In the figure shown, what is the value of \(v+x+y+z+w\)?
Show Hint
Memorize the result that the sum of the angles at the tips of a five-pointed star is always \(180^{\circ}\). This is a frequently tested concept in geometry problems on competitive exams. The general formula for the sum of the vertex angles of an n-pointed star is \((n-4) \times 180^{\circ}\).
Step 1: Understanding the Concept:
The problem asks for the sum of the angles at the five vertices (tips) of a five-pointed star (a pentagram). This is a classic result in Euclidean geometry. Step 2: Key Formula or Approach:
We can solve this using the exterior angle theorem, which states that the measure of an exterior angle of a triangle is equal to the sum of the measures of its two remote interior angles. Step 3: Detailed Explanation:
Let's label two of the intersection points in the star to create a triangle that contains the angle \(v\). Let these points be P and Q, such that the triangle at the top is \(\triangle vPQ\). The sum of angles in this triangle is:
\[ v + \angle vPQ + \angle vQP = 180^{\circ} \]
Now, let's analyze the angles \(\angle vPQ\) and \(\angle vQP\).
1. The angle \(\angle vPQ\) is an exterior angle to the triangle containing the vertices with angles \(x\) and \(z\). According to the exterior angle theorem, the measure of \(\angle vPQ\) is the sum of the two remote interior angles, \(x\) and \(z\).
\[ \angle vPQ = x + z \]
2. Similarly, the angle \(\angle vQP\) is an exterior angle to the triangle containing the vertices with angles \(y\) and \(w\). According to the exterior angle theorem, the measure of \(\angle vQP\) is the sum of the two remote interior angles, \(y\) and \(w\).
\[ \angle vQP = y + w \]
Now substitute these two expressions back into the equation for the sum of angles in \(\triangle vPQ\):
\[ v + (x + z) + (y + w) = 180^{\circ} \]
Rearranging the terms, we get:
\[ v + x + y + z + w = 180^{\circ} \]
Step 4: Final Answer:
The sum of the angles at the vertices of the five-pointed star is 180 degrees.
