Question

statue a company is making different size statues that are in the shape of hour glasses. use the figure to find the missing measure, x°. the figure is not drawn to scale.

Explanation:

Step1: Recall angle - sum property of a triangle

The sum of interior angles of a triangle is 180°.

Step2: Consider the top - triangle

Let the third angle of the top - triangle be \(y\). For the top - triangle with angles \(70^{\circ}\), \(x^{\circ}\), and \(y\), we have \(70 + x+y=180\), so \(y = 180-(70 + x)=110 - x\).

Step3: Consider the bottom - triangle

The vertical angles are equal. The vertical angle to \(y\) in the bottom - triangle and the other two angles of the bottom - triangle are related by the angle - sum property. The vertical angle to \(y\) in the bottom - triangle is also \(y\), and the other two angles of the bottom - triangle are equal to the non - vertical angles of the top - triangle. Let's use the fact that the vertical angles are equal. The vertical angle to \(y\) and the other two angles of the bottom - triangle give \(y+62 + z=180\) (where \(z\) is the third angle of the bottom - triangle). But since the two triangles are related by vertical angles, we can also use the property that the sum of angles around the intersection point of the two lines forming the hour - glass shape.
Another way is to note that the two triangles are similar (by AA similarity, as the vertical angles are equal).
We know that for the two triangles, the sum of angles in each triangle is 180°.
For the top triangle: \(70+x + y=180\) and for the bottom triangle \(62 + y+z = 180\). Since the vertical angles are equal (\(y\) is the vertical angle), we can also use the fact that the sum of angles in the two - triangle system.
We know that the sum of angles in the two triangles: \(2\times180\) (total of both triangles) and the vertical angles are counted twice.
Since the vertical angles are equal, we can directly set up the equation for one of the triangles.
For the top triangle, using the angle - sum property of a triangle:
Let's consider the fact that the vertical angles are equal. The vertical angle between the two triangles is common.
We know that \(70+x+(180 - 62)=180\) (because the non - \(x\) and non - 70 angle in the top triangle and the non - 62 angle in the bottom triangle are vertical angles).
\[70+x + 118=180\]
\[x=180-(70 + 118)\]
\[x = 180 - 188\] is wrong.
The correct way:
Since the vertical angles are equal, for the top triangle with angles \(70^{\circ}\), \(x^{\circ}\), and the vertical - angle \(v\) and the bottom triangle with angles \(62^{\circ}\), \(v\), and the third angle.
We know that \(70+x=62+(180 - v)\) and \(v\) (vertical angle) cancels out when we use the fact that the sum of angles in a triangle is 180°.
The two triangles are related such that \(70+x=62+(180 - 70 - x)\)
\[70+x=62 + 110 - x\]
\[x+x=62 + 110 - 70\]
\[2x=102\]
\[x = 51\]

Answer:

51