Line Logic Worksheet Line Logic
Geometry: Theorems vs. Converses
Student:
Date:
Part 1: The Logical Trap
"If it rains, the ground is wet. I look outside and see the ground is wet. Therefore, it definitely rained."
Is this logic sound? Why or why not?
Conditional Statement
If \( P \), then \( Q \).
Converse Statement
If \( Q \), then \( P \).
Part 2: Proving Parallelism
As you watch the video (0:32-1:45), fill in the logic for the "Converse" theorems. How do they differ from the "Forward" theorems?
Theorem Type Converse Logic (If... then...) Alt. Interior If alternate interior angles are congruent, then lines are _____________. Alt. Exterior If ________________ angles are congruent, then lines are _____________. Corresponding If ________________ angles are congruent, then _____________________. Same-Side Int. If same-side interior angles are ________________, then ________________.
Part 3: The Path to Parallel
A B C D
Given:
\( AB \cong DC \)
\( AD \cong BC \)
Prove:
\( AB \parallel DC \)
Statements Reasons 1. \( AB \cong DC \); \( AD \cong BC \) 1. Given 2. 2. Reflexive Property 3. \(\triangle ABD \cong \triangle CDB\) 3. 4. \(\angle ABD \cong \angle CDB\) 4. 5. \( AB \parallel DC \) 5.
Part 4: Sorting Strategy
After the sorting activity, summarize the "Universal Test" for parallelism:
Parallel Pathfinding Slides Parallel
Pathfinding
The Logic of Converses in Geometry
CONVERSE
LOGIC
PROOF
Warm Up Logic
"If it rains, the ground is wet."
The ground is wet. Did it rain?
Initial Claim
If \( P \), then \( Q \).
The Converse
If \( Q \), then \( P \).
Proving Parallelism
Embedded media
Focus:
Listen for the narrator's specific order of logic.
"If angles are congruent... then lines are parallel."
Activity: Theorem Sort
1
Take a scenario card from your envelope.
2
Identify the relationship between the angles shown.
3
Decide: Is there enough info to prove the lines are parallel?
The Sort Piles
Parallel
Cannot Determine
The Big Distinction
The Theorem
If || lines...
...then angles are congruent.
The Converse
If angles are congruent...
...then lines are ||.
Create a Poster that makes this difference CRYSTAL CLEAR.
Theorem Sort Cards Theorem Sort Cards
Activity: Parallel Pathfinding
Instructions:
Cut out the cards below. Sort them into two piles: Lines are Parallel or Cannot Determine . For every "Parallel" card, identify the specific Converse Theorem you used.
\( 105^\circ \) \( 105^\circ \)
Scenario A
Alternate Interior Angles are congruent.
\( 75^\circ \) \( 75^\circ \)
Scenario B
Vertical angles are congruent.
\( 80^\circ \) \( 100^\circ \)
Scenario C
Interior angles sum to \( 180^\circ \).
\( 110^\circ \) \( 110^\circ \)
Scenario D
Same-side Interior angles are congruent.
\( 50^\circ \) \( 50^\circ \)
Scenario E
Corresponding angles are congruent.
\( 60^\circ \) \( 60^\circ \)
Scenario F
Alt. Exterior angles are congruent.
Converse Compass Anchor Chart Visual Reference Guide
Converse Compass
Navigating between Parallel Lines & Angle Proofs
The Theorem
Starting Condition
"If lines are PARALLEL..."
Conclusion
"...then angles are CONGRUENT."
Use this when you KNOW the lines are parallel and need to find a missing angle measure.
Proves Parallel!
The Converse
Starting Condition
"If angles are CONGRUENT..."
Conclusion
"...then lines are PARALLEL."
Use this when you need to PROVE that two lines are parallel to each other.
The 4 Tests for Parallelism
AI
Alt. Interior
Angles Congruent
AE
Alt. Exterior
Angles Congruent
CO
Corresponding
Angles Congruent
SS
Same-Side Int.
Supplementary!
Logic Rule: P \(\rightarrow\) Q is NOT the same as Q \(\rightarrow\) P unless specifically proven.
Parallel Pathfinding Answer Key Answer Key
Parallel Pathfinding: Lesson Materials
Teacher Use
Geometry
Line Logic Worksheet
Part 1: The Logical Trap
Sample Answer: The logic is unsound. While rain makes the ground wet, other things (sprinklers, a spilled bucket, melting snow) can also make the ground wet. Seeing the ground is wet (the "consequent") does not guarantee it rained (the "antecedent").
Part 2: Converse Logic Table
Theorem Type Correct Response Alt. Interior If alt. interior angles are \(\cong\), then lines are parallel . Alt. Exterior If alt. exterior angles are \(\cong\), then lines are parallel . Corresponding If corresponding angles are \(\cong\), then lines are parallel . Same-Side Int. If same-side interior angles are supplementary (\(180^\circ\)) , then lines are parallel .
Part 3: Two-Column Proof
Statement: \( AB \cong DC \); \( AD \cong BC \) | Reason: GivenStatement: \( DB \cong DB \) | Reason: Reflexive PropertyStatement: \(\triangle ABD \cong \triangle CDB\) | Reason: SSS PostulateStatement: \(\angle ABD \cong \angle CDB\) | Reason: CPCTCStatement: \( AB \parallel DC \) | Reason: Converse of Alt. Interior Angle Thm.
Theorem Sort Cards
Lines are Parallel
Scenario A: Parallel (Converse of Alt. Interior)Scenario C: Parallel (Converse of Same-Side Interior)Scenario E: Parallel (Converse of Corresponding)Scenario F: Parallel (Converse of Alt. Exterior)
Cannot Determine
Scenario B: Cannot Determine (Vertical angles being congruent is always true, but doesn't relate the two lines).Scenario D: Cannot Determine (Same-side interior angles must be 
