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4-gons (Quadrilaterals)

Track quiz

Ten ideas, eleven checks. The midpoint theorem gets two, since it runs both ways.

Ten claims about four corners, in one cheat sheet
ClaimThe argument that earns it
Forcing the nameA safe test is passed by every parallelogram. No other shape passes it.
The diagonal splitA landing diagonal gives alternate angles, which force both side pairs parallel.
Both pairs of sides equalSSS on the shared diagonal, then the same alternate-angle steps.
One pair equal and parallelSAS on the diagonal, then the other alternate pair.
Diagonals that bisectSAS at the crossing, then the one-pair route finishes it.
Midpoint theoremJoin two midpoints of a triangle. The segment is parallel to the third side. It is exactly half that side.
Medians and the centroidThe three medians meet at one point, the centroid. It splits every median 2 to 1 from the corner.
The midpoint 4-gonJoin the four side midpoints of any quadrilateral in order. They make a parallelogram.
Fold linesIn every parallelogram the diagonals bisect each other. A half-turn about their crossing swaps each corner with the opposite one. So every parallelogram turns onto itself. A slanted one has no fold line.
Tiling with any 4-gonFour copies of one quadrilateral meet at a point. Each copy gives one corner. There is no gap and no overlap.

A test is passed by every parallelogram. What else must be checked before it is called safe?

  • Nothing. Passing on every parallelogram is enough
  • That no other shape passes it
  • That it is easy to measure with a ruler

Diagonal AC of ABCD lands triangle ABC on triangle CDA exactly. What makes AB parallel to DC?

  • The shared side AC, since both triangles contain that side
  • AB and DC, since they are matching parts of the triangles
  • Angles BAC and DCA, since they are alternate and equal

A quadrilateral has sides 6, 4, 6, 4 in order. A second has sides 6, 4, 5, 4. Which claim is right?

  • Only the first is a parallelogram, since both pairs match
  • Both are parallelograms, since one pair of sides repeats in each
  • Neither is forced, since four lengths prove nothing

In ABCD, AB = DC = 7 and AB is parallel to DC. What is ABCD?

  • Only a trapezium, since one pair is parallel
  • Nothing yet, since the other pair is unknown
  • A parallelogram, since one pair is equal and parallel

The diagonals of a shape cross at O. OA = OC and OB = OD. Which step earns the first parallel pair?

  • The diagonals are equal, so AB is parallel to DC
  • SAS lands triangles AOB and COD, using equal angles at O
  • Angle OAB equals angle ODC, in the same shape

In triangle ABC, D and E are midpoints of AB and AC. What can you say about DE?

  • It runs parallel to BC and measures half of AB
  • It measures half of BC only if angle A is right
  • It is parallel to BC and half of BC

A line through D, the midpoint of AB, runs parallel to BC. Where does it meet AC?

  • At E, the midpoint of AC
  • Halfway between B and C
  • At the point one third along AC

In triangle ABC, medians BE and CF meet at G. P and Q are the midpoints of GB and GC. What shows that G halves the diagonals FQ and EP?

  • G is the middle of the triangle, so it halves everything
  • FE and PQ are equal and parallel, so FEQP is a parallelogram
  • G halves BE, so it must halve the diagonals too

ABCD has corners (0,0), (8,0), (10,6) and (2,4). Its side midpoints are (4,0), (9,3), (6,5) and (1,2). What shape do they make?

  • Nothing in particular, since the first shape was not a parallelogram
  • A smaller copy of the first shape, since midpoints shrink it
  • A parallelogram, since each pair of sides is half a diagonal

A shape turns onto itself after a half-turn. It has no fold line anywhere. Which name from the family fits?

  • A slanted parallelogram, since its diagonals bisect each other
  • A rectangle, since rectangles lean over
  • A kite, since kites have one axis

A tile has one corner pushed inwards. One angle is then more than 180°. What happens when it tiles a floor?

  • It cannot tile, since tiling needs tidy corners
  • It tiles, since the four corners total 360°
  • It needs a fix, since an angle over 180° is a fault

4-gons (Quadrilaterals) is the Class 9 topic this track follows. You have spent nine lessons earning claims instead of borrowing them.

Next: Surface Area and Volume. It asks two questions about cuboids, cylinders, cones and spheres.

Jump to lesson

This track

  1. 0 Start here
  2. 1 Which claims force the name
  3. 2 One diagonal, two triangles
  4. 3 Both pairs equal prove it
  5. 4 Three more ways in
  6. 5 The midpoint theorem
  7. 6 Checkpoint: earning the name
  8. 7 Three medians, one point
  9. 8 Four halfway points, one parallelogram
  10. 9 Fold lines and half-turns
  11. 10 Any four-gon covers a floor
  12. 11 Checkpoint: midpoints, symmetry and tiles
  13. 12 The big picture
  14. 13 Track quiz

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