Question

1. lines ad and ec meet at point b.

which of these must be true?
select all that apply.
a. a 180° clockwise rotation using center b takes d to a.
b. the image of d after a 180° rotation using center b lies on
ray ba.
c. if a 180° rotation using center b takes c to e, then it also
takes e to c.
d. angle abc is congruent to angle dbe.
e. angle abe is congruent to angle abc.

Explanation:

Step1: Analyze rotation properties

A \(180^{\circ}\) rotation about center \(B\) swaps points across \(B\). So a \(180^{\circ}\) clock - wise rotation using center \(B\) takes \(D\) to \(A\) (since \(B\) is the mid - point of \(AD\)).

Step2: Check image of \(D\) after \(180^{\circ}\) rotation

The image of \(D\) after a \(180^{\circ}\) rotation about \(B\) lies on ray \(BA\) because of the straight - line nature of \(AD\) through \(B\).

Step3: Consider rotation of \(C\) and \(E\)

If a \(180^{\circ}\) rotation about \(B\) takes \(C\) to \(E\) (as \(B\) is the mid - point of \(CE\)), then by the property of \(180^{\circ}\) rotation (inverse of itself), it also takes \(E\) to \(C\).

Step4: Examine angle congruence

Angle \(ABC\) and angle \(DBE\) are vertical angles. Vertical angles are congruent.

Step5: Analyze angle \(ABE\) and \(ABC\)

Angle \(ABE\) and angle \(ABC\) are not congruent in general. \(ABE\) and \(ABC\) are adjacent angles formed by intersecting lines \(AD\) and \(EC\) at \(B\), and their measures are not necessarily equal.

Answer:

A. A \(180^{\circ}\) clockwise rotation using center \(B\) takes \(D\) to \(A\).
B. The image of \(D\) after a \(180^{\circ}\) rotation using center \(B\) lies on ray \(BA\).
C. If a \(180^{\circ}\) rotation using center \(B\) takes \(C\) to \(E\), then it also takes \(E\) to \(C\).
D. Angle \(ABC\) is congruent to angle \(DBE\).