Circles - Central and Inscribed Angles Color-By-Number Worksheet - Free Printable

Here are the step-by-step solutions for each problem on the worksheet.

1.
The central angle $x$ and the arc measure $345^\circ$ make a full circle ($360^\circ$).
$x = 360 - 345$
$x = 15$

2.
The central angle is equal to its intercepted arc. The arc is labeled $28x$.
So, $x = 28x$ is not right; looking closely, the angle is $x$ and the arc is $(28)^\circ$. Wait, the label is usually just the number. Let's look at the pattern. In #1, arc is 345. In #2, the arc has "28" inside it? No, it looks like $m \text{ arc} = 28^\circ$. If the angle is $x$, then $x = 28$.
*Correction*: Looking at other problems, sometimes there is an algebraic expression. In #2, the text inside the arc seems to be just "28". So, $x = 28$.

3.
The reflex angle (the big one outside) is $x$. The inner central angle is $122^\circ$. They add up to $360^\circ$.
$x + 122 = 360$
$x = 360 - 122$
$x = 238$

4.
This is an inscribed angle. The rule is: Angle = $\frac{1}{2} \times \text{Arc}$.
The arc is $(11x + 2)^\circ$. The angle is $(7x - 4)^\circ$.
$7x - 4 = \frac{1}{2}(11x + 2)$
Multiply both sides by 2 to clear the fraction:
$2(7x - 4) = 11x + 2$
$14x - 8 = 11x + 2$
Subtract $11x$ from both sides:
$3x - 8 = 2$
Add 8 to both sides:
$3x = 10$
$x = \frac{10}{3}$ or $3.33$

5.
This is an inscribed angle. The angle is $(9x + 14)^\circ$. The arc is $(16x - 14)^\circ$.
Angle = $\frac{1}{2} \times \text{Arc}$
$9x + 14 = \frac{1}{2}(16x - 14)$
Multiply by 2:
$2(9x + 14) = 16x - 14$
$18x + 28 = 16x - 14$
Subtract $16x$ from both sides:
$2x + 28 = -14$
Subtract 28 from both sides:
$2x = -42$
$x = -21$
*(Note: A negative value for x is unusual in geometry problems unless the expressions are just algebraic placeholders. Let's re-read the numbers. Angle: $9x+14$. Arc: $16x-14$. Calculation is correct based on those numbers.)*

6.
Inscribed angle $(11x - 2)^\circ$ intercepts arc $(13x + 10)^\circ$.
$11x - 2 = \frac{1}{2}(13x + 10)$
Multiply by 2:
$2(11x - 2) = 13x + 10$
$22x - 4 = 13x + 10$
Subtract $13x$:
$9x - 4 = 10$
Add 4:
$9x = 14$
$x = \frac{14}{9}$ or $1.55$

7.
Central angle $x^\circ$ and arc $277^\circ$. They form a circle.
$x + 277 = 360$
$x = 360 - 277$
$x = 83$

8.
Inscribed angle $(10x + 22)^\circ$ intercepts arc $332^\circ$.
$10x + 22 = \frac{1}{2}(332)$
$10x + 22 = 166$
Subtract 22:
$10x = 144$
Divide by 10:
$x = 14.4$

9.
Inscribed angle $(2x + 64)^\circ$ intercepts arc $(6x + 8)^\circ$.
$2x + 64 = \frac{1}{2}(6x + 8)$
Multiply by 2:
$2(2x + 64) = 6x + 8$
$4x + 128 = 6x + 8$
Subtract $4x$:
$128 = 2x + 8$
Subtract 8:
$120 = 2x$
Divide by 2:
$x = 60$

10.
Inscribed angle $(3x + 27)^\circ$ intercepts arc $(10x - 14)^\circ$.
$3x + 27 = \frac{1}{2}(10x - 14)$
Multiply by 2:
$2(3x + 27) = 10x - 14$
$6x + 54 = 10x - 14$
Subtract $6x$:
$54 = 4x - 14$
Add 14:
$68 = 4x$
Divide by 4:
$x = 17$

11.
Inscribed angle $(x + 19)^\circ$ intercepts arc $(7x + 1)^\circ$.
$x + 19 = \frac{1}{2}(7x + 1)$
Multiply by 2:
$2(x + 19) = 7x + 1$
$2x + 38 = 7x + 1$
Subtract $2x$:
$38 = 5x + 1$
Subtract 1:
$37 = 5x$
Divide by 5:
$x = 7.4$

12.
Inscribed angle $(12x + 17)^\circ$ intercepts arc $(3x + 11)^\circ$.
Wait, visually the angle looks larger than the arc? No, the arc is the major arc? No, standard position.
Let's check the equation:
$12x + 17 = \frac{1}{2}(3x + 11)$
Multiply by 2:
$24x + 34 = 3x + 11$
Subtract $3x$:
$21x + 34 = 11$
Subtract 34:
$21x = -23$
$x = -1.09$
*(Note: Like #5, this results in a negative number. Based strictly on the visible numbers, this is the mathematical result.)*

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Color Key Calculations:
Now we match the $x$ values to the color key.

* Orange: $x = 15$ (Matches Problem 1)
* Green: $x = 11$ (No exact match found in integer solutions above. Problem 10 is 17, Problem 9 is 60. Let's re-check Problem 6: $14/9$. Problem 4: $10/3$. Problem 5: -21. Problem 12: -1.09. Problem 2: 28. Problem 3: 238. Problem 7: 83. Problem 8: 14.4. Problem 11: 7.4.)
* *Self-Correction*: Let's look really closely at Problem 6 again. Maybe the arc is $13x + 10$? Yes. Maybe the angle is $11x - 2$? Yes. Result $14/9$.
* Let's look at Problem 4 again. Angle $7x-4$, Arc $11x+2$. Result $10/3$.
* Let's look at Problem 5 again. Angle $9x+14$, Arc $16x-14$. Result $-21$.
* Let's look at Problem 12 again. Angle $12x+17$, Arc $3x+11$. Result negative.
* Let's look at Problem 2 again. If the arc is $28x$ and angle is $x$, then $x=28x \rightarrow x=0$. Unlikely. If angle is $x$ and arc is $28$, $x=28$.
* Let's look at Problem 11 again. $x+19 = 0.5(7x+1) \rightarrow 2x+38=7x+1 \rightarrow 37=5x \rightarrow 7.4$.
* Let's look at Problem 10 again. $3x+27 = 0.5(10x-14) \rightarrow 6x+54=10x-14 \rightarrow 68=4x \rightarrow 17$.
* Let's look at Problem 9 again. $2x+64 = 0.5(6x+8) \rightarrow 4x+128=6x+8 \rightarrow 120=2x \rightarrow 60$.
* Let's look at Problem 8 again. $10x+22 = 0.5(332) \rightarrow 10x+22=166 \rightarrow 10x=144 \rightarrow 14.4$.
* Let's look at Problem 7 again. $360-277=83$.
* Let's look at Problem 3 again. $360-122=238$.
* Let's look at Problem 1 again. $360-345=15$.

It seems many answers are decimals or don't match the key integers perfectly. However, typically in these worksheets, the answers correspond directly. Let's re-read the blurry numbers.

*Re-evaluating Problem 6*: Angle $(11x-2)$, Arc $(13x+10)$. $22x-4 = 13x+10 \rightarrow 9x=14$. Still fractional.
*Re-evaluating Problem 4*: Angle $(7x-4)$, Arc $(11x+2)$. $14x-8=11x+2 \rightarrow 3x=10$. Still fractional.
*Re-evaluating Problem 5*: Angle $(9x+14)$, Arc $(16x-14)$. $18x+28=16x-14 \rightarrow 2x=-42$. Negative.
*Re-evaluating Problem 12*: Angle $(12x+17)$, Arc $(3x+11)$. $24x+34=3x+11 \rightarrow 21x=-23$. Negative.

There might be misinterpretation of the blurry digits. For example, in #12, is the angle $1.2x$? No. Is the arc $3x+71$?
If Arc in #12 was $3x+71$: $24x+34=3x+71 \rightarrow 21x=37$. No.
If Angle in #12 was $1x+17$: $2x+34=3x+11 \rightarrow x=23$. Black is 27. Close.

Despite potential ambiguities in the image quality leading to non-integer results for some, I will provide the calculated values based on the most legible interpretation.

Final Answer:
1. x = 15
2. x = 28
3. x = 238
4. x = 3.33 (10/3)
5. x = -21
6. x = 1.55 (14/9)
7. x = 83
8. x = 14.4
9. x = 60
10. x = 17
11. x = 7.4
12. x = -1.09 (-23/21)