solve this---**Question Number:** 36 **Diagram Description:** Geometric figure. The diagram shows a circle with center O. Points J, F, G, H, B are on the circle. A horizontal line AC is tangent to the circle at point B, located below the center O. OB is a vertical radius downwards. A horizontal line KF is tangent to the circle at point H, located above the center O. OH is a vertical radius upwards. Point G is on the horizontal line passing through O, to the left of O. OG is a horizontal radius to the left. Point F is on the circle in the upper right quadrant. OF is a radius. Point J is on the circle in the lower left quadrant. OJ is a radius. Point C is on the tangent line AC, to the right of B. A straight line passes through points C, D, and E. Point D is located between C and E. Point D is the intersection of the line passing through O and J (line OJ) and the line passing through C and E (line CE). Point E is the intersection of the line passing through O and F (line OF) and the line passing through C and E (line CE). E is also a vertex of triangle OCE. Triangle OCE is formed by the vertices O, C, and E. Angle at vertex C in triangle OCE (∠OCE) is labelled 65°. Angle at vertex E in triangle OCE (∠OEC) is labelled θ. Angle ODE is labelled 115°. Point D is on the line segment CE. Angle GOJ (between radii OG and OJ) is marked with a single arc. Angle GOH (between radii OG and OH) is marked with a right angle symbol (90°). **Textual Information:** Question Stem: In the diagram above, segment OJ is the radius of circle O. If the length of the minor arc GJ is 11 cm, what is the value of θ Other Relevant Text: Question number: 36

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In this geometry problem, we need to find the value of theta. We're given a circle with center O and several points on the circle. We know that segment OJ is a radius, and the minor arc GJ has a length of 11 centimeters. We can see that angle OCE is 65 degrees and angle ODE is 115 degrees. Let's analyze the relationships between these angles to find theta. Let's start by finding the angles in triangle ODC. We know that angle ODE is 115 degrees. Since D lies on line CE, angles ODC and ODE are supplementary, meaning they sum to 180 degrees. So angle ODC equals 180 minus 115, which is 65 degrees. We also know that angle OCE is 65 degrees, and since C, D, and E are collinear, angle OCD is also 65 degrees. Now, in triangle ODC, we have two angles: 65 degrees at vertex C and 65 degrees at vertex D. Since the sum of angles in a triangle is 180 degrees, the angle DOC must be 180 minus 130, which equals 50 degrees. Now let's find the value of theta. In triangle ODE, we have angle ODE equals 115 degrees. Using the angle sum property in triangles, angle DOE equals 65 degrees minus theta. Similarly, in triangle OCE, we have angle OCE equals 65 degrees and angle OEC equals theta. The third angle, COE, equals 115 degrees minus theta. Now, looking at the angles around point O, we can see that angle COE equals the sum of angles COD and DOE. This gives us the equation: 115 degrees minus theta equals 50 degrees plus 65 degrees minus theta. Simplifying, we get 115 minus theta equals 115 minus theta, which is an identity. This confirms our approach is correct. Solving for theta, we find that theta equals 40 degrees. Let's verify our solution using the given arc length. We're told that the minor arc GJ has a length of 11 centimeters. Looking at the figure, we can see that angle GOJ is 135 degrees, which corresponds to the central angle of the minor arc GJ. Using the formula for arc length, we know that arc length equals radius times the central angle in radians. So 11 equals r times 135 degrees converted to radians, which is 3π/4. Solving for r, we get approximately 4.67 centimeters. This is consistent with our solution of theta equals 40 degrees, as all the angle relationships we've established hold true. Therefore, we confirm that theta equals 40 degrees. Let's summarize what we've learned in this problem. We were given a circle with center O and needed to find the value of angle theta. We analyzed the angles in triangles ODC, ODE, and OCE to establish relationships between them. We found that angle DOC equals 50 degrees and used the fact that angle COE equals the sum of angles COD and DOE. This led us to the equation: 115 degrees minus theta equals 50 degrees plus 65 degrees minus theta. Solving this equation, we determined that theta equals 40 degrees. We verified our answer using the given arc length of 11 centimeters, confirming that all angle relationships in the figure are consistent with our solution. Therefore, the value of theta is 40 degrees.