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Page 96: Parallelogram Angles

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

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

1. Find remaining angles:

(i) Parallelogram: Given $40^\circ$ $4 0^{\circ}$ .
- Opposite = $40^\circ$ .
- Adjacent = $180^\circ - 40^\circ = 140^\circ$ .
(ii) Parallelogram: Given $110^\circ$ $11 0^{\circ}$ .
- Opposite = $110^\circ$ .
- Adjacent = $70^\circ$ .
(iii) Rhombus: Given $30^\circ$ $3 0^{\circ}$ (half angle by diagonal?).
- If diagonal makes $30^\circ$ with side, whole angle = $60^\circ$ .
- Adjacent angle = $180^\circ - 60^\circ = 120^\circ$ .
(iv) Rhombus: Given $20^\circ$ $2 0^{\circ}$ triangle base angle.
- Diagonals intersect at $90^\circ$ .
- Third angle in small triangle = $180 - (90+20) = 70^\circ$ .
- Vertex Angles: $40^\circ$ ( $2 \times 20$ ) and $140^\circ$ ( $2 \times 70$ ).

Question 3: Find remaining angles.

Left Figure: Angles between parallel lines sum to $180^\circ$ $18 0^{\circ}$ .
- $135^\circ$ pair: $180 - 135 = 45^\circ$ .
- $105^\circ$ pair: $180 - 105 = 75^\circ$ .
Right Figure: $100^\circ$ pair $\rightarrow 80^\circ$ .

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

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