Find Angle GJO Given Angles 21°, 33°, 42° in a Circle

Answer

Based on the geometric properties of the circle provided, the size of angle $\angle GJO$ is $21^\circ$. This result is derived using the "Angles in the Same Segment" theorem, which states that angles subtended by the same arc at the circumference are equal.

Explanation

The image shows a circle containing four points ($G, H, I, J$) on the circumference. There are two intersecting chords, $GI$ and $JH$, which meet at point $O$. Your handwritten notes correctly identify several supplementary and vertical angles around point $O$, but to find $\angle GJO$, we must look at the arcs.

1. Identify the arc subtending the required angle The angle we need to find is $\angle GJO$. This angle is an inscribed angle that "opens up" to arc $GO$. ⚠️ This step is required on exams to justify which circle theorem you are applying.

2. Find other angles subtended by the same arc Look for any other angle on the circumference that subtends the same arc $GO$. Following the lines from $G$ and $O$ to the circumference, we see that $\angle GHO$ (or $\angle GHI$) also subtends arc $GO$. However, the value for $\angle GHO$ is not immediately given. Instead, let's look at arc $HI$.

3. Apply the "Angles in the Same Segment" theorem The theorem states that angles subtended by the same arc at the circumference are equal. $\text{Angle } GJO = \text{Angle } GHO \quad (\text{subtended by arc } GO)$ $\text{Angle } OGH = \text{Angle } OJI \quad (\text{subtended by arc } HI)$ $\text{Angle } OIJ = \text{Angle } OGH \text{ ? No, check the shared arc.}$ Let's re-examine arc $GH$. Both $\angle GIH$ and $\angle GJH$ subtend arc $GH$. Let's re-examine arc $JI$. Both $\angle JGI$ and $\angle JHI$ subtend arc $JI$. Let's re-examine arc $HI$. Both $\angle HGI$ and $\angle HJI$ subtend arc $HI$. From the given information: $\angle OGH = 21^\circ$ This angle $\angle OGH$ (or $\angle IGH$) subtends arc $HI$. Another angle subtending arc $HI$ is $\angle HJI$ (or $\angle OJI$). $\angle OJI = \angle OGH = 21^\circ$ This formula establishes that inscribed angles starting and ending at the same points on the circle's edge are identical in measure.

4. Solve for the specific requested angle The question asks for $\angle GJO$. We look at arc $GH$. The angles subtended by arc $GH$ are $\angle GIH$ and $\angle GJH$. From the prompt, $\angle OIJ$ is given as $42^\circ$. This angle subtends arc $GJ$. Another angle subtending arc $GJ$ is $\angle GHJ$. Wait, let's look at arc $GI$. The angles subtending arc $GI$ are $\angle GJI$ and $\angle GHI$. Actually, looking specifically at arc $HI$: $\angle HGI = 21^\circ$ (given). Therefore, $\angle HJI = 21^\circ$ (angles in the same segment). Note that $\angle HJI$ is the same as $\angle OJI$. Searching for $\angle GJO$: it subtends arc $GO$. Looking at the diagram, $\angle GHO$ also subtends arc $GO$. Since the triangles are formed by chords: $\angle GJO = \angle GHO$ In Triangle $GHO$, we can find $\angle GHO$ if we know the other angles. However, there is a simpler path. Look at arc $IJ$. The angles subtending it are $\angle IGJ$ and $\angle IHJ$. $\angle IGJ = 33^\circ \implies \angle IHJ = 33^\circ$ Look at arc $GH$. The angles subtending it are $\angle GIH$ and $\angle GJH$. The angle $\angle GIH$ is not explicitly given, but we have $\angle OIJ = 42^\circ$ (subtends arc $GJ$). Therefore, $\angle GHJ$ must also be $42^\circ$. Finally, for $\angle GJO$, we look at arc $GO$. This is equal to $\angle GHO$. In Triangle $OGH$: $\angle GHO = 180^\circ - \text{Angle } GOH - \text{Angle } OGH$ From your handwriting, $\angle GOH = 117^\circ$. $\angle GHO = 180^\circ - 117^\circ - 21^\circ = 42^\circ$ The sum of interior angles in any triangle must equal 180 degrees.

5. Final Calculation Since $\angle GJO = \angle GHO$: $\angle GJO = 42^\circ$ This numerical value represents the degree measure of the angle at vertex J.

Final Answer

$\boxed{42^\circ}$

Common Mistakes

• Confusing Arc Subtension: Students often assume opposite angles in the center intersection (vertical angles) are equal to the circumference angles. Always trace the "arms" of the angle back to the circle's edge to identify the arc.
• Assuming Isosceles Triangles: Unless the problem states the lines pass through the center (radii), do not assume $OG = OH$ or $OJ = OI$. Only the angles subtended by the same arc are guaranteed to be equal.