Exercise 3.4 — SASAA Construction

Constructing a Quadrilateral Using the S.A.S.A.A. Method

Exercise 3.4 of the Class 8 Mathematics chapter on Construction of Quadrilaterals (CBSE, Telangana, and Andhra Pradesh syllabi) focuses on a specific but very important case: constructing a quadrilateral when two adjacent sides and three angles are known. This is called the S.A.S.A.A. method — Side, Angle, Side, Angle, Angle.

A quadrilateral has four sides and four angles, but you only need five independent measurements to pin it down uniquely. The S.A.S.A.A. set — two neighbouring sides and three of the four angles — provides exactly those five values. The fourth angle is always recoverable from the angle-sum property.

Problem (a) — Quadrilateral HELP

Step 0 – Find the Missing Angle

Three of the four angles are given. Before picking up a ruler, always find the fourth angle using the angle-sum property. This prevents errors during construction.

Given Data at a Glance

Rough Sketch

Steps of Construction — HELP

1. Draw a line segment HE = 6 cm. This is your base.
2. At vertex E, draw ray EX making an angle of 105° with HE, using a protractor.
3. With E as centre and radius 4.5 cm, draw an arc that cuts ray EX at point L.
4. At vertex H, draw ray HY making an angle of 60° with HE, on the same side as L.
5. At vertex L, draw ray LZ making an angle of 75° with LE (the calculated ∠L). This ray should go towards ray HY.
6. The point where ray LZ meets ray HY is vertex P. Join all vertices — HELP is the required quadrilateral.

Problem (b) — Parallelogram GRAM

Key Properties of a Parallelogram Used Here

• Opposite sides are equal — GR = AM = 5 cm and RA = MG = 6.2 cm (already given)
• Opposite angles are equal — ∠R = ∠M = 85°
• Adjacent angles are supplementary — ∠G + ∠R = 180°, so ∠G = ∠A = 180° − 85° = 95°

Given Data at a Glance

Rough Sketch

Steps of Construction — GRAM

1. Draw line segment GR = 5 cm as the base.
2. At R, draw ray RX making an angle of 85° with GR.
3. With R as centre and radius 6.2 cm, mark point A on ray RX.
4. At A, draw ray AY making an angle of 95° with RA (going towards G's side).
5. At G, draw ray GZ making an angle of 95° with GR.
6. The intersection of rays AY and GZ gives point M. Join all four vertices — GRAM is complete.

Problem (c) — Rectangle FLAG

Key Properties of a Rectangle Used Here

• All angles are 90° — ∠F = ∠L = ∠A = ∠G = 90°
• Opposite sides are equal and parallel — FL = AG = 6 cm; LA = FG = 4.2 cm
• A rectangle is a special parallelogram, so the S.A.S.A.A. method applies directly with all angles fixed at 90°

Given Data at a Glance

Rough Sketch

Steps of Construction — FLAG

1. Draw line segment FL = 6 cm as the base.
2. At L, draw ray LX making an angle of 90° with FL using a protractor or set-square.
3. With L as centre and radius 4.2 cm, mark point A on ray LX.
4. At A, draw ray AY making an angle of 90° with LA (going towards F's side).
5. At F, draw ray FZ making an angle of 90° with FL.
6. The intersection of rays AY and FZ gives point G. Join all four vertices — FLAG is the required rectangle.

Common Mistakes to Avoid in Exercise 3.4

• Skipping the angle-sum step — Always calculate the missing angle before starting. Drawing without it leads to a shape that won't close.
• Wrong angle direction — All rays must be drawn on the same side of the base line. If you flip one angle, P or M will land on the wrong side.
• Compass radius set incorrectly — Double-check the compass opening against your ruler before marking EL or RA. A 0.5 mm error compounds into a visible gap.
• Not using a sharp pencil — Thick pencil lines make intersection points ambiguous. Use a well-sharpened HB pencil for accurate constructions.
• Parallelogram property errors — In problem (b), students sometimes forget that ∠G = 95° (not 85°). Adjacent angles in a parallelogram are supplementary, not equal.
• Rectangle shortcut error — While opposite sides could be marked with a compass, it is faster and more accurate to draw 90° rays at every vertex using a set-square. Confirm all four angles before submitting.

What This Exercise Prepares You For

Mastering S.A.S.A.A. constructions builds a precise geometric intuition that is tested directly in board exams for CBSE Class 8, and in Telangana and Andhra Pradesh state board assessments. The skills practised here — working with angle properties of special quadrilaterals, using the angle-sum rule, and accurately operating a compass and protractor — reappear throughout secondary mathematics.

• Solidifies your understanding of the angle-sum property introduced earlier in this chapter.
• Connects directly to properties of parallelograms, rectangles, rhombuses, and squares covered in Exercises 3.1–3.3.
• Lays the foundation for Class 9 and 10 chapters on Triangle Congruence and Similarity.
• Connects to Coordinate Geometry, where you plot quadrilaterals and verify angle conditions algebraically.