Question

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
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
f. reflect across line ce. then angle cbd is the image of angle eba

Explanation:

Step1: Analyze option a

Since \(c + b\) and \(d + c\) are not equal in general (no geometric property shown for this equality), option a is false.

Step2: Analyze option b

\(d + b\) is not equal to \(180^{\circ}\) in general. There is no indication from the geometric figure (no straight - line or supplementary - angle - forming property for \(d\) and \(b\) shown), so option b is false.

Step3: Analyze option c

When we rotate clockwise by \(\angle ABC\) about center \(B\), the transformation rules for rotation (a rigid transformation that preserves angles and side - lengths) make \(\angle CBD\) the image of \(\angle EBA\).

Step4: Analyze option d

A \(180^{\circ}\) rotation about center \(B\) (a rotation that is equivalent to a point - reflection about \(B\)) does not map \(\angle EBA\) to \(\angle CBD\) in the way described. A \(180^{\circ}\) rotation would reverse the orientation in a different manner than what is required for the angle - image relationship.

Step5: Analyze option e

Reflecting across the angle bisector of \(\angle ABC\) (a reflection that is a non - rotational symmetry) does not map \(\angle EBA\) to \(\angle CBD\) as per the geometric figure's properties.

Step6: Analyze option f

Reflecting across line \(CE\) (a reflection that is a non - rotational symmetry) does not map \(\angle EBA\) to \(\angle CBD\) as per the geometric figure's properties.

Answer:

c. Rotate clockwise by angle \(ABC\) using center \(B\). Then angle \(CBD\) is the image of angle \(EBA\)