Topic Review — Geometric Reasoning — Solutions

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1. Angle relationships fluency

   1. Complement of 34°: 90 − 34 = 56°
   2. Supplement of 97°: 180 − 97 = 83°
   3. Co-interior partner to 74°: 180 − 74 = 106° (co-interior angles, parallel lines)
   4. 48° intersection: Vertically opposite = 48°; adjacent angles = 132° each
   5. Third angle, 55°+78°: 180 − 55 − 78 = 47° (angle sum of triangle)
   6. Exterior 142°, one interior 89°: 142 − 89 = 53° (exterior angle theorem)

2. Triangle and quadrilateral angles

   1. Isosceles apex 28°: Base angles = (180−28)÷2 = 76° each
   2. Parallelogram 53°: Opposite = 53°; adjacent = 127° each
   3. Rhombus 118°: Opposite = 118°; adjacent = 62° each
   4. Trapezium co-interior 91°+y=180: y = 89°
   5. 2x+3x+4x+51=360: 9x=309; x=34.33°; angles≈68.7°, 103°, 137.3°, 51°
   6. Equilateral (3m−3)=60: 3m=63; m=21

3. Congruence and proof

   1. Two equal angles and included side: AAS (the side between the two angles)
   2. ▵ABC ≡ ▵CDA in parallelogram: AB=CD, BC=DA (opposite sides equal); AC=CA (common). ∴ SSS. Or: use SAS with the alternate angles.
   3. Base angles of isosceles are equal: Let AM bisect ∠A. In ▵ABM and ▵ACM: AB=AC (given); ∠BAM=∠CAM (AM bisects ∠A); AM=AM (common). ∴ ▵ABM≡▵ACM (SAS). Hence ∠B=∠C (corresponding parts).
   4. XM ⊥ YZ proof: In ▵XYM and ▵XZM: XY=XZ (given); YM=ZM (M midpoint); XM=XM (common). ∴ ▵XYM≡▵XZM (SSS). ∠XMY=∠XMZ (corresponding parts); ∠XMY+∠XMZ=180° (straight line). ∴ ∠XMY=90°, so XM⊥YZ.

4. Quadrilateral properties

   1. Diagonals bisect each other: Parallelogram, Rectangle, Rhombus, or Square
   2. Rectangle PR=SQ: 5k−3=3k+9: 2k=12; k=6; diagonal=27
   3. Rhombus side 13, diagonal 10: Half diagonal=5; other half=√(13²−5²)=√(169−25)=√144=12; full diagonal=24 cm
   4. Rhombus vs square: A rhombus has all sides equal but angles need not be 90°. A square has all sides equal AND all angles 90°. A rhombus becomes a square when all its angles are right angles.

5. Problem solving

   1. ∠APD with P between AB∥CD: Draw EP through P parallel to AB and CD. ∠APE=72° (alternate, AB∥EP); ∠EPC=38° (alternate, EP∥CD). ∠APD=72+38=110°
   2. ABCD with AB=BC, AD=DC, ∠B=90°: Kite (two pairs of adjacent equal sides). ∠A+∠D+∠B+∠C=360; ∠A+130+90+∠C=360; ∠A+∠C=140. In a kite, ∠B≠∠D, but ∠A=∠C=70°.
   3. Angles ratio 1:2:3: Angles=30°, 60°, 90°. Exterior at 90° vertex=180−90=90°. Verify: 30+60=90° ✓
   4. Square ABCD side 8, E midpoint BC: AB=DC=8 (sides of square); ∠B=∠C=90° (angles of square); BE=CE=4 (E midpoint). ∴ ▵ABE≡▵DCE (SAS). AE=√(AB²+BE²)=√(64+16)=√80=4√5 cm