Question

given: c is the mid - point of \\( \overline{ad} \\) \\( \overline{ab} \cong \overline{db} \\) prove: \\( \triangle abc \cong \triangle dbc \\) given: b is the mid - point of \\( \overline{dc} \\) \\( \overline{ab} \perp \overline{dc} \\) prove: \\( \triangle abd \cong \triangle abc \\)

Explanation:

Question 25

To determine the congruence reason for \( \triangle ABC \cong \triangle DBC \), we analyze the given statements:

1. \( AC \cong DC \) (from midpoint definition).
2. \( AB \cong DB \) (given).
3. \( BC \cong BC \) (reflexive property, common side).

We have three pairs of congruent sides (\( AC \cong DC \), \( AB \cong DB \), \( BC \cong BC \)). The SSS (Side - Side - Side) congruence criterion states that if three sides of one triangle are congruent to three sides of another triangle, the triangles are congruent. So the reason is SSS.

To find the congruence reason for \( \triangle ABD \cong \triangle ABC \), we look at the statements:

1. \( DB \cong CB \) (from midpoint definition).
2. \( \angle ABD \cong \angle ABC \) (both are right angles, as \( AB\perp DC \)).
3. \( AB \cong AB \) (reflexive property, common side).

We have two sides and the included angle congruent (\( DB \cong CB \), \( \angle ABD \cong \angle ABC \), \( AB \cong AB \)). The SAS (Side - Angle - Side) congruence criterion states that 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. So the reason is SAS.

Answer:

SSS

Question 26