Question

here is a diagram. 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° 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: Analyze option A

Subtract \(c\) from both sides of \(c + b=d + c\), we get \(b = d\). Since vertical - angles are equal (\(\angle ABE\) and \(\angle CBD\) are vertical angles, \(\angle EBC\) and \(\angle ABD\) are vertical angles), \(b = d\) is True.

Step2: Analyze option B

\(\angle EBC\) and \(\angle CBD\) form a linear - pair. \(\angle EBC=d\) and \(\angle CBD = b\), so \(d + b=180^{\circ}\) is True.

Step3: Analyze option C

A clock - wise rotation by \(\angle ABC\) using center \(B\) will not map \(\angle ABE\) to \(\angle CBD\). The rotation angle is wrong for this mapping.

Step4: Analyze option D

A \(180^{\circ}\) rotation about point \(B\) will map \(\angle EBA\) to \(\angle CBD\) because a \(180^{\circ}\) rotation about a point swaps the positions of angles on opposite sides of the center of rotation.

Step5: Analyze option E

Reflecting across the angle - bisector of \(\angle ABC\) will not map \(\angle ABE\) to \(\angle CBD\) as the angle - bisector is not a line of symmetry for this mapping.

Step6: Analyze option F

Reflecting across line \(CE\) will map \(\angle EBA\) to \(\angle CBD\) since line \(CE\) is a line of symmetry for the angles formed by the intersection of the two lines at point \(B\).

Answer:

A. \(c + b=d + c\)
B. \(d + b = 180\)
D. Rotate \(180^{\circ}\) using center \(B\). Then angle \(CBD\) is the image of angle \(EBA\).
F. Reflect across line \(CE\). Then angle \(CBD\) is the image of angle \(EBA\).