Essay
Your teacher will grade your response for the following question to ensure that you receive proper credit for your answer.
What are the missing reasons in the proof?
Given: parallelogram upper A upper B upper C upper D with diagonal modifying above upper B upper D with bar
Prove: triangle upper A upper B upper D congruent to triangle upper C upper D upper B
Parallelogram upper A upper B upper C upper D is shown with diagonal segment upper B upper D drawn.
A partially completed two column proof is given with the first column titled 'Statements' and the second column titled 'Reasons.'
The first column contains:
1 period modifying above upper A upper D with bar parallel to modifying above upper B upper C with bar
2 period angle upper A upper D upper B congruent to angle upper C upper B upper D
3 period modifying above upper A upper B with bar parallel to modifying above upper C upper D with bar
4 period angle upper A upper B upper D congruent to angle upper C upper D upper B
5 period modifying above upper D upper B with bar congruent to modifying above upper D upper B with bar
6 period triangle upper A upper B upper D congruent to triangle upper C upper D upper B
The second column contains:
1 period Definition of parallelogram
2 period Alternate Interior Angles Theorem
3 period question-mark
4 period question-mark
5 period question-mark
6 period question-mark
(4 points)
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To complete the missing reasons in the proof, we need to provide justifications for statements 3, 4, 5, and 6.
Possible reasons for statement 3 could be:
3. Parallel lines cut by a transversal form congruent alternate interior angles.
Possible reasons for statement 4 could be:
4. Corresponding angles of parallel lines are congruent.
Possible reasons for statement 5 could be:
5. A segment is congruent to itself.
Possible reasons for statement 6 could be:
6. If two triangles have congruent corresponding angles and a congruent included side, then the triangles are congruent (ASA Congruence Theorem).
To find the missing reasons in the proof, we need to examine the given statements and see what logical steps were made to arrive at each statement.
1. Statement: segment AD is parallel to segment BC
Reason: Definition of parallelogram
2. Statement: angle ADB is congruent to angle CBD
Reason: Alternate Interior Angles Theorem
3. Statement: segment AB is parallel to segment CD (missing reason)
Reason: ?
4. Statement: angle ABD is congruent to angle CDB (missing reason)
Reason: ?
5. Statement: segment DB is congruent to segment DB (missing reason)
Reason: ?
6. Statement: triangle ABD is congruent to triangle CDB (missing reason)
Reason: ?
From the given information, it seems that some steps are missing their corresponding reasons. You should review your notes or textbook to identify the missing reasons and complete the proof.
To determine the missing reasons in the proof, we need to analyze the given statements and identify the appropriate geometric theorems or properties that can be used to support those statements.
Let's start with the given information:
1. Statement: AD || BC (AD is parallel to BC)
Reason: Definition of a parallelogram.
2. Statement: ∠ADB ≅ ∠CBD (angle ADB is congruent to angle CBD)
Reason: Alternate Interior Angles Theorem (if two parallel lines are intersected by a transversal, then the alternate interior angles are congruent).
3. Statement: AB || CD (AB is parallel to CD)
Reason: Unknown (question-mark).
4. Statement: ∠ABD ≅ ∠CDB (angle ABD is congruent to angle CDB)
Reason: Unknown (question-mark).
5. Statement: DB ≅ DB (side DB is congruent to itself)
Reason: Unknown (question-mark).
6. Statement: ΔABD ≅ ΔCDB (triangle ABD is congruent to triangle CDB)
Reason: Unknown (question-mark).
Based on the given statements, we can determine the missing reasons using relevant geometric theorems and properties:
3. Statement: AB || CD (AB is parallel to CD)
Reason: Corresponding Angles Theorem (if two parallel lines are intersected by a transversal, then the corresponding angles are congruent).
4. Statement: ∠ABD ≅ ∠CDB (angle ABD is congruent to angle CDB)
Reason: Vertical Angles Theorem (when two lines intersect, the angles opposite each other are congruent or equal).
5. Statement: DB ≅ DB (side DB is congruent to itself)
Reason: Reflexive Property of Congruence (any segment or angle is congruent to itself).
6. Statement: ΔABD ≅ ΔCDB (triangle ABD is congruent to triangle CDB)
Reason: Side-Angle-Side (SAS) Congruence Postulate (if two sides and the included angle of one triangle are congruent to the corresponding parts of another triangle, then the triangles are congruent).
Now, the completed two-column proof would look like this:
Statements | Reasons
---------------------------------
1. AD || BC | Definition of parallelogram
2. ∠ADB ≅ ∠CBD | Alternate Interior Angles Theorem
3. AB || CD | Corresponding Angles Theorem
4. ∠ABD ≅ ∠CDB | Vertical Angles Theorem
5. DB ≅ DB | Reflexive Property of Congruence
6. ΔABD ≅ ΔCDB | Side-Angle-Side (SAS) Congruence Postulate
This completes the missing reasons in the proof.