Question

1. y is a midpoint. which conjecture tells you \\(\triangle ayb \cong \triangle ryn\\)?

(image of a rectangle with vertices a, y, r, n and b, with y as midpoint, and triangles ayb and ryn marked with congruency marks)

Explanation:

Step1: Analyze the given information

Since \(Y\) is a mid - point, \(AY = RY\). Also, \(\angle BAY=\angle NRY = 90^{\circ}\) (assuming \(BA\perp AR\) and \(NR\perp AR\) from the figure's shape which looks like a rectangle - like structure with right - angles at \(A\) and \(R\)) and \(BA = NR\) (marked as equal in the figure).

Step2: Recall the congruence postulate

The Side - Angle - Side (SAS) congruence postulate states that if two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, then the two triangles are congruent.
In \(\triangle AYB\) and \(\triangle RYN\), we have \(AY = RY\) (side), \(\angle BAY=\angle NRY\) (angle), and \(BA = NR\) (side).

Answer:

The SAS (Side - Angle - Side) congruence conjecture tells that \(\triangle AYB\cong\triangle RYN\).