Why this works. The quadrilateral is fully determined by two adjacent sides and their included angles: $AD=4\text{ cm}$, $AB=3\text{ cm}$, with angles $\angle DAB=130^\circ$ and $\angle ADC=110^\circ$, plus side $DC=6\text{ cm}$. Constructing from those data locates $B$ and $C$ uniquely; then $BC$ follows by joining the points.

If you start with $CD$ instead: Draw $CD=6\text{ cm}$. At $D$, draw the $110^\circ$ ray and measure $DA=4\text{ cm}$ to locate $A$. At $A$, draw the $130^\circ$ ray and measure $AB=3\text{ cm}$ to locate $B$. Finally join $B$ to $C$.

🧠 PROBLEM-SOLVING Strategy

Accurate Constructions with Ruler, Set Square & Protractor

Use these steps to draw perpendiculars, parallels, and quadrilaterals to scale.

1. Plan from knowns. Highlight all given lengths (e.g., $3\ \text{cm}$, $4\ \text{cm}$, $7.5\ \text{cm}$, $8\ \text{cm}$, $10.5\ \text{cm}$) and angles (e.g., $35^\circ$, $52^\circ$, $60^\circ$, $65^\circ$, $70^\circ$, $75^\circ$, $80^\circ$, $110^\circ$, $120^\circ$, $122^\circ$, $125^\circ$, $130^\circ$).
2. Perpendicular through a point on a line.
   • Set square: place one edge flush with the line; draw along the other edge to get $90^\circ$.
   • Protractor: put the centre on the point, align the baseline to the line, mark $90^\circ$, draw the ray.
3. Perpendicular from a point to a line. With the protractor centre at the point, align to the line and mark the two $90^\circ$ directions; draw the segment to meet the line at right angles and label the foot (e.g., $P$).
4. Parallel through a point.
   • Sliding set-square: hold a ruler along the given line; slide the set square against it and draw through the point.
   • Offset method (fixed gap): draw a perpendicular to the given line, measure the required separation (e.g., $5\ \text{cm}$) on the perpendicular, then draw a second perpendicular at that mark. The new line is parallel and at the correct distance.
5. Building a quadrilateral from side–angle data.
   1. Draw a convenient base (e.g., $10.5\ \text{cm}$).
   2. At each end, use the protractor to set the given base angles (e.g., $65^\circ$ and $80^\circ$).
   3. Measure along the rays the adjacent side lengths (e.g., $7.5\ \text{cm}$ and $8\ \text{cm}$) to locate the next vertices.
   4. Join the remaining pair of vertices to close the figure. For concave cases, ensure the interior angle (e.g., $122^\circ$) opens inward.
6. Reasoning checks.
   • Straight line totals $180^\circ$; angles around a point total $360^\circ$.
   • Opposite (vertical) angles are equal; corresponding/alternate angles are equal for parallel lines; co-interior angles sum to $180^\circ$.
   • For a quadrilateral, interior angles sum to $360^\circ$. If three angles are $120^\circ$, the fourth would be $0^\circ$ (degenerate) — useful sanity check.
7. Accuracy tips. Work to within $\pm 1\ \text{mm}$ and $\pm 1^\circ$. Keep pencil sharp, read the correct protractor scale ($0^\circ$–$180^\circ$), and label as you go (e.g., $A,B,C,D$, $P$).

📘 What we've learned

• How to draw perpendicular lines using a set square or protractor (align and mark $90^\circ$).
• How to draw a line parallel to a given line, including the fixed-distance method via a perpendicular offset (e.g., $5\ \text{cm}$).
• Angle facts at an intersection: vertically opposite angles are equal; adjacent (linear pair) sum to $180^\circ$.
• Angle facts with parallel lines cut by a transversal: corresponding and alternate interior angles are equal; co-interior angles sum to $180^\circ$.
• Sums of angles: on a straight line $180^\circ$ and around a point $360^\circ$.
• Interior angles of any quadrilateral add to $360^\circ$; three angles of $120^\circ$ force the fourth to be $0^\circ$ (degenerate), so such a quadrilateral is impossible.
• Practical construction of quadrilaterals from side–angle data using ruler and protractor (e.g., build from a known base and two base angles/side lengths).
• That a quadrilateral can be fixed by about five independent measurements (e.g., one side, the two adjacent sides, and the two end angles).
• How to use right-triangle trigonometry or measurement to check lengths dropped perpendicular to a line (e.g., find $XP$ and $YP$ with a known angle and side).
• Checked accuracy by comparing measured results against expected values (e.g., diagonals/segments close to target lengths).