Mastery Guide: Angles, Triangles, Quadrilaterals, Symmetry & 3D Nets

(i) All Adjacent Angles:
Adjacent pairs share a common vertex and common arm: $(\angle 1, \angle 2)$, $(\angle 2, \angle 3)$, $(\angle 3, \angle 4)$, and $(\angle 4, \angle 1)$.

(ii) All Supplementary Angles:
Pairs of adjacent angles that form a flat straight line ($180^\circ$): $(\angle 1, \angle 2)$, $(\angle 2, \angle 3)$, $(\angle 3, \angle 4)$, and $(\angle 4, \angle 1)$.

(i) Angles $50^\circ, 60^\circ, 70^\circ$: $50^\circ + 60^\circ + 70^\circ = 110^\circ + 70^\circ = \mathbf{180^\circ}$ ✓ (Verified)

(ii) Angles $90^\circ, 45^\circ, 45^\circ$: $90^\circ + 45^\circ + 45^\circ = 135^\circ + 45^\circ = \mathbf{180^\circ}$ ✓ (Verified)

(iii) Angles $30^\circ, 40^\circ, 110^\circ$: $30^\circ + 40^\circ + 110^\circ = 70^\circ + 110^\circ = \mathbf{180^\circ}$ ✓ (Verified)

(iv) Angles $60^\circ, 60^\circ, 60^\circ$: $60^\circ + 60^\circ + 60^\circ = 120^\circ + 60^\circ = \mathbf{180^\circ}$ ✓ (Verified)

(i) Given angles $35^\circ$ and $65^\circ$: $\text{Third Angle} = 180^\circ - (35^\circ + 65^\circ) = 180^\circ - 100^\circ = \mathbf{80^\circ}$

(ii) Given angles $45^\circ$ and $85^\circ$: $\text{Third Angle} = 180^\circ - (45^\circ + 85^\circ) = 180^\circ - 130^\circ = \mathbf{50^\circ}$

(iii) Given angles $25^\circ$ and $115^\circ$: $\text{Third Angle} = 180^\circ - (25^\circ + 115^\circ) = 180^\circ - 140^\circ = \mathbf{40^\circ}$

(i) Isosceles triangle with top vertex angle $80^\circ$ and two equal base angles $x$:
$$x + x + 80^\circ = 180^\circ \implies 2x = 180^\circ - 80^\circ = 100^\circ \implies x = 100^\circ \div 2 = \mathbf{50^\circ}$$

(ii) Right-angled triangle with one acute angle $42^\circ$ and unknown angle $x$:
$$x + 42^\circ + 90^\circ = 180^\circ \implies x = 180^\circ - 132^\circ = \mathbf{48^\circ}$$

(iii) Triangle with interior angles $50^\circ$ and $60^\circ$, unknown interior angle $x$, and adjacent exterior angle $y$:
$$x = 180^\circ - (50^\circ + 60^\circ) = 180^\circ - 110^\circ = \mathbf{70^\circ}$$ $$y = 180^\circ - x = 180^\circ - 70^\circ = \mathbf{110^\circ}$$

Answer: Sajid must turn $255^\circ$ more to face North again.

(a) $4\text{ cm}, 4\text{ cm}, 4\text{ cm}$: Equilateral Triangle (All 3 sides equal)

(b) $5\text{ cm}, 5\text{ cm}, 3\text{ cm}$: Isosceles Triangle (2 sides equal)

(c) $3\text{ cm}, 4\text{ cm}, 5\text{ cm}$: Scalene Triangle (All 3 sides different)

(d) $6\text{ cm}, 6\text{ cm}, 2\text{ cm}$: Isosceles Triangle (2 sides equal)

(a) $60^\circ, 70^\circ, 50^\circ$: Acute-angled Triangle (All angles $< 90^\circ$)

(b) $90^\circ, 40^\circ, 50^\circ$: Right-angled Triangle (Has a $90^\circ$ right angle)

(c) $120^\circ, 35^\circ, 25^\circ$: Obtuse-angled Triangle (Has an angle $> 90^\circ$)

(d) $60^\circ, 60^\circ, 60^\circ$: Acute-angled / Equiangular Triangle (All angles equal to $60^\circ$)

(a) Square: All 4 sides equal, all 4 angles $90^\circ$, 2 pairs of parallel sides, 4 lines of symmetry.

(b) Rectangle: Opposite sides equal and parallel, all 4 angles $90^\circ$, 2 lines of symmetry.

(c) Parallelogram: Opposite sides equal and parallel, opposite angles equal, 0 lines of symmetry.

(d) Rhombus: All 4 sides equal, opposite sides parallel, opposite angles equal, 2 lines of symmetry.

(e) Trapezium: Exactly 1 pair of opposite sides parallel.

(f) Kite: 2 pairs of adjacent equal sides, diagonals intersect at $90^\circ$, 1 line of symmetry.

Shapes with reflective symmetry are symmetric when folded along a central vertical or horizontal line (e.g., isosceles triangle, rectangle, butterfly shape, regular star). Asymmetric irregular polygons have no lines of symmetry.

(i) Square: Exactly $\mathbf{4\text{ lines of symmetry}}$ (1 vertical, 1 horizontal, 2 diagonal).

(ii) Rectangle: Exactly $\mathbf{2\text{ lines of symmetry}}$ (1 vertical, 1 horizontal).

(i) Exactly 2 lines of symmetry: Rectangle and Rhombus.

(ii) 0 lines of symmetry: Parallelogram (and general Scalene Trapezium).

(iii) Exactly 1 line of symmetry: Kite (and Isosceles Trapezium).

Reflect each vertex and segment across the bold red symmetry line by counting equal grid units to produce the matching mirror image.

(i) Equilateral Triangle: $\mathbf{3\text{ lines of symmetry}}$ (from each vertex to opposite midpoint).

(ii) Isosceles Triangle: $\mathbf{1\text{ line of symmetry}}$ (from apex vertex to base midpoint).

(iii) Scalene Triangle: $\mathbf{0\text{ lines of symmetry}}$ (no equal sides or angles).

A cube has 6 square faces. Nets that fold without overlapping faces to create a closed cube are valid cube nets (such as the standard cross net and T-net with 4 squares in a row and 1 square on each side).

(a) Net with 6 connected rectangles: Cuboid

(b) Net with 1 central square and 4 attached triangles: Square-based Pyramid

(c) Net with 6 connected squares in cross shape: Cube

(d) Net with 2 triangles and 3 rectangles: Triangular Prism

(e) Net with 5 squares/rectangles forming an open box: Open Box / Cuboid Net

(a) Angle sum around the point is: (iv) $360^\circ$

(b) Which represent angles on a straight line? (ii) $60^\circ, 120^\circ$ (since $60^\circ + 120^\circ = 180^\circ$)

(c) The complement of $20^\circ$ is: (iii) $70^\circ$ (since $90^\circ - 20^\circ = 70^\circ$)

(d) Sum of measures of three angles in a triangle is: (iii) $180^\circ$

(e) Two angles will be called supplementary angles if their sum is equal to: (i) $180^\circ$

(f) Which of the following shapes is not a quadrilateral? (iii) Pentagon (has 5 sides, not 4)

(g) Two angles in a triangle are $30^\circ, 60^\circ$. Measure of third angle is: (iv) $90^\circ$ ($180^\circ - 90^\circ = 90^\circ$)

(h) A triangle with .......... equal sides is called an isosceles triangle: (ii) 2

(i) Which of the following is not the net of a cube? (iii) 2×3 solid grid (overlapping faces cannot fold into a cube)

(j) Which is showing adjacent angles? (i) (angles sharing a common vertex and common arm with non-overlapping interiors)

(k) The order of rotational symmetry of a regular hexagon is: 6

Figure (a): Angles $l$ and $m$ share a common vertex and common middle arm → Adjacent Angles.

Figure (b): Angles $p$ and $q$ share a vertex and common dividing ray → Adjacent Angles.

Figure (c): $40^\circ$ and $50^\circ$ share the common ray → Adjacent Complementary Angles ($40^\circ + 50^\circ = 90^\circ$).

Figure (d): Angles $a$ and $b$ on the straight line share a common arm → Adjacent Supplementary Angles.

With respect to sides (3 types): (1) Equilateral Triangle, (2) Isosceles Triangle, (3) Scalene Triangle.

With respect to angles (3 types): (1) Acute-angled Triangle, (2) Right-angled Triangle, (3) Obtuse-angled Triangle.

(a) Scalene right triangle: No reflective symmetry ($0$ lines of symmetry).

(b) Symmetrical plaque: Has reflective symmetry ($2$ lines of symmetry: vertical and horizontal).

(c) Arrowhead / Chevron: Has reflective symmetry ($1$ horizontal line of symmetry).

(d) Division sign $\div$: Has reflective symmetry ($2$ lines of symmetry: vertical and horizontal).

Draw a net of 6 squares ($4\text{ cm} \times 4\text{ cm}$) on cardboard → Fold into a Cube (6 faces, 12 edges, 8 vertices).
Draw a net of 1 square base with 4 triangular flaps → Fold into a Square Pyramid (5 faces, 8 edges, 5 vertices).

(i) Straight line linear pair with $130^\circ$:
$$m = 180^\circ - 130^\circ = \mathbf{50^\circ}$$

(ii) Angles around a point: $79^\circ, 175^\circ,$ and $n$:
$$n = 360^\circ - (79^\circ + 175^\circ) = 360^\circ - 254^\circ = \mathbf{106^\circ}$$

(iii) Straight line with $50^\circ$, right angle ($90^\circ$), and angle $a$:
$$50^\circ + 90^\circ + a = 180^\circ \implies a = 180^\circ - 140^\circ = \mathbf{40^\circ}$$

(iv) Angles around a point with $231^\circ$ and 3 equal angles $c$:
$$231^\circ + 3c = 360^\circ \implies 3c = 360^\circ - 231^\circ = 129^\circ \implies c = 129^\circ \div 3 = \mathbf{43^\circ}$$

Answer: The value of angle $g$ is $36^\circ$.