Answer :
Let's carefully analyze the given question:
### Translation on a Coordinate Plane
When a polygon GHIJ translates 8 units to the left to form polygon G'H'I'J', each vertex of the polygon moves 8 units left along the x-axis. In a translation, each corresponding segment and angle of the polygon remains congruent (the same length and measure, respectively) because a translation is an isometry, meaning it preserves distances and angles.
### Evaluating Each Option Given in the Question:
#### Option A. \( GH = G'H' \)
This statement states that the length of segment GH is equal to the length of segment G'H'.
Since translations preserve distances, \( GH \) will indeed be equal to \( G'H' \). This statement is true.
#### Option B. \( G'G = 8 \) units
This statement states that the distance between the original point G and the translated point G' is 8 units.
Since the polygon translates 8 units to the left, the distance between point G and point G' (which is the same point G moved 8 units left) is indeed 8 units. This statement is true.
#### Option C. \( m\angle HIJ = m\angle H'I'J' \)
This statement states that the measure of angle \( HIJ \) in the original polygon is equal to the measure of angle \( H'I'J' \) in the translated polygon.
Since translations also preserve angles, \( m\angle HIJ \) will indeed be equal to \( m\angle H'I'J' \). This statement is true.
#### Option D. \( m\angle HIYJ = m\angle HIJ \)
This statement seems to introduce an unfamiliar angle notation \( m\angle HIYJ \). If we assume any additional segment or angle (which does not exist among G, H, I, and J or any notations typically used for angles), it implies an irrelevant or unidentified transformation or an extra vertex Y, which is not clarified. As a result, there is some ambiguity in the interpretation of \( m\angle HIYJ \), which is inappropriate without further information.
Given the context of typical notation in geometry and translations on the coordinate plane, Option D doesn't seem to follow the direct consequences or standard interpretation of a pure translation.
### Conclusion:
By logical deduction and understanding the properties of translations on a coordinate plane, the equation that is not necessarily true is:
Option D. [tex]\( m\angle HIYJ = m\angle HIJ \)[/tex]
### Translation on a Coordinate Plane
When a polygon GHIJ translates 8 units to the left to form polygon G'H'I'J', each vertex of the polygon moves 8 units left along the x-axis. In a translation, each corresponding segment and angle of the polygon remains congruent (the same length and measure, respectively) because a translation is an isometry, meaning it preserves distances and angles.
### Evaluating Each Option Given in the Question:
#### Option A. \( GH = G'H' \)
This statement states that the length of segment GH is equal to the length of segment G'H'.
Since translations preserve distances, \( GH \) will indeed be equal to \( G'H' \). This statement is true.
#### Option B. \( G'G = 8 \) units
This statement states that the distance between the original point G and the translated point G' is 8 units.
Since the polygon translates 8 units to the left, the distance between point G and point G' (which is the same point G moved 8 units left) is indeed 8 units. This statement is true.
#### Option C. \( m\angle HIJ = m\angle H'I'J' \)
This statement states that the measure of angle \( HIJ \) in the original polygon is equal to the measure of angle \( H'I'J' \) in the translated polygon.
Since translations also preserve angles, \( m\angle HIJ \) will indeed be equal to \( m\angle H'I'J' \). This statement is true.
#### Option D. \( m\angle HIYJ = m\angle HIJ \)
This statement seems to introduce an unfamiliar angle notation \( m\angle HIYJ \). If we assume any additional segment or angle (which does not exist among G, H, I, and J or any notations typically used for angles), it implies an irrelevant or unidentified transformation or an extra vertex Y, which is not clarified. As a result, there is some ambiguity in the interpretation of \( m\angle HIYJ \), which is inappropriate without further information.
Given the context of typical notation in geometry and translations on the coordinate plane, Option D doesn't seem to follow the direct consequences or standard interpretation of a pure translation.
### Conclusion:
By logical deduction and understanding the properties of translations on a coordinate plane, the equation that is not necessarily true is:
Option D. [tex]\( m\angle HIYJ = m\angle HIJ \)[/tex]
