M2L5 Congruence - Investigating (Part 1)

Page 1: Introduction

  • Topic: M2L5 Congruence - Integrated Mathematics 9

Page 2: Understanding Congruence

  • Question: Have you heard the word "congruent" before?

  • Task: Define it in your own words with an example (angles or shapes).

Page 3: Learning Objectives

  • Review prior learning on congruence.

  • Goals for the day:

    • Investigating triangle congruence

    • Checking understanding of concepts.

Page 4: Defining Congruent Figures

  • Update definitions with keywords:

    • Size

    • Shape

    • Position

    • Orientation

    • Translation

    • Rotation

    • Reflection

Page 5: Properties of Congruent Figures

  • Congruent Figures:

    • Two shapes are congruent when they are:

      • Identical in size and shape

      • Do NOT require the same position or orientation

    • When congruent:

      • Write corresponding vertices in the same order

      • Use the symbol ≅

Page 6: Analyzing Congruent Statements




  • Evaluate three statements for correctness:a) ABCD ≅ PQRS (correct)b) ABCD ≅ RSPQ (not correct)c) ABCD ≅ PSRQ (not correct)

  • Explanation required for incorrect statements.

Page 7: Naming Polygons

  • Names of polygons:

    • Write vertices in consecutive order (e.g., PQRS can be named QRSP or SRQP, but not PRQS).

    • In congruence statements, the order indicates corresponding parts (e.g., ΔABC ≅ ΔDEF).

Page 8: Reiteration of Congruent Statements

  • Similar to Page 6:

    • Evaluate three statements for congruence:

      • ABCD ≅ PQRS (correct)

      • ABCD ≅ RSPQ (not correct)

      • ABCD ≅ PSRQ (not correct)

Page 9: Quadrilateral Congruence Practice

  • Practice with quadrilaterals ABCD and PQRS:



    1. Identify a side of the same length as:i) [BC]ii) [RS]



    2. Identify an angle of the same measure as:i) angle ABCii) angle RSP

Page 10: Triangle Measurements Practice

  • Practice with triangles:

    • Length of [BC]: 3 cm



    • Angles measured:i) 30°ii) 70°



    • Determine sizes:i) measure of angle BACii) measure of angles LNM & LNM.

Page 11: Investigation Introduction

  • Time to investigate congruence in triangles.

Page 12: Triangle Congruence Investigation

  • Team activity:

    • Determine what information confirms two triangles are congruent.

Page 13: Triangle Construction Activity

  • Task: Draw a triangle ABC with given measures (AB = 3cm, BC = 2cm).

Page 14: Using a Protractor

  • Introduction to using a protractor for measurements.

Page 15: Investigation Focus

  • Investigation on what information is sufficient to uniquely describe triangles.

Page 16: Investigation Instructions

  • Instructions for building triangles with specific criteria:

    • Each team gets 4 sets of criteria.

    • Use rulers, protractors, spaghetti, and paper to construct triangles.

    • Approval from the teacher before submission.

Page 17: Protractor Usage Continued

  • Detailed instructions on how to use a protractor effectively.

Page 18: Gallery Walk Activity

  • Compare triangles made by groups across different building projects.

    • Document observations from each project.

Page 19: Creating Criteria for Congruency

  • Brainstorm a set of criteria to ensure all groups create congruent triangles.

  • Discuss minimum criteria needed and rationale.

Page 20: Findings on Triangle Congruence

  • Congruence criteria summarized:

    • SSS: Three pairs of congruent sides

    • SAS: Two sides and an included angle

    • ASA: Two angles and an included side

    • AAS: Two angles and a side opposite them

    • HL: Hypotenuse-Leg of right triangle.

Page 21: Non-Congruence Conditions

  • What does NOT guarantee triangle congruence:

    • Side-Side-Angle (SSA) can yield two different triangles.

    • All angles (AAA) do not determine triangle size.

Page 22: Why SSA and AAA are Not Congruence Shortcuts

  • SSA can result in multiple triangles with the same side-angle measures.

  • AAA does not ensure congruence as it does not account for size differences.

Page 23: Triangle Congruence Theorems

  • Overview of theorems on triangle congruence:

    • SSS

    • SAS

    • ASA

    • AAS

    • HL

    • Triangle Congruence Theorems

    • 1. **SSS (Side-Side-Side): Two triangles are congruent if all three pairs of corresponding sides are equal in length. This means that if you measure each side of one triangle and find them to be equal to the corresponding sides of another triangle, the triangles are congruent.

    • 2. **SAS (Side-Angle-Side)**: Two triangles are congruent if two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle. The included angle is the angle formed between the two sides.

    • 3. **ASA (Angle-Side-Angle)**: Two triangles are congruent if two angles and the side between those angles are equal. This means that if you know the measures of two angles and the side between them in one triangle, you can prove the other triangle is congruent to it.

    • 4. **AAS (Angle-Angle-Side)**: Two triangles are congruent if two angles and a side not between those angles are equal. This means that the triangle's dimensions can be determined by the angles and any side, even if the side does not connect the angles.

    • 5. **HL (Hypotenuse-Leg): In right triangles, if the hypotenuse and one leg of one right triangle are equal to the hypotenuse and one leg of another right triangle, then the triangles are congruent. This theorem is specific to right triangles and takes advantage of the properties of right angles.

Page 24: Final Notes and Diagrams

  • If time allows, create notes with diagrams to aid memory.

  • Offer assistance if needed.