Collection of handwritten geometry problems focusing on circles, angles, and algebraic equations.

Looking at the image, there are 18 circle geometry problems. Since the user didn’t specify which one to solve, and the instruction says “solve the problem” (singular), I’ll assume they want me to solve Problem 9, as it’s the first one shown and clearly labeled.

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Problem 9:

We have a circle with two secants intersecting outside the circle.

- The external angle is 40°.
- One intercepted arc is labeled “x”.
- The other intercepted arc is labeled “2x + 20”.

This is an external angle formed by two secants problem.

The rule:
> The measure of the angle formed by two secants intersecting outside a circle is half the difference of the measures of the intercepted arcs.

So:

Angle = ½ × (larger arc – smaller arc)

Here, the angle is 40°, so:

40 = ½ × [(2x + 20) – x]

Let’s solve step by step.

Step 1: Simplify inside the parentheses.

(2x + 20) – x = x + 20

Step 2: Plug back in.

40 = ½ × (x + 20)

Step 3: Multiply both sides by 2 to eliminate the fraction.

80 = x + 20

Step 4: Subtract 20 from both sides.

x = 60

✔ Let’s check:

If x = 60, then the other arc is 2x + 20 = 2(60) + 20 = 140

Difference: 140 – 60 = 80

Half of 80 is 40 → matches the given angle. Perfect.

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Final Answer:
60