Question

1. write a two - column proof for the following and be sure to code:

given: \\( \overline { b d } \perp \overline { a c } \\), \\( \overline { b d } \\) bisects \\( \overline { a c } \\)
prove: \\( \angle a b d \cong \angle c b d \\)
statements

1. \\( \overline { b d } \perp \overline { a c } \\)
2. \\( \angle b d c \cong \angle b d a = 90 ^ { \circ } \\)
3. \\( \angle \underline {} \cong \angle \underline {} \\)
4. \\( \overline { b d } \\) bisects \\( \overline { a c } \\)

5.

1. \\( \overline { b d } \cong \overline { b d } \\)

7.

1. \\( \angle a b d \cong \angle c b d \\)

reasons

1. given
2. definition of perpendicular

3.

1. given

5.
6.
7.
8.

Explanation:

Step1: Identificar ángulos congruentes

$\angle BDC\cong\angle BDA$

Step2: Justificar ángulos congruentes

Por definición de ángulo recto (todos los ángulos rectos son congruentes)

Step3: Identificar segmentos congruentes

$\overline{CD}\cong\overline{AD}$

Step4: Justificar segmentos congruentes

Por definición de bisectriz de segmento

Step5: Justificar segmento congruente consigo mismo

Por propiedad reflexiva de la congruencia

Step6: Identificar triángulos congruentes

$\triangle BDC\cong\triangle BDA$

Step7: Justificar triángulos congruentes

Por criterio SAS (lado-ángulo-lado)

Step8: Justificar ángulos congruentes

Por CPCTC (partes correspondientes de triángulos congruentes son congruentes)

Answer:

Statements	Reasons
2. $\angle BDC\cong\angle BDA = 90^{\circ}$	2. Definition of Perpendicular
3. $\angle BDC\cong\angle BDA$	3. All right angles are congruent
4. $\overline{BD}$ bisects $\overline{AC}$	4. Given
5. $\overline{CD}\cong\overline{AD}$	5. Definition of segment bisector
6. $\overline{BD}\cong\overline{BD}$	6. Reflexive Property of Congruence
7. $\triangle BDC\cong\triangle BDA$	7. SAS (Side - Angle - Side)
8. $\angle ABD\cong\angle CBD$	8. CPCTC (Corresponding Parts of Congruent Triangles are Congruent)