Question

prove: △abd ≅ △cbd
proof:
statements | reasons
1.? | 1. given
2.? | 2. definition of midpoint
3.? | 3. reflexive property of congruence
4.? | 4. sss
(image of a quadrilateral with points a, b, c, d; d is the midpoint of ac, ab ≅ cb, ad ≅ dc, bd ≅ bd, △abd ≅ △cbd)

Explanation:

Step1: Identify Given Info

From the diagram and hints, the given should be \( AB \cong CB \), \( D \) is midpoint of \( AC \), and \( BD \cong BD \). So Statement 1: \( AB \cong CB \), \( D \) is the midpoint of \( AC \), \( BD \cong BD \) (Given).

Step2: Apply Midpoint Definition

By definition of midpoint, if \( D \) is midpoint of \( AC \), then \( AD \cong DC \). So Statement 2: \( AD \cong DC \) (Definition of midpoint).

Step3: Apply Reflexive Property

Reflexive property says a segment is congruent to itself, so \( BD \cong BD \). Statement 3: \( BD \cong BD \) (Reflexive Property of Congruence).

Step4: Apply SSS Congruence

SSS (Side - Side - Side) congruence: if three sides of one triangle are congruent to three sides of another triangle, triangles are congruent. So \( \triangle ABD \cong \triangle CBD \) by SSS. Statement 4: \( \triangle ABD \cong \triangle CBD \) (SSS).

Answer:

Statements	Reasons
2. \( AD \cong DC \)	Definition of midpoint
3. \( BD \cong BD \)	Reflexive Property of Congruence
4. \( \triangle ABD \cong \triangle CBD \)	SSS