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Chapter 12 · Period 1 of 16 · 50 min

Defining a Quadrilateral — What Counts and What Does Not

What Exactly is a Quadrilateral? · Deductive

This unit builds the formal definition of a quadrilateral from scratch, so ground every requirement in one of the excluded figures — students find it easier to accept a clause when they see what it rules out. A very common error is treating CUTS (self-intersecting) as a valid quadrilateral because it has four vertices and four line segments; make the 'exactly one side' clause of Definition 1 explicit. Students wishing to read further may look at Exercise Set 12.1 Q3, p.56, which asks about convexity under single conditions such as AB ∥ DC.

0-12Open by drawing the tiling-floor image on the board and posing the chapter's opening question — can any quadrilateral tile the plane? Then display figures ABCD, NOPE, SILY, DART, CUTS, OPENS and BENT (board diagram) and ask students in pairs: which of these should be called a quadrilateral and why? Take brief responses and list the emerging requirements on the board.
12-30Work through Definition 1 deductively: start from the definition of a triangle as points on segments joining three non-collinear vertices, and build each requirement for a quadrilateral — no three vertices collinear, all four vertices in one plane, no self-intersection — showing which figure each requirement excludes. State Definition 1 formally. Introduce the naming conventions (BCDA, DCBA etc.) and the terms vertices, sides, diagonals, adjacent sides, adjacent angles, internal angles. Then define convex quadrilateral (every internal angle less than 180°) and the diagonal-intersection test for convexity, using DART as the non-convex example. Briefly show the Venn diagram classifying 4-gons.
30-42Think-and-reflect pause: pose the think_reflect questions from the section — how OPENS is excluded by Definition 1; whether a quadrilateral ABCD can be defined in the same way as a triangle; which quadrilaterals can tile the plane. Allow individual thinking, then a short pair discussion; take two or three responses.
42-50Students work individually on Exercise Set 12.1 Q1, p.55 — listing adjacent and opposite sides and angles for ABCD and writing vertex-free definitions of opposite sides and opposite angles. Review answers as a class; clarify the difference between adjacent and opposite in the quadrilateral context.
Assessment

Competency (C-4.1). Understand and classify geometric shapes, including quadrilaterals, using definitions and properties.

Learning outcome. Students can apply Definition 1 to classify a given figure as a valid quadrilateral, a self-intersecting quadrilateral, or a non-quadrilateral, stating which requirement it violates.

Four distinct points W, X, Y, Z lie in a plane, no three of them collinear. The segments WX, XY, YZ and ZW are drawn. A fifth point E, which is not any of W, X, Y, Z, is found to lie on both segment WX and segment YZ. Which statement correctly classifies figure WXYZ?

  • A WXYZ is a non-convex quadrilateral because one of its internal angles exceeds 180°.
  • B WXYZ is a self-intersecting quadrilateral because a point E other than a vertex lies on two of its sides simultaneously.
  • C WXYZ is a valid (non-self-intersecting) quadrilateral because it has four distinct vertices and four straight sides.
  • D WXYZ is not a quadrilateral at all because E is a fifth point, making the figure have five vertices.

B. WXYZ is a self-intersecting quadrilateral because a point E other than a vertex lies on two of its sides simultaneously.

What each wrong option reveals

  • A Conflates self-intersection with non-convexity — a non-convex quadrilateral has an internal angle exceeding 180° but does not self-intersect; the student has applied the wrong classification criterion.
  • C Confuses the necessary conditions for a quadrilateral with the sufficient ones — having four vertices and four sides does not by itself guarantee non-self-intersection; the student has not applied the 'exactly one side' clause of Definition 1.
  • D Misidentifies E as a vertex rather than an intersection point of sides; the student does not distinguish between the vertices of a quadrilateral and incidental crossing points of its edges.
Chapter 12 · 16 periods of 50 min

Also saved as 10 periods or 13 periods.

  • 2Classifying 4-gons and Locating the Fourth Vertex
    What Exactly is a Quadrilateral?
  • 3Parallelogram Properties and Their Converses — Theorems 1–2
    Parallelograms
  • 4Converses of Parallelogram Properties — Theorems 3, 4 and 5
    Parallelograms
  • 5Applying Parallelogram Properties — Rhombus, Rectangle, Square
    Parallelograms
  • 6Introducing Parallelogram Applications — Linkages and the Midpoint Idea
    Applications of Parallelograms
  • 7Proving the Midpoint Theorem — Theorem 6
    An Important Property of Triangles
  • 8Proving the Converse of the Midpoint Theorem — Theorem 7
    An Important Property of Triangles
  • 9More Applications of the Midpoint Theorem — Trisection and Trapezium
    An Important Property of Triangles
  • 10Proving the Centroid Theorem — Theorem 8
    A Surprising Property of Medians
  • 11Varignon's Theorem — Midpoints of a Quadrilateral's Sides Form a Parallelogram
    A Surprising Property of Medians
  • 12Reconstructing a Quadrilateral from Its Varignon Parallelogram
    A Surprising Property of Medians
  • 13Tiling the Plane with Any 4-gon — Fitting Four Copies Around a Point
    Tiling the Plane Using Any 4-gon
  • 14Tiling by the Varignon Grid — Method 2 and Its Justification
    Tiling the Plane Using Any 4-gon
  • 15Harder Problems Across the Chapter — Midpoints and Parallelograms
    Tiling the Plane Using Any 4-gon
  • 16Whole-Chapter Synthesis — Connecting Quadrilateral Properties, Theorems and Tilings
    Synthesis
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