 6 the whole $\angle MQR$? Or part of it?
Looking at the lines: There is a line from R to Q. There is a line from S to Q.
Angle 6 is $\angle MQR$? No, there is a line segment RQ. And SQ.
Angle 6 is $\angle MQR$? No, Angle 6 is $\angle MQS$? No.
Usually, numbers are placed in the smallest regions.
Region 6 is bounded by QM and QR? Or QM and QS?
Let's look at Region 7. Bounded by QR and QS?
Let's look at Region 8. Bounded by QS and QP?

Let's assume:
Angle 6 = $\angle MQR$? No, if there is a chord QS, then $\angle MQR$ is split into $\angle MQS$ and $\angle SQR$.
Let's look at the position of '6'. It is between QM and QR?
Wait, does the line go from R to Q? Yes.
Does the line go from S to Q? Yes.
So at vertex Q, we have rays QM, QR, QS, QP.
Order of rays clockwise: QM, QR, QS, QP?
Let's check the arcs.
Arc $MR = 85$. Arc $RS = 70$. Arc $SQ = 55$.
Angle $\angle MQR$ intercepts Arc $MR$? No, Arc $MSR$?
Inscribed angle $\angle MQR$ intercepts Arc $MR$?
The vertices on the circle are M, R, S, Q.
Angle $\angle MQR$ connects M and R. The arc is Arc $MR$ (minor) or Arc $MSR$ (major)?
Since Q is on the major arc side relative to MR?
Arc $MR = 85$. The rest is $275$.
Angle $\angle MQR = \frac{85}{2} = 42.5^\circ$.

Angle $\angle RQS$ connects R and S. Intercepts Arc $RS = 70$.
Angle $\angle RQS = \frac{70}{2} = 35^\circ$.

Angle $\angle SQP$? P is outside.

Let's map the numbers to these angles.
Angle 6 looks like $\angle MQR$? Or $\angle MQS$?
In the diagram, the line RQ divides the space. The line SQ divides the space.
Number 6 is in the sector between QM and QR?
Number 7 is in the sector between QR and QS?
Number 8 is in the sector between QS and QP?

If so:
Angle 6 = $\angle MQR$. Intercepts Arc $MR$? No.
Wait, if the order on the circle is M, R, S, Q...
Then Chord MR, Chord RS, Chord SQ, Chord QM.
Diagonals are MS and RQ.
Angle $\angle MQR$ is formed by Chord QM and Chord QR.
It intercepts Arc $MR$? No, it intercepts Arc $M...R$?
The arc NOT containing Q. That is Arc $MR$ (85) + Arc $RS$ (70)? No.
The endpoints are M and R. The arc is Arc $MR$.
So Angle $\angle MQR = \frac{1}{2} \text{Arc } MR = 42.5^\circ$.
BUT, looking at the diagram, Angle 6 and Angle 7 are adjacent.
Angle 6 is likely $\angle MQS$? No, R is between M and S?
Let's look at Vertex M again. Rays: ML(tangent), MR, MS, MQ, MN(tangent).
Order: L, R, S, Q, N.
So Chords are MR, MS, MQ.

At Vertex Q: Rays QM, QR, QS, QP.
Order on circle: M, R, S, Q.
So geometrically, R is between M and S?
Arc $MR=85$, Arc $RS=70$. So yes, R is between M and S.
Therefore, Ray QR is between QM and QS.
So Angle $\angle MQS$ is the big angle composed of Angle 6 and Angle 7?
Or is Angle 6 $\angle MQR$ and Angle 7 $\angle RQS$?
Let's assume:
Angle 6 = $\angle MQR$. Intercepts Arc $MR$?
Wait, the inscribed angle theorem says angle is half the intercepted arc.
Angle $\angle MQR$ intercepts Arc $MR$?
The chord is MR. The angle is at Q.
Yes, it intercepts Arc $MR$.
Arc $MR = 85$.
So Angle 6 = $42.5^\circ$.

Angle 7 = $\angle RQS$. Intercepts Arc $RS$.
Arc $RS = 70$.
So Angle 7 = $35^\circ$.

Angle 8 = $\angle SQP$.
This is part of the triangle PQS? Or just the angle between chord QS and secant QP.
We need to find Angle 10 or 11 first.

Let's go to Vertex S.
Rays: SP(secant), SR, SM, SQ.
Order on circle: R, S, Q.
Arc $RS = 70$. Arc $SQ = 55$.
Angle $\angle RSQ$? No.
Angle $\angle MSR$? Intercepts Arc $MR$? No, Arc $MR$ is opposite?
Angle $\angle MSR$ intercepts Arc $MR$? No, endpoints M, R.
Angle at S intercepts Arc $MR$?
Arc $MR = 85$.
So Angle $\angle MSR = \frac{85}{2} = 42.5^\circ$.
Which number is this?
At S, we have numbers 9, 10, 8? No, 8 is at Q.
Numbers at S: 9, 10? And maybe another?
Diagram shows:
Angle 9 is $\angle MSR$?
Angle 10 is $\angle RSP$?
Angle 8 is at Q.

Let's refine the mapping at S.
Rays: SM, SR, SP. (And SQ).
Angle $\angle MSQ$? Intercepts Arc $MQ$? No, Arc $MR+RS$?
Endpoints M, Q. Arc $MQ$ (minor) is 150? No, Arc $MQ$ via N is 150.
Arc $MSQ$? No.
Arc $M...Q$ not containing S. That is Arc $MQ = 150$.
So Angle $\angle MSQ = \frac{150}{2} = 75^\circ$.
Angle $\angle MSQ$ is composed of Angle $\angle MSR$ + Angle $\angle RSQ$?
Angle $\angle MSR$ intercepts Arc $MR$? No.
Vertices M, S, R. Angle at S. Intercepts Arc $MR$?
Arc $MR = 85$. So Angle $\angle MSR = 42.5^\circ$.
Angle $\angle RSQ$ intercepts Arc $RQ$?
Arc $RQ$ = Arc $RS$ + Arc $SQ$? No.
Endpoints R, Q. Arc $RQ$ (minor) = Arc $RS$ + Arc $SQ$?
No, R and Q are separated by S?
Arc $RS = 70$. Arc $SQ = 55$.
So Arc $RQ$ (containing S) is $125$.
The arc NOT containing S is Arc $RMQ$?
Arc $RM$ (85) + Arc $MQ$ (150) = 235?
Wait. Total 360.
Arc $RQ$ (minor) = Arc $RS$ + Arc $SQ$? No, that's the path through S.
The chord RQ cuts off two arcs.
One is Arc $RSQ$? No, Arc $R...Q$ via S is $70+55=125$.
The other is Arc $RMQ$ via M is $85+150=235$.
Angle $\angle RSQ$ is an inscribed angle?
Vertex S. Rays SR, SQ.
Intercepts Arc $RQ$? The one NOT containing S.
So it intercepts Arc $RMQ$?
Arc $RMQ = 360 - 125 = 235$.
Angle $\angle RSQ = \frac{235}{2} = 117.5^\circ$.

Let's re-evaluate the angles at S based on the diagram labels.
Label 9 is inside $\triangle MSR$?
Label 10 is inside $\triangle PSR$?
Label 8 is at Q.

Let's look at Angle 9.
It is $\angle MSR$?
If Angle 9 = $\angle MSR$, it intercepts Arc $MR$?
Wait, Angle $\angle MSR$ vertex S. Endpoints M, R.
Arc $MR = 85$.
So Angle 9 = $42.5^\circ$.

Angle 10.
It is $\angle RSP$?
We need Angle $\angle PRS$ or something else.
Let's use Triangle POS? No.
Let's use Triangle PQS or PRS.

Let's go back to Angle 11.
Given as $40^\circ$.

Angle 12.
Vertex R. Exterior to circle? No, R is on the circle.
Angle 12 is $\angle PRM$? Or $\angle LRP$?
Line L-R-P? No, L-M is tangent. P-R-M is a secant line.
So P, R, M are collinear.
Therefore, Angle 12 and Angle 15 are supplementary?
Angle 15 is $\angle LRM$? No, L is on tangent.
Let's look at Vertex R.
Lines: PM (secant), RQ, RS, RL? No, L is far away.
Line LR? No, L is connected to M.
Is there a line L-R?
Diagram shows a line from L to R?
"Circle O with tangent LM".
Line L-R-P?
If L-R-P is a straight line, then L-R-P is a secant?
But it says "Tangent LM".
Usually, tangents touch at one point.
If L-R-P is a line, it intersects at R and P? No, P is outside.
If L-R-P is a secant, it intersects at R and some other point?
The diagram shows line segment L-R-S? No.
Line segment L-R? And R is on the circle.
If L-R is part of a secant L-R-?-P?
Let's look at the line starting at P.
It goes through R and ends at M?
"P ... R ... M". Yes, P-R-M is a secant.
What about L?
L is connected to M (tangent).
Is L connected to R?
There is a line segment L-R.
Is L-R-P a straight line?
If so, L-R-P is a secant intersecting at R and... where else?
It seems L-R-P is a single straight line secant intersecting the circle at R and... S?
Let's check the bottom line. P-S-Q.
So we have two secants from P:
1. P-R-M? (Intersects at R and M).
2. P-S-Q? (Intersects at S and Q).

What is the line L-R?
It connects L (on tangent at M) to R?
If L-R-P is a straight line, then the secant is P-R-L?
But the secant intersects the circle at R and M?
If P-R-M is a line, and L-R-P is a line, then L, R, M are collinear?
But M and R are distinct points on the circle.
A line can only intersect a circle at 2 points.
If P-R-M is a secant, the line contains P, R, M.
If L is on the tangent at M, and L-R is drawn...
Is L-R-P a straight line?
Visually, L-R and R-P look like they form a straight line.
IF L-R-P is a straight line, then the secant from P passes through R and... does it hit the circle again?
It hits at R. Where is the second point?
If the line is tangent at R? No.
If the line is secant P-R-X?
Let's assume the standard configuration:
Secant 1: P-R-M. (Points P, R, M collinear).
Secant 2: P-S-Q. (Points P, S, Q collinear).
Line L-M is tangent at M.
Line L-R?
If L-R-P is a straight line, then P, R, L are collinear.
This would mean the secant is P-R-L?
But we already established P-R-M is a secant.
Contradiction unless R=M (impossible) or L, R, M are collinear (chord RM extended to L?).
If L, R, M are collinear, then LM is a secant, not a tangent.
But the problem states "Tangent LM".
Therefore, L, R, M are NOT collinear.
So L-R-P is NOT a straight line with P-R-M.

Let's look at the line labeled L.
It originates at L, goes through R?
Or is L just a point on the tangent?
And there is a chord/line from R to L?
If L-R is a segment, and L-M is tangent...
And P-R-M is a secant...
Then Angle 12 is $\angle PRS$? Or $\angle LRP$?

Let's look at Angle 1.
Vertex L.
Triangle LMR?
If L-M is tangent and L-R is a secant?
If L-R intersects the circle at R and... S?
Let's assume the line is L-R-S-P?
If L-R-S-P is a straight line, then it's a secant intersecting at R and S.
Then we have:
Secant 1 from L: L-R-S.
Tangent from L: L-M.
This makes sense for Angle 1.
Angle 1 is formed by Tangent LM and Secant LRS.
Formula: $\text{Angle } 1 = \frac{1}{2} (\text{Far Arc} - \text{Near Arc})$.
Far Arc = Arc $MS$? Near Arc = Arc $MR$?
Let's check the arcs.
Arc $MR = 85$.
Arc $RS = 70$.
Arc $SM$?
Arc $SM$ = Arc $SR$ + Arc $RM$? No.
Arc $SM$ (minor) = Arc $SQ$ + Arc $QM$? No.
Arc $SM$ via Q? $55 + 150 = 205$.
Arc $SM$ via R? $70 + 85 = 155$? No.
Points order: M, R, S, Q.
Arc $MR = 85$.
Arc $RS = 70$.
Arc $SQ = 55$.
Arc $QM = 150$.

If Secant is L-R-S, it intersects at R and S.
Far Arc = Arc $MS$?
Which Arc MS? The one not containing R.
Arc $MQS$? $150 + 55 = 205$.
Near Arc = Arc $MR$? $85$.
Angle 1 = $\frac{1}{2} (205 - 85) = \frac{120}{2} = 60^\circ$.

Let's verify this interpretation.
If L-R-S is a line, then P is on the extension?
Diagram shows P-R and R-L?
If P-R-M is a secant, and L-R-S is a secant...
Do they intersect at R? Yes.
Is P-R-L a straight line?
If P-R-M is a line, and L-R-S is a line...
Then Angle $\angle PRS$ and Angle $\angle LRM$ are vertical?

Let's look at Angle 12.
Vertex R.
Between Line PR and Line RS?
If P-R-M is a line and L-R-S is a line...
Then Angle 12 is $\angle PRS$?
Angle $\angle PRS$ and Angle $\angle MRL$ are vertical.
Angle $\angle MRS$ is an inscribed angle?
Angle $\angle MRS$ intercepts Arc $MS$?
Arc $MS$ (via Q) = $150 + 55 = 205$.
Angle $\angle MRS = \frac{205}{2} = 102.5^\circ$.
Then Angle $\angle PRS$ (Angle 12) is supplementary to $\angle MRS$?
Because P-R-M is a straight line.
Angle 12 + Angle $\angle MRS$ = 180.
Angle 12 = $180 - 102.5 = 77.5^\circ$.

Alternatively, Angle 12 is an exterior angle to $\triangle PRS$? No.
Let's check Angle 15.
Angle 15 is $\angle LRM$?
If L-R-S is a line, Angle 15 and Angle $\angle MRS$ are supplementary?
Angle 15 + Angle $\angle MRS$ = 180?
Angle 15 = $180 - 102.5 = 77.5^\circ$.

Let's check Angle 1 using Triangle LMR?
If L-R-S is a line, Triangle LMR is not formed by secant LRS.
Triangle L-M-R?
Angle at M: Angle between Tangent LM and Chord MR.
Angle $\angle LMR = \frac{1}{2} \text{Arc } MR = 42.5^\circ$.
Angle at R: Angle 15 = $77.5^\circ$.
Angle at L: Angle 1.
Sum = $180$.
Angle 1 = $180 - 42.5 - 77.5 = 60^\circ$.
Matches the secant-tangent formula result ($60^\circ$).
So this model is consistent.

Summary of Geometry:
- Tangent LM at M.
- Secant L-R-S (Line LRS).
- Secant P-R-M (Line PRM).
- Secant P-S-Q (Line PSQ).

Calculations:

1. Angle 1:
Calculated above: $60^\circ$.

2. Angle 2:
Angle between Tangent LM and Chord MR?
Wait, Angle 2 is labeled inside the circle?
Diagram: Angle 2 is $\angle LMR$? No, L is outside.
Angle 2 is $\angle RMS$?
Let's look at the position of "2".
It is between Chord MR and Chord MS?
If so, Angle 2 = $\angle RMS$.
Intercepts Arc $RS$? No, Arc $RS$ is 70.
Angle $\angle RMS$ intercepts Arc $RS$?
Vertex M. Rays MR, MS.
Intercepts Arc $RS = 70$.
Angle 2 = $35^\circ$.

Wait, earlier I calculated Angle 2 as 42.5 assuming it was $\angle LMR$.
Let's look at the label "2" again.
It is adjacent to "85°".
The arc 85 is Arc $MR$.
Angle 2 is inside the triangle MRS?
If Angle 2 is $\angle RMS$, it is $35^\circ$.
If Angle 2 is $\angle LMR$, it is $42.5^\circ$.
Usually, numbers inside the circle refer to inscribed angles.
Number 1 is outside (Vertex L).
Number 2 is at Vertex M, inside the circle.
So Angle 2 = $\angle RMS$?
But there is also Angle 3 and 4.
Angle 3 is $\angle SMR$? No.
Let's look at the sectors at M.
Sector 1: Outside (Tangent-Chord). Label 2?
Sector 2: Inside (Chord-Chord). Label 3?

Let's re-read the diagram carefully.
Label "2" is between Tangent LM and Chord MR?
Or between Chord MR and Chord MS?
The arc 85 is marked for Arc $MR$.
The angle subtending Arc $MR$ from the tangent is $\frac{85}{2} = 42.5$.
The angle subtending Arc $MR$ from the circumference (e.g., $\angle MSR$) is $42.5$.

Let's look at Label "3".
It is between Chord MR and Chord MS?
If so, Angle 3 = $\angle RMS$. Intercepts Arc $RS$? No.
Angle $\angle RMS$ intercepts Arc $RS$?
Vertex M. Endpoints R, S.
Arc $RS = 70$.
Angle 3 = $35^\circ$.

Label "4".
Between Chord MS and Chord MQ?
Angle 4 = $\angle SMQ$. Intercepts Arc $SQ = 55$.
Angle 4 = $27.5^\circ$.

Label "5".
Between Chord MQ and Tangent MN?
Angle 5 = $\angle QMN$. Intercepts Arc $MQ = 150$.
Angle 5 = $75^\circ$.

Check Sum at M:
Angle(Tangent-Chord MR) + Angle 3 + Angle 4 + Angle(Tangent-Chord MQ)?
Wait, Angle 2 is likely the Tangent-Chord angle $\angle LMR$.
If Angle 2 = $\angle LMR = 42.5^\circ$.
Then Angle 3 = $\angle RMS = 35^\circ$.
Angle 4 = $\angle SMQ = 27.5^\circ$.
Angle 5 = $\angle QMN = 75^\circ$.
Sum: $42.5 + 35 + 27.5 + 75 = 180^\circ$.
This fits perfectly on the straight line tangent.

So:
Angle 2 = $42.5^\circ$
Angle 3 = $35^\circ$
Angle 4 = $27.5^\circ$
Angle 5 = $75^\circ$

3. Angle 6, 7, 8 at Q:
We established:
Angle 6 = $\angle MQR$. Intercepts Arc $MR$?
Wait, earlier I said Angle 6 = $\angle MQR$.
Does $\angle MQR$ intercept Arc $MR$?
Vertex Q. Endpoints M, R.
Arc $MR = 85$.
Angle 6 = $42.5^\circ$.

Angle 7 = $\angle RQS$. Intercepts Arc $RS = 70$.
Angle 7 = $35^\circ$.

Angle 8 = $\angle SQP$.
We need Angle $\angle SQP$.
Consider $\triangle PQS$.
Angle at P = 40.
Angle at S = Angle 10?
Angle at Q = Angle 8?

Let's find Angle 10 first.

4. Angles at S:
Label 9: $\angle MSR$?
Vertex S. Endpoints M, R.
Arc $MR = 85$.
Angle 9 = $42.5^\circ$.

Label 10: $\angle RSP$?
Line P-S-Q is straight.
Angle $\angle RSQ$?
Angle $\angle RSQ$ intercepts Arc $RQ$?
Arc $RQ$ (not containing S) = Arc $RM$ + Arc $MQ$?
Arc $RM = 85$. Arc $MQ = 150$.
Sum = 235.
Angle $\angle RSQ = \frac{235}{2} = 117.5^\circ$.

Angle 10 and Angle $\angle RSQ$ are supplementary?
Because P-S-Q is a line.
Angle 10 + Angle $\angle RSQ$ = 180.
Angle 10 = $180 - 117.5 = 62.5^\circ$.

5. Back to Angle 8:
In $\triangle PQS$:
Sum of angles = 180.
Angle P = 40.
Angle S = Angle 10 = 62.5.
Angle Q = Angle 8.
Angle 8 = $180 - 40 - 62.5 = 77.5^\circ$.

Let's double check Angle 8 using exterior angle theorem?
Angle $\angle RQS$ (Angle 7) = 35.
Angle $\angle MQR$ (Angle 6) = 42.5.
Angle $\angle MQS$ = $35 + 42.5 = 77.5$.
Angle $\angle MQP$?
Line P-S-Q.
Angle 8 is $\angle SQP$? No, Angle 8 is $\angle PQS$?
Wait, Angle 8 is inside the triangle PQS?
Yes, Angle 8 is $\angle PQS$.
So Angle 8 = $77.5^\circ$.

Wait, is Angle 8 $\angle PQS$ or $\angle RQP$?
Diagram: 6, 7, 8 are adjacent.
6 is $\angle MQR$.
7 is $\angle RQS$.
8 is $\angle SQP$.
Yes.

So:
Angle 6 = $42.5^\circ$
Angle 7 = $35^\circ$
Angle 8 = $77.5^\circ$

6. Angles at R:
Label 12: $\angle PRS$?
We calculated Angle 12 = $77.5^\circ$ earlier.
Let's verify.
Angle $\angle MRS$ (inscribed) intercepts Arc $MS$?
Arc $MS$ (via Q) = $150 + 55 = 205$.
Angle $\angle MRS = 102.5^\circ$.
Angle 12 is supplementary to $\angle MRS$ (Linear Pair on Line PRM).
Angle 12 = $180 - 102.5 = 77.5^\circ$.

Label 13: $\angle SRQ$?
Inscribed angle intercepting Arc $SQ$?
Arc $SQ = 55$.
Angle 13 = $\frac{55}{2} = 27.5^\circ$.

Label 14: $\angle QRM$?
Inscribed angle intercepting Arc $QM$?
Arc $QM = 150$.
Angle 14 = $\frac{150}{2} = 75^\circ$.

Check Sum at R on Line PRM:
Angle 12 + Angle 13 + Angle 14?
$77.5 + 27.5 + 75 = 180$.
Perfect.

So:
Angle 12 = $77.5^\circ$
Angle 13 = $27.5^\circ$
Angle 14 = $75^\circ$

7. Remaining Angles:
Label 15: $\angle LRM$?
Vertical to Angle 12? No.
Line L-R-S and Line P-R-M intersect at R.
Angle 15 and Angle $\angle PRS$ (Angle 12) are vertical?
No, Angle 15 is $\angle LRM$?
If L-R-S is a line, and P-R-M is a line...
Angle $\angle LRM$ and Angle $\angle PRS$ are vertical?
No, $\angle LRM$ and $\angle SRP$?
L-R-S is a line. P-R-M is a line.
Angle $\angle LRP$ and Angle $\angle SRM$ are vertical.
Angle $\angle LRM$ and Angle $\angle PRS$ are vertical?
Yes.
So Angle 15 = Angle 12 = $77.5^\circ$.

Label 11: Given as $40^\circ$.

Label 16, 17, 18, 19 at Center O?
No, O is the center.
Angles 16, 17, 18, 19 are around the center?
Diagram shows lines from O to vertices?
No, O