Question

Suppose that CB is perpendicular to AD, BC is congruent to BC, and we want to prove that triangles △ABC and △DBC are congruent. Which additional piece of information would guarantee their congruence?

Answer 1

To guarantee the congruence of triangles △ABC and △DBC, the additional information needed is that angle A is congruent to angle C.

Step-by-step explanation:

In order to prove that triangles △ABC and △DBC are congruent, we need to show that they have at least three corresponding congruent parts. Given that CB is perpendicular to AD and BC is congruent to BC, we can conclude that angle B is congruent to angle D. This is because the right angles formed by the perpendicular lines are congruent, and the reflexive property ensures that BC is congruent to itself.

Therefore, the additional piece of information needed to guarantee the congruence of the triangles is that angle A is congruent to angle C. This can be proven by showing that AD is congruent to CB.

Once we have established that triangles △ABC and △DBC have all three pairs of congruent parts - two angles and a side, we can conclude that they are congruent by the Side-Angle-Side (SAS) congruence criterion.