10.3 Angles, Coordinates, Translation & Reflection

3. Angle Theorems & Calculations

Fundamental Angle Facts

1. Angles on a Straight Line: Adjacent angles on a straight line add to $180^\circ$.
2. Angles Around a Point: Angles around a single point sum to $360^\circ$.
3. Vertically Opposite Angles: When two straight lines intersect, the non-adjacent opposite angles are equal ($a = c$ and $b = d$).
4. Angles in a Triangle: Sum to $180^\circ$.
5. Angles in a Quadrilateral: Sum to $360^\circ$.

Optional extension: angles in parallel lines

When a straight transversal line crosses two parallel lines, three primary angle relationships emerge:

Use the stated parallel lines and named angle positions; a sketch must identify the relevant sides of the transversal.

1. Alternate Angles (the "Z-Rule"): Angles lying on opposite sides of the transversal between the two parallel lines are equal: $\angle a = \angle b$.
2. Corresponding Angles (the "F-Rule"): Angles in identical relative positions at each intersection are equal: $\angle c = \angle d$.
3. Allied / Co-Interior Angles (the "C-Rule"): Interior angles on the same side of the transversal sum to $180^\circ$: $$\angle x + \angle y = 180^\circ$$

---

4. Coordinates, Translations & Reflections

Cartesian Coordinates in the First Quadrant

The first-quadrant working area used in primary-school grids has non-negative coordinates ($x \ge 0, y \ge 0$), including the boundary axes. Strictly, points inside the quadrant have both coordinates positive.

• The horizontal axis is the $x$-axis; the vertical axis is the $y$-axis.
• The point of intersection $(0, 0)$ is the origin.
• In any coordinate pair $(x, y)$, the $x$-coordinate is stated first, followed by the $y$-coordinate: "along the corridor, then up the stairs".

Deducing Missing Coordinates of Polygons

Entrance questions frequently give three coordinates of a rectangle, square, or parallelogram and ask for the fourth vertex. Use horizontal and vertical displacement vectors:

• If vertex $A = (2, 3)$ and vertex $B = (8, 3)$, the horizontal displacement is $+6$ units along the $x$-axis.
• If vertex $C = (10, 7)$, the fourth vertex $D$ of a parallelogram lies 6 units to the left along the $x$-axis at the same height: $$D = (10 - 6, 7) = (4, 7)$$

8-Point Compass & Directional Bearings

                  North (000°)
                       │
        North-West     │     North-East
          (315°)       │       (045°)
             \         │         /
              \        │        /
   West ───────┼───────┼───────┼─────── East (090°)
  (270°)      /        │        \
             /         │         \
        South-West     │     South-East
          (225°)       │       (135°)
                       │
                  South (180°)

• Each 45-degree increment shifts between adjacent 8-point compass directions.
• Clockwise turns add to the angle; anti-clockwise turns subtract from the angle. (e.g., turning clockwise from West ($270^\circ$) through $135^\circ$: $270^\circ + 135^\circ = 405^\circ \equiv 405^\circ - 360^\circ = 045^\circ = \text{North-East}$).

2D Transformations: Translations & Reflections

1. Translations

A translation shifts a shape horizontally, vertically, or both without changing its size, orientation, or internal angles.

• Moving right adds to the $x$-coordinate: $(x + a, y)$.
• Moving left subtracts from the $x$-coordinate: $(x - a, y)$.
• Moving up adds to the $y$-coordinate: $(x, y + b)$.
• Moving down subtracts from the $y$-coordinate: $(x, y - b)$.

2. Reflections

A reflection flips a shape across a specified mirror line. Every vertex on the reflected image lies at the exact same perpendicular distance from the mirror line as the corresponding original vertex, but on the opposite side:

• Reflection across a vertical line $x = k$:
  • The $y$-coordinate remains unchanged: $y_{\text{image}} = y_{\text{original}}$.
  • The distance to the mirror line is $k - x_{\text{original}}$. The reflected $x$-coordinate is: $$x_{\text{image}} = k + (k - x_{\text{original}}) = 2k - x_{\text{original}}$$
• Reflection across a horizontal line $y = c$:
  • The $x$-coordinate remains unchanged: $x_{\text{image}} = x_{\text{original}}$.
  • The reflected $y$-coordinate is: $$y_{\text{image}} = 2c - y_{\text{original}}$$

---

Measuring angles and identifying solids from drawings

An acute angle is less than 90°, a right angle is 90°, an obtuse angle is between 90° and 180°, and a reflex angle is between 180° and 360°. Estimate first: an angle visibly wider than a right angle cannot be 35°. On a printed protractor diagram, put its centre at the vertex and align the baseline with one arm. Choose the scale that starts at zero on that arm. If the other arm meets 50 on one scale and 130 on the other, your initial acute/obtuse estimate helps choose correctly. Do not assume a picture is drawn accurately unless the question says to measure it.

To draw a 65° angle during ordinary practice, draw a baseline, place the protractor at its endpoint, mark 65° on the correct scale and join that mark to the vertex. In the live online assessment, follow its instructions and supplied diagram; this classroom exercise teaches the measurement idea rather than permission to bring extra equipment.

For a 3D drawing, distinguish the solid from the flat shapes used to represent its faces. A cube viewed from a corner may show three diamond-like regions, but its actual faces are squares. A cylinder may be drawn with elliptical ends, yet those ends represent circles seen at an angle. A square-based pyramid has a square base and triangular side faces. Dashed lines can mark hidden edges; count them when the task asks about the whole solid, not just its visible edges.

5. Worked Examples

Worked Example 1: an isosceles triangle angle (extension)

Triangle PQR has PQ = PR and angle PQR = 56°. The equal sides meet at P, so the base angles at Q and R are both 56°. Using the triangle angle sum, angle QPR = 180° − 56° − 56° = 68°. The input angle is stated explicitly; do not infer it from an unlabelled sketch.

Worked Example 2: Finding a Missing Parallelogram Vertex

Problem: Three vertices of a parallelogram $ABCD$ are plotted in the first quadrant of a Cartesian grid: $A(2, 4)$, $B(9, 4)$, and $C(12, 9)$. What are the coordinates of vertex $D$?

• Step 1 (Analyze vector displacement between known opposite sides):
  • Side $AB$ is horizontal, running from $(2, 4)$ to $(9, 4)$. Length $= 9 - 2 = 7$ units.
  • Because opposite sides of a parallelogram are congruent and parallel, side $DC$ must also be horizontal, having a length of $7$ units and pointing in the same direction from $D$ to $C$.
• Step 2 (Calculate coordinates of $D$ from $C$):
  • Since $C$ is at $(12, 9)$, vertex $D$ must lie $7$ units to the left of $C$ at the same vertical height: $$x_D = 12 - 7 = 5$$ $$y_D = 9$$
• Step 3 (Verify via alternate side $AD$):
  • Side $BC$ moves from $(9, 4)$ to $(12, 9)$, which is a shift of $+3$ units horizontally and $+5$ units vertically.
  • Applying this shift to $A(2, 4)$: $$D = (2 + 3, 4 + 5) = (5, 9)$$
  • Both methods yield the exact same point.
• Answer: $(5, 9)$.

Core missing-angle checks

Adjacent angles filling a straight line total 180°. If one is 67°, the other is 113°, because 180 − 67 = 113. Angles filling a whole turn total 360°. If three are 80°, 130° and 90°, their total is 300° and the remaining angle is 60°. A rectangle has four right angles and equal opposite sides: if one horizontal side is 9 cm, the opposite horizontal side is also 9 cm, regardless of how the rectangle is tilted in the picture.