Question

1 which postulate or theorem proves the triangles are congruent? if the triangles are not congruent, state
ot congruent.\ code only the necessary parts to prove congruence for each picture.
2 complete the proof
given: $overline{ab} \perp overline{dc}$, point $b$ is the midpoint of $overline{dc}$
prove: $\triangle abd \cong \triangle abc$
explain what additional information, if any, would you need to prove $\triangle abc \cong \triangle dec$ by each postulate or theorem.
3 sss
4 sas

Explanation:

Step1: Analyze the first statement's justification

The first statement is "Point B is the midpoint of \(\overline{DC}\)", so the justification is "Given" (since it's part of the given information).

Step2: Determine the statement for "Definition of Midpoint"

By the definition of a midpoint, if B is the midpoint of \(\overline{DC}\), then \(\overline{DB} \cong \overline{BC}\). So the statement here is \(\overline{DB} \cong \overline{BC}\).

Step3: Identify the statement for "Given" (perpendicularity)

The given information also includes \(\overline{AB} \perp \overline{DC}\), so the statement is \(\overline{AB} \perp \overline{DC}\).

Step4: Justify "∠ABD & ∠ABC are right angles"

Since \(\overline{AB} \perp \overline{DC}\), by the "Definition of Perpendicular", the angles formed are right angles. So the justification is "Definition of Perpendicular".

Step5: Justify "∠ABD ≅ ∠ABC"

All right angles are congruent, so the justification is "All Right Angles are Congruent".

Step6: Determine the statement for "Reflexive Property of Congruence"

The reflexive property of congruence for a segment is \(\overline{AB} \cong \overline{AB}\), so the statement is \(\overline{AB} \cong \overline{AB}\).

Step7: Justify "ΔABD ≅ ΔABC"

We have \(\overline{DB} \cong \overline{BC}\), \(\angle ABD \cong \angle ABC\), and \(\overline{AB} \cong \overline{AB}\), so by SAS (Side - Angle - Side) congruence postulate, the triangles are congruent. So the justification is "SAS".

Statements	Justifications
\(\overline{DB} \cong \overline{BC}\)	Definition of Midpoint
\(\overline{AB} \perp \overline{DC}\)	Given
∠ABD & ∠ABC are right angles	Definition of Perpendicular
∠ABD ≅ ∠ABC	All Right Angles are Congruent
\(\overline{AB} \cong \overline{AB}\)	Reflexive Property of Congruence
ΔABD ≅ ΔABC	SAS

Answer:

The completed proof table is as shown above. For the specific blanks:

1. First blank (justification for "Point B is the midpoint of \(\overline{DC}\)"): Given
2. Second blank (statement for "Definition of Midpoint"): \(\boldsymbol{\overline{DB} \cong \overline{BC}}\)
3. Third blank (statement for "Given" - perpendicularity): \(\boldsymbol{\overline{AB} \perp \overline{DC}}\)
4. Fourth blank (justification for "∠ABD & ∠ABC are right angles"): Definition of Perpendicular
5. Fifth blank (justification for "∠ABD ≅ ∠ABC"): All Right Angles are Congruent
6. Sixth blank (statement for "Reflexive Property of Congruence"): \(\boldsymbol{\overline{AB} \cong \overline{AB}}\)
7. Seventh blank (justification for "ΔABD ≅ ΔABC"): SAS