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Figure It Out: Parallelograms

April 10, 2024
1 min read
solutions parallelogram rhombus

Page 96: Parallelogram Angles

Question: If one angle is 30∘30^\circ, find the others. Solution:

  • Adjacent angles sum to 180∘180^\circ. ∠B=180∘−30∘=150∘\angle B = 180^\circ - 30^\circ = 150^\circ.
  • Opposite angles are equal. ∠C=30∘\angle C = 30^\circ, ∠D=150∘\angle D = 150^\circ.

Page 102: Figure It Out

1. Find remaining angles:

  • (i) Parallelogram: Given 40∘40^\circ.
    • Opposite = 40∘40^\circ.
    • Adjacent = 180∘−40∘=140∘180^\circ - 40^\circ = 140^\circ.
  • (ii) Parallelogram: Given 110∘110^\circ.
    • Opposite = 110∘110^\circ.
    • Adjacent = 70∘70^\circ.
  • (iii) Rhombus: Given 30∘30^\circ (half angle by diagonal?).
    • If diagonal makes 30∘30^\circ with side, whole angle = 60∘60^\circ.
    • Adjacent angle = 180∘−60∘=120∘180^\circ - 60^\circ = 120^\circ.
  • (iv) Rhombus: Given 20∘20^\circ triangle base angle.
    • Diagonals intersect at 90∘90^\circ.
    • Third angle in small triangle = 180−(90+20)=70∘180 - (90+20) = 70^\circ.
    • Vertex Angles: 40∘40^\circ (2×202 \times 20) and 140∘140^\circ (2×702 \times 70).

Page 107: Trapeziums

Question 3: Find remaining angles.

  • Left Figure: Angles between parallel lines sum to 180∘180^\circ.
    • 135∘135^\circ pair: 180−135=45∘180 - 135 = 45^\circ.
    • 105∘105^\circ pair: 180−105=75∘180 - 105 = 75^\circ.
  • Right Figure: 100∘100^\circ pair →80∘\rightarrow 80^\circ.

Question 5: Two rectangles PAIR and RODS. Find ∠IOD\angle IOD.

  • Properties: Diagonals of rectangle bisect and are equal.
  • In rectangle RODS, diagonals intersect at O. ΔROD\Delta ROD isosceles logic applies.
  • Need specific diagram values to solve fully, but generally involves subtracting angles from 90∘90^\circ or using isosceles triangle properties at the intersection.