Question

if there is enough information to conclude that the pair of triangles is congruent, write a valid congruence statement. if not, state any additional congruences that are needed.
31.
32.

Explanation:

Step1: Check congruence criteria

For problem 31:
We have \(AD = DC\) (given by the single tick marks on \(AD\) and \(DC\)), \(\angle ADB=\angle CDB = 90^{\circ}\) (right - angles), and \(AB = CB\) (given by the double tick marks). By the Hypotenuse - Leg (HL) congruence criterion (for right - triangles, if the hypotenuse and one leg are congruent, the triangles are congruent), \(\triangle ABD\cong\triangle CBD\).

Step2: Check congruence criteria

For problem 32:
We have \(AB = CD\) (given by the single tick marks), \(BD=BD\) (common side), and \(\angle ABD=\angle CDB\) (given by the angle marks). By the Side - Angle - Side (SAS) congruence criterion (if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, the triangles are congruent), \(\triangle ABD\cong\triangle CDB\).

Answer:

1. \(\triangle ABD\cong\triangle CBD\)
2. \(\triangle ABD\cong\triangle CDB\)