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🔷 Chapter 12: Quadrilaterals • Exercise Set 12.1

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Exercise Set 12.1 4Exercise Set 12.2 4Exercise Set 12.3 3Exercise Set 12.4 1End-of-Chapter Exercises 7All Exercises 19
Q1

Let ABCD be a quadrilateral.

  • (i) List all sides of ABCD adjacent to side AB. List all sides opposite to AB.
  • (ii) List all angles of ABCD adjacent to ∠A. List all angles opposite to ∠A.
  • (iii) Define a pair of opposite sides and a pair of opposite angles without using the names of the vertices.
  • (i) Adjacent to AB: Sides BC and DA. Opposite to AB: Side CD.
  • (ii) Adjacent to ∠A: Angles ∠B and ∠D. Opposite to ∠A: Angle ∠C.
  • (iii) Opposite sides are two sides of a quadrilateral that do not share a common vertex. Opposite angles are two internal angles that do not share a common side.
Q2

Precisely define the internal angle of a quadrilateral at a given vertex. Your answer should work for a non-convex quadrilateral too.

Answer: At any vertex of a quadrilateral, the two adjacent sides form an angle. The internal angle is the angle formed by these two rays that sweeps across the interior region of the quadrilateral. To define it precisely without ambiguity (especially for non-convex cases): It is the angle between the two adjacent sides at the vertex such that a line segment connecting points immediately near the vertex on these two sides lies completely within the quadrilateral's interior. If the quadrilateral is non-convex (has a "dent"), this interior angle will be greater than 180°.

Convex (All angles < 180°) > 180° Non-Convex (Reflex interior angle)
Convex vs. Non-Convex quadrilateral showing internal reflex angle > 180°
Q3

In a quadrilateral ABCD, suppose AB || DC. Can ABCD be non-convex? What if we instead assume AB = CD? What if we instead assume ∠A = ∠C?

  • If AB || DC: No, it cannot be non-convex. If two opposite sides are parallel, the consecutive interior angles must sum to 180° (e.g., ∠A + ∠D = 180°). Since their sum is 180°, neither angle can be greater than 180°. Thus, it must be convex.
  • If AB = CD: Yes, it can be non-convex. A self-intersecting "crossed" quadrilateral or a non-convex dart shape can have a pair of opposite sides equal.
  • If ∠A = ∠C: Yes, it can be non-convex. A dart shape (non-convex quadrilateral) can be symmetric such that its two non-reflex opposite angles are equal.
Q4

Consider three non-collinear points A, B, C and draw the lines AB, BC, CA. For every possible location of point D in the plane outside these lines, decide if ABCD is self-intersecting, non-convex, or convex.

Answer: The three lines extending from the triangle ABC divide the plane into 7 regions (the central triangle, and 6 outer regions). Depending on where point D is placed, the resulting sequence of vertices A-B-C-D forms different types of quadrilaterals.

B C A Convex Self-Int. Non-convex
  • If D is placed in the regions directly "behind" the vertices (e.g., opposite the triangle from A, B, or C), ABCD will be a self-intersecting (crossed) quadrilateral.
  • If D is placed in the regions bounded by the extended sides (the "wedges" pointing outwards from the edges), ABCD will be non-convex (dart-shaped).
  • If D is placed such that all internal angles remain less than 180°, it forms a convex quadrilateral.