Quadrilaterals

Skills and methods in this topic, easy to hard

1. 1Properties of a parallelogramOpposite sides and angles equal, diagonals bisect each other, next-door angles make 180°

   Method∠A = 70°: ∠C = 70°, ∠B = 180° − 70° = 110°

   Another way: Two congruent triangles△ABC ≅ △CDA (ASA), AB = CD, ∠B = ∠D

2. 2Rectangles and squaresRectangle: equal diagonals, AE = BE = CE = DE; square: diagonals at 45° to the sides

   Method∠EAB = 35°: ∠EBA = 35°, ∠AEB = 180° − 70° = 110°

   Another way: Use the right angle∠EBC = 90° − 35° = 55°

3. 3RhombusesDiagonals at right angles, bisecting the angles: four congruent right triangles

   Method∠ABC = 70°: ∠ABE = 35°, ∠BAE = 90° − 35° = 55°

   Another way: An isosceles triangle∠BAC = (180° − 70°) ÷ 2 = 55°

4. 4Conditions for a parallelogramOpposite sides equal, opposite angles equal, diagonals bisect, one pair equal and parallel

   MethodAB // DC, 2x + 1 = x + 5, x = 4

   Another way: Put it back in2 × 4 + 1 = 9, 4 + 5 = 9

5. 5Proofs with quadrilateralsFind congruent triangles: alternate angles, a common side, opposite sides of the parallelogram

   Method∠EAB = ∠ECD, ∠EBA = ∠EDC, AB = CD: △ABE ≅ △CDE (ASA)

   Another way: Work back from the endTo prove AE = CE: look at △ABE and △CDE

6. 6The mid-point theoremThe line joining two mid-points is parallel to the third side and half as long

   MethodBC = 12: MN = 12 ÷ 2 = 6

   Another way: Similar trianglesAM : AB = 1 : 2, MN = 12 × 12 = 6

7. 7The intercept theoremEqual pieces on one crossing line mean equal pieces on any other

   MethodAB = BC: DE = EF; 2x + 1 = x + 4, x = 3

   Another way: Even lines cut equal parts10 cm across 5 gaps: 10 ÷ 5 = 2 cm

8. 8Congruent quadrilateralsFour equal sides aren't enough; four sides and a diagonal are

   MethodA 5 cm square and a 5 cm rhombus: equal sides, not congruent

   Another way: Two triangles along a diagonalAB, BC, AC equal and CD, DA, AC equal: SSS twice

Worked examples

From different skills in this topic. Have a go first, then open each one for the answer and the working.

1. Example 1Properties of a parallelogram →

   ABCD is a parallelogram. x =

   ▶Show the answer and working

   Answer: 55°

   ∠C and ∠A are opposite: x = 55° (opp. ∠s of // gram).

2. Example 2Proofs with quadrilaterals →

   ABCD is a parallelogram. One line of a proof is: ∠BAC = ∠DCA ( ? ). Which reason goes in the gap?

   • A. opp. ∠s of // gram
   • B. alt. ∠s, AD // BC
   • C. alt. ∠s, AB // DC
   • D. vert. opp. ∠s

   Choose the right answer

   ▶Show the answer and working

   Answer: C. alt. ∠s, AB // DC

   AC cuts the parallels AB and DC, and ∠BAC and ∠DCA are on either side of it: alt. ∠s, AB // DC.

3. Example 3Congruent quadrilaterals →

   A square with 7 cm sides and a square with 10 cm sides are congruent.

   True or false?

   ▶Show the answer and working

   Answer: False

   A square's angles are all 90°; two squares with the same side fit exactly, with different sides they don't.