Question

1. select all true statements about the figure. a c + b = d + c b d + b = 180 c rotate clockwise by angle abc using center b. then angle cbd is the image of angle abe. d rotate 180 degrees using center b. then angle cbd is the image of angle eba. e reflect across the angle bisector of angle abc. then angle cbd is the image of angle abe. f reflect across line ce. then angle cbd is the image of angle eba.

Explanation:

Step1: Recall angle - relationship properties

Vertical angles are equal. Adjacent angles that form a straight - line sum to 180 degrees. Here, \(a\) and \(c\) are vertical angles, \(b\) and \(d\) are vertical angles. So \(a = c\) and \(b = d\). Then \(c + b=d + c\) (since \(b = d\)), so statement A is true. Also, \(d\) and \(b\) are adjacent angles that form a straight - line, so \(d + b=180\), statement B is true.

Step2: Analyze rotation and reflection properties

For rotation: A 180 - degree rotation about a point maps an angle to its opposite. For reflection: Reflection across an angle bisector divides the angle into two congruent parts.

• For statement C: Rotating clockwise by angle \(ABC\) about center \(B\) does not necessarily mean that angle \(CBD\) is the image of angle \(ABE\).
• For statement D: Rotating 180 degrees using center \(B\), angle \(CBD\) is the image of angle \(EBA\) because a 180 - degree rotation about a point maps an angle to its opposite.
• For statement E: Reflecting across the angle bisector of angle \(ABC\) does not imply that angle \(CBD\) is the image of angle \(ABE\).
• For statement F: Reflecting across line \(CE\) does not mean that angle \(CBD\) is the image of angle \(EBA\).

Answer:

A. \(c + b=d + c\)
B. \(d + b = 180\)
D. Rotate 180 degrees using center \(B\). Then angle \(CBD\) is the image of angle \(EBA\)