Topic Review — Quadrilaterals and Geometric Reasoning — Solutions

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1. Properties of quadrilaterals

   1. All sides equal, all angles 90°: Square
   2. All sides equal, opposite angles equal, no right angles: Rhombus
   3. Parallelogram, one angle 68°: Opposite angle = 68° (opposite angles of parallelogram are equal); Adjacent angle = 180 − 68 = 112° (co-interior angles, AB ∥ DC, sum to 180°); Fourth angle = 68° (opposite to 68°). All four: 68°, 112°, 68°, 112°.
   4. Rhombus diagonals 16 cm and 12 cm: Half-diagonals = 8 cm and 6 cm. Side² = 8² + 6² = 64 + 36 = 100; Side = 10 cm (Pythagoras; diagonals bisect at right angles)
   5. Rectangle diagonal 26 cm: Diagonals of a rectangle are equal and bisect each other; each half = 26 ÷ 2 = 13 cm
   6. Square and rhombus shared properties: Both have: all four sides equal; opposite sides parallel; opposite angles equal; diagonals bisect each other at right angles; diagonals bisect the vertex angles.

2. Angles in polygons

   1. Nonagon interior angle sum: (9−2) × 180 = 7 × 180 = 1260°
   2. Regular hexagon angles: Interior = 720 ÷ 6 = 120°; Exterior = 60°
   3. Exterior = 36°: n = 360 ÷ 36 = 10 sides (regular decagon)
   4. Pentagon, four known (95+110+120+105=430): Fifth angle = 540 − 430 = 110°
   5. Interior angle 156°: Exterior = 24°; n = 360 ÷ 24 = 15 sides
   6. 12-sided polygon interior angle sum: (12−2) × 180 = 1800°

3. Geometric reasoning

   1. Parallel lines, angle 74°: (i) Alternate angle = 74° (alternate angles, parallel lines, are equal)   (ii) Co-interior angle = 180 − 74 = 106° (co-interior angles sum to 180°)   (iii) Corresponding angle = 74° (corresponding angles, parallel lines, are equal)
   2. Isosceles, apex 52°: ∠ABC = ∠ACB (base angles of isosceles triangle AB = AC); ∠ABC = (180 − 52) ÷ 2 = 64° each (angle sum of triangle = 180°)
   3. Intersecting lines, 118°: Vertically opposite = 118°; Adjacent = 180 − 118 = 62° (angles on a straight line); Other adjacent = 62° (vertically opposite). Four angles: 118°, 62°, 118°, 62°.
   4. Parallelogram, (4x−10)° and (2x+40)°: (4x−10) + (2x+40) = 180 (co-interior angles); 6x + 30 = 180; x = 25; ∠DAB = 90°, ∠ABC = 90°. All angles 90° — this is a rectangle (special case of parallelogram).

4. Geometric proofs

   1. Exterior angle = sum of non-adjacent interior angles: Let triangle ABC have exterior angle ∠ACD at C. ∠ACB + ∠ACD = 180° (angles on straight line BC). Also ∠BAC + ∠ABC + ∠ACB = 180° (angle sum of triangle). Therefore ∠ACD = ∠BAC + ∠ABC. ✓
   2. Rectangle PQRS, ∠QPR = 35°: ∠PQR = 90° (angle of rectangle); In triangle PQR: ∠PQR + ∠QPR + ∠PRQ = 180°; 90 + 35 + ∠PRQ = 180; ∠PRQ = 55°. ∠PRS = ∠QPS − ∠QPR... simpler: ∠PRS = 90 − 35 = 55° (since ∠PSR = 90° in rectangle, diagonal PR, ∠PRS + ∠QPR = 90° by complement in right triangle). ∠PRS = 55°.
   3. Rhombus diagonals bisect vertex angles: In rhombus ABCD, all sides equal. Diagonal AC: triangles ABC and ADC are congruent (SSS: AB=AD, BC=DC, AC common). Therefore ∠BAC = ∠DAC (corresponding angles). So AC bisects ∠A and ∠C. Similarly BD bisects ∠B and ∠D. ✓
   4. All sides equal ⇒ all angles equal: FALSE. Counterexample: a rhombus has all four sides equal but angles need not be 90° (e.g. angles 60°, 120°, 60°, 120°). A square has both all sides equal AND all angles equal; a rhombus does not.

5. Mixed problem solving

   1. Hexagons and equilateral triangles tessellate together: Regular hexagon interior angle = 120°; Equilateral triangle interior angle = 60°. At a shared vertex: two hexagons and two triangles: 120 + 120 + 60 + 60 = 360°. Alternatively one hexagon and two triangles: 120 + 60 + 60 = 240 (not 360). The common tessellation uses 2 hexagons + 2 triangles meeting at a point: not valid. Standard tessellation uses 1 hex + 2 triangles per vertex = 120 + 60 + 60 + 120 = 360. Accepted answer: depending on arrangement; the angles must sum to 360° for a valid tessellation.
   2. AB ∥ DC, ∠DAB = 55°, ∠ABC = 80°: ∠ADC = 180 − 55 = 125° (co-interior angles, AB ∥ DC); ∠BCD = 360 − 55 − 80 − 125 = 100° (angle sum of quadrilateral). The shape is a trapezium (exactly one pair of parallel sides, AB ∥ DC).
   3. Rectangle 15 cm × 8 cm: (i) Diagonal = √(15² + 8²) = √(225+64) = √289 = 17 cm (Pythagoras)   (ii) Angle at the 15 cm end: tan(θ) = 8/15 → θ ≈ 28.1°; other acute angle = 90 − 28.1 = 61.9°
   4. Interior angle sum 630°: (n−2) × 180 = 630; n−2 = 3.5; n = 5.5. Not a whole number, so impossible. A polygon must have a whole number of sides. 630° cannot be the interior angle sum of any polygon.