Let's analyze each statement to determine which is not necessarily true when polygon GHIJ translates 8 units to the left to form polygon [tex]\( G' H' I J' \)[/tex]:
### Statement A: [tex]\( GH = G' H' \)[/tex]
When a polygon is translated on the coordinate plane, all points of the polygon move the same distance and in the same direction. A translation does not alter the shape or the size of the polygon.
- Since both [tex]\( G \)[/tex] and [tex]\( H \)[/tex] are translated by the same distance (8 units to the left), the distance between [tex]\( G \)[/tex] and [tex]\( H \)[/tex] remains the same. Therefore, [tex]\( GH = G' H' \)[/tex].
Thus, Statement A is true.
### Statement B: [tex]\( G'G = 8 \)[/tex] units
Translation means shifting every point of the polygon equally. In this case, each point is moving 8 units to the left.
- Therefore, the distance between the original point [tex]\( G \)[/tex] and the translated point [tex]\( G' \)[/tex] will be exactly 8 units.
Thus, Statement B is true.
### Statement C: [tex]\( m \angle HIJ = m \angle H' IJ' \)[/tex]
Angles in polygons remain the same after translation. However, in this statement, the notation [tex]\( \angle HIJ \)[/tex] compared to [tex]\( \angle H' IJ' \)[/tex] involves checking if an angle remains the same after translation. Here, the notation suggests checking if:
- The angle [tex]\( \angle HIJ \)[/tex] formed by the original points [tex]\( H, I, J \)[/tex]
- Equals the angle [tex]\( \angle H'IJ' \)[/tex] involving [tex]\( H', I, J' \)[/tex], which may imply additional translations not indicated in the problem statement
Compare the angles before and after translation:
- If [tex]\( H, I, \)[/tex] and [tex]\( J \)[/tex] translate to [tex]\( H', I', \)[/tex] and [tex]\( J' \)[/tex], the angle [tex]\( H'IJ' \)[/tex] does not correctly correspond to a simple translation.
Thus, Statement C is not necessarily true.
### Conclusion
After evaluating all the statements, we have determined that:
- Statement A is true.
- Statement B is true.
- Statement C is not necessarily true.
Therefore, the equation that is not necessarily true is the assertion in Statement C: [tex]\( m \angle HIJ = m \angle H'IJ' \)[/tex].
Answer: 3.
