there is a rectangle with an x reaching to each of it corners and in the middle of it is labeled O

Question

in a roof that the diagonals of rectangle ABCD are congruent, which statement comes just after the statements that line AD= line BC, line AB=lineAB, and angle DAB= angle CBA?
A. triangle BAB = triangle DOC
B. triangle DAB = triangle DAO
C. triangle DAB = triangle AOB
D. triangleDAB = triangle CBA

Answer 1

To analyze the properties of the rectangle ABCD and the information provided, we start with the relevant properties of rectangles:

1. Opposite sides are equal: \( \text{line AD} = \text{line BC} \) and \( \text{line AB} = \text{line CD} \).
2. The diagonals are congruent: in a rectangle, the diagonals \( AC \) and \( BD \) are equal in length.
3. The angles formed by the sides of the rectangle and its diagonals are right angles, so \( \angle DAB = \angle CBA = 90^\circ \).

Given the equal sides and angles, we can apply the congruence of triangles in the rectangle. The statements provided substantiate that triangle properties can be established based on the shared segments and angles.

Here are the options provided:

A. triangle BAB = triangle DOC B. triangle DAB = triangle DAO C. triangle DAB = triangle AOB
D. triangle DAB = triangle CBA

Among these, the statement that logically follows after establishing that \( \text{line AD} = \text{line BC} \), \( \text{line AB} = \text{line CD} \), and \( \angle DAB = \angle CBA \) is:

C. triangle DAB = triangle AOB

This option reflects that both triangles share the vertex O (where the diagonals intersect) and that they have congruent corresponding sides and angles, leading to their congruence. Thus, C is the correct continuation of the logical deductions made regarding the triangles in the rectangle.