9/25 Introduction to Triangle Congruence and Proofs

Congruence Statements and Quadrilateral Proofs

• Warm-up Exercise: Quadrilateral Congruence

  • Problem Statement: Polygon $ABCD \cong \text{Polygon } WXYZ$. Students were tasked with finding the value of $x$ and the measure of $\angle BCD$.

  • Given Data:

    • $\angle C = 9x + 10$

    • $\angle Y = 10x + 1$

  • Step-by-Step Solution:

    • Identify corresponding angles: Since the polygons are congruent, $\angle C$ must be equal to $\angle Y$.

    • Equation: $9x + 10 = 10x + 1$

    • Subtract $10x$ from both sides: $-x + 10 = 1$

    • Subtract $10$ from both sides: $-x = -9$

    • Result: $x = 9$

  • Determining Angle Measures:

    • Substitute $x = 9$ into the expression for $\angle BCD$: $9 \times 9 + 10 = 81 + 10 = 91^{\circ}$.

    • Verify with $\angle XYZ$: $10 \times 9 + 1 = 90 + 1 = 91^{\circ}$.

    • Both angles are confirmed at $91^{\circ}$.

• Homework Review: Triangle Vertex Ordering

  • Question Number 7: Focused on naming congruent triangles correctly based on vertex markings.

  • Identifying Corresponding Vertices:

    • $\angle A$ (one arc with one line) corresponds to $\angle E$.

    • $\angle B$ (one arc with two lines) corresponds to $\angle D$.

    • The remaining vertex is $C$.

  • Correct Congruence Statement: $\triangle ABC \cong \triangle EDC$.

  • Side Verification:

    • Side $AB$ (one line) $\cong$ Side $DE$.

    • Side $AC$ (three lines) $\cong$ Side $CE$.

    • Side $BC$ (two lines) $\cong$ Side $CD$.

Side-Side-Side ($SSS$) and Side-Angle-Side ($SAS$) Postulates

• Reflexive Property of Congruence

  • Standard Definition: A segment is always congruent to itself (e.g., $CB \cong CB$ or $DB \cong DB$).

  • This is frequently used in proofs where triangles share a common side.

• Postulate 4.3.1: Side-Side-Side ($SSS$)

  • Definition: If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent.

  • Two-Column Proof Example ($SSS$):

    • Given: $AB \cong CD$ and $AC \cong BD$.

    • Statement 1: $AB \cong CD$ (Reason: Given).

    • Statement 2: $AC \cong BD$ (Reason: Given).

    • Statement 3: $CB \cong CB$ (Reason: Reflexive Property of Congruence).

    • Conclusion: $\triangle ABC \cong \triangle DCB$ (Reason: $SSS$ Postulate).

• Postulate 4.3.2: Side-Angle-Side ($SAS$)

  • Definition: If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent.

  • Crucial Condition: The angle MUST be the "included angle," meaning it is located exactly between the two congruent sides.

  • Two-Column Proof Example ($SAS$):

    • Given: $AB \cong BC$ and $EB \cong BD$.

    • Statement 1: $AB \cong BC$ (Reason: Given).

    • Statement 2: $EB \cong BD$ (Reason: Given).

    • Statement 3: $\angle 1 \cong \angle 2$ (Reason: Vertical Angle Theorem).

    • Conclusion: $\triangle ABE \cong \triangle CBD$ (Reason: $SAS$ Postulate).

Analysis of Congruence Criteria in Diagrams

• Evaluating Information Sufficiency:

  • Case A: Three pairs of corresponding sides are marked. Result: $SSS$.

  • Case B: Side $10\,\text{mm}$, side $13\,\text{mm}$, and an included angle of $110^{\circ}$ are matched. Result: $SAS$.

  • Case C: Two sides matched, but the marked angle is not the included angle. Result: Not enough information.

  • Case D: Two pairs of sides matched, and the triangles share a common side. Result: $SSS$ due to the shared common side (Reflexive Property).

Detailed Proof Construction: Midpoints and Bisectors

• Midpoint and Side-Side-Side ($SSS$) Proof (Section 2.5/2.6)

  • Goal: Prove $\triangle ABM \cong \triangle DCM$.

  • Given Information:

    1. $AB \cong CD$.

    2. $M$ is the midpoint of $AD$.

    3. $M$ is the midpoint of $BC$.

  • Proof Steps:

    • 1. $AB \cong CD$ (Given).

    • 2. $M$ is the midpoint of $AD$ and $BC$ (Given).

    • 3. $BM \cong CM$ (Definition of Midpoint).

    • 4. $DM \cong AM$ (Definition of Midpoint).

    • 5. $\triangle ABM \cong \triangle DCM$ (Reason: $SSS$).

  • Vertex Matching Check: $A$ corresponds to $D$, $B$ corresponds to $C$, and $M$ corresponds to $M$ (vertical angle location).

Angle-Side-Angle ($ASA$) and Angle-Angle-Side ($AAS$)

• Postulate: Angle-Side-Angle ($ASA$)

  • Definition: Two angles and the included side (the side between the two angles) must be congruent.

  • Example: $\angle D \cong \angle A$, $\angle C \cong \angle F$, and Side $AC \cong DF$.

• Theorem: Angle-Angle-Side ($AAS$)

  • Definition: Two angles and a non-included side must be congruent.

  • Derivation: This theorem is derived from the fact that if two angles in a triangle are congruent, the third angle must also be congruent (Third Angles Theorem). Thus, $AAS$ effectively becomes $ASA$.

• Wait-Time/Discussion Proof: Angle Bisector

  • Goal: Prove $\triangle DAB \cong \triangle DCB$.

  • Given: $DB$ bisects $\angle ABC$; $\angle 1 \cong \angle 2$.

  • Steps:

    • 1. $DB$ bisects $\angle ABC$ (Given).

    • 2. $\angle 3 \cong \angle 4$ (Definition of Angle Bisector).

    • 3. $\angle 1 \cong \angle 2$ (Given).

    • 4. $DB \cong DB$ (Reflexive Property).

    • 5. Conclusion: $\triangle DAB \cong \triangle DCB$ (Reason: $ASA$).

Special Proof: Right Angles and CPCTC

• Right Angle Congruence Proof:

  • Given: $\angle B$ and $\angle E$ are right angles; $C$ is the midpoint of side $AD$.

  • Reasoning:

    • $\angle B \cong \angle E$ (All right angles are congruent).

    • $\angle 1 \cong \angle 2$ (Vertical angles).

    • $AC \cong CD$ (Definition of Midpoint).

    • Conclusion: $\triangle ABC \cong \triangle DEC$ via $AAS$ (side is not included between congruent angles).

• Segment Addition and Substitution Proof:

  • Given: $WX \cong VZ$, $WY \cong VY$, $YZ \cong YX$.

  • Process:

    • Use the Addition Property of Congruence: If $WY \cong VY$ and $YX \cong YZ$, then $WY + YZ = VY + YX$.

    • Segment Addition Postulate: $WZ = WY + YZ$ and $VX = VY + YX$.

    • Substitution: $WZ \cong VX$.

    • Final Triangle Congruence via $SSS$.

• CPCTC (Corresponding Parts of Congruent Triangles are Congruent):

  • Usage: Once triangles are proven congruent, any of their side or angle pairs are automatically congruent.

  • Example: If $\triangle ABF \cong \triangle CBD$, then $CE \cong AE$ by CPCTC.

Classroom Logistics and Knowledge Check

• Knowledge Check (Midterm-Style):

  • Schedule: Opens tomorrow; closes Tuesday at midnight.

  • Constraints: $120$ minutes (2 hours) total. Three attempts allowed.

  • Content: Chapters 2, 3, and 4 (including Chapter 3 from the previous week).

  • Format: $15$ to $20$ questions. Includes fill-in-the-blank and proof-selection dropboxes. Uses randomized versions for each student.

  • System Details: Timer starts immediately upon clicking "Start." No option to pause and return later.

• Grading and Attendance:

  • Discussion and Exit Tickets: Account for $15\%$ of the total grade.

  • Knowledge Checks: Two per semester, each worth $22.5\%$ of the total grade.

  • Policy: Missing a knowledge check significantly reduces the likelihood of passing. Final exams can sometimes replace a missed knowledge check at the professor's discretion.

• Exit Ticket Issues:

  • The exit ticket for June 25th was cancelled due to a broken link in the system (repeatedly defaulting to June 24th/18th formats).

  • Students received credit without completion due to the technical failure.

• Questions & Discussion:

  • Question (Student): In the number line example ($0$ to $5$ then $2$ more), what is modeled?

  • Answer: Addition ($5 + 2 = 7$).

  • Question (Ivan): "Do you need the camera on for the test?"

  • Response: No camera is required for the knowledge check. The student can complete it in their own time but must complete it within the timer once started.

Supplemental Concepts: Definitions

• Segment Bisector: A line, ray, or segment that divides a given segment into two congruent parts.

• Reflexive Property: Identifying shared components in overlapping geometric figures.

• Theorem Distinction: Postulates are assumed truths; theorems (like $AAS$) must be proven using postulates (like $ASA$).