When a geometry diagram is complex, break it down into simpler shapes and identify the basic rules that apply to each (e.g., sum of angles in a triangle, angles on a straight line). Solve for one variable first, then use that result to find the others.
Step 1: Understanding the Concept:
This problem requires us to find the values of angles \(x\) and \(y\) from a geometric diagram by applying properties of angles, such as those in a triangle and vertically opposite angles. Step 2: Key Formula or Approach:
1. The sum of angles in a triangle is 180\(^\circ\).
2. Vertically opposite angles formed by intersecting lines are equal.
3. Angles on a straight line add up to 180\(^\circ\). Step 3: Detailed Explanation:
Let's analyze the diagram. It shows two intersecting lines. A triangle is formed, with one of its vertices at the intersection point.
- Inside the triangle, we see angles labeled \(x\) and \(2x\). There is also a right-angle symbol, indicating an angle of 90\(^\circ\). The angle labeled \(x\) in the diagram is one of the triangle's internal angles.
- The sum of the angles in this triangle is 180\(^\circ\). Therefore, we can set up an equation:
\[ x + 2x + 90^\circ = 180^\circ \]
\[ 3x = 180^\circ - 90^\circ \]
\[ 3x = 90^\circ \]
\[ x = 30^\circ \]
- Now, we need to find the value of \(y\). The angle \(y\) and the angle \(x\) inside the triangle are vertically opposite to each other at the intersection point.
- Therefore, \(y\) must be equal to the sum of the other two internal angles of the triangle, but this seems incorrect. Let's re-examine the diagram.
- A more plausible interpretation is that the angle \(y\) and the angle represented by the sum `(angle in triangle) + (right angle)` are vertically opposite. However, the most direct interpretation is that \(x\) and \(y\) are adjacent angles on a straight line along with another angle.
- Let's assume the standard interpretation where the two intersecting lines form four angles. The angle \(x\) inside the triangle is vertically opposite to another angle \(x\) outside the triangle. Angle \(y\) is shown as another angle at that same intersection. If \(x\) and \(y\) are adjacent angles, they sum to 180\(^\circ\).
\[ x + y = 180^\circ \]
With \(x = 30^\circ\), we can find \(y\):
\[ 30^\circ + y = 180^\circ \]
\[ y = 150^\circ \]
Now we can compare the quantities in Column A and Column B. For Column A:
\(y = 150\). For Column B:
\(4x = 4 \times 30 = 120\). Comparison:
Column A is 150 and Column B is 120. Since \(150>120\), the quantity in Column A is greater. Step 4: Final Answer:
By solving for \(x\) using the triangle and then for \(y\) using the straight-line property, we find that \(y=150\) and \(4x=120\). Therefore, Column A is greater.
