Practical Geometry Ex 10.3: Constructing Triangles (SAS Criterion)

Understanding the SAS (Side-Angle-Side) Criterion for Triangle Construction

In geometry, the SAS (Side-Angle-Side) criterion is a fundamental rule that helps us construct a unique triangle. It states that if the lengths of two sides and the measure of the angle included between these two sides are given, then a unique triangle can be constructed. "Included angle" means the angle formed by the two given sides at their common vertex. Imagine you have two sticks of specific lengths and you want to join them at a particular angle. Once you've joined them, there's only one way to connect their free ends to form a closed triangle. This is the essence of the SAS criterion. It's crucial for accuracy to always start with a rough sketch. This helps you visualize the triangle and plan your construction steps, ensuring you don't miss any details or make silly errors. For example, if you're given sides AB = 5 cm, BC = 6 cm, and angle B = 60°, you'll first draw the 6 cm side (BC), then measure 60° at point B, and finally mark 5 cm along the ray from B. Connecting the endpoints completes your triangle.

Step-by-Step Construction using SAS Criterion

1. Step 1: Draw a Rough Sketch — Before you begin with your tools, always draw a rough sketch of the triangle you need to construct. Label the vertices and mark the given side lengths and the included angle. This helps you visualize and plan your construction.
2. Step 2: Draw the Base Side — Using a ruler, draw one of the given sides as the base of your triangle. For instance, if you are given sides AB, BC, and the included angle B, you can draw BC as the base.
3. Step 3: Construct the Included Angle — At the endpoint of the base where the included angle lies (e.g., point B if angle B is given), use a protractor to construct the given angle. Draw a ray from this point extending sufficiently.
4. Step 4: Mark the Second Side — With the compass, measure the length of the second given side. Place the compass needle at the vertex where the angle was constructed (e.g., point B) and draw an arc that cuts the ray drawn in Step 3. This intersection point will be the third vertex of your triangle (e.g., point A).
5. Step 5: Complete the Triangle — Finally, use your ruler to join the two endpoints (e.g., point A and point C) to complete the triangle. Your constructed triangle will now have the specified side lengths and included angle.

Worked Examples: Constructing Triangles with SAS

• Example 1: Construct triangle PQR where PQ = 4 cm, QR = 6 cm and ∠PQR = 75°. Step 1: Draw a rough sketch of triangle PQR with the given measurements. Step 2: Draw a line segment QR of length 6 cm. Step 3: At point Q, construct an angle of 75° using a protractor. Draw a ray QX making ∠RQX = 75°. Step 4: With Q as the center and radius 4 cm (length of PQ), draw an arc intersecting the ray QX at point P. Step 5: Join P to R. Triangle PQR is the required triangle.
• Example 2: Construct triangle ABC where AB = 5.5 cm, AC = 4 cm and ∠BAC = 60°. Step 1: Draw a rough sketch of triangle ABC with AB = 5.5 cm, AC = 4 cm, and the included angle ∠BAC = 60°. Step 2: Draw a line segment AB of length 5.5 cm. Step 3: At point A, construct an angle of 60° using a compass or protractor. Draw a ray AY such that ∠BAY = 60°. Step 4: With A as the center and radius 4 cm (length of AC), draw an arc intersecting the ray AY at point C. Step 5: Join C to B. Triangle ABC is the required triangle.
• Example 3: Construct triangle XYZ such that XY = 7 cm, YZ = 5 cm and ∠XYZ = 90°. Step 1: Draw a rough sketch for visualising the triangle XYZ with XY = 7 cm, YZ = 5 cm, and the right angle at Y. Step 2: Draw a line segment XY of length 7 cm. Step 3: At point Y, construct an angle of 90° (a right angle) using a protractor or compass. Draw a ray YP such that ∠XYP = 90°. Step 4: With Y as the center and radius 5 cm (length of YZ), draw an arc intersecting the ray YP at point Z. Step 5: Join Z to X. Triangle XYZ is the required right-angled triangle.

Exam Tip: Ensuring Accuracy in Practical Geometry

Accuracy is key in practical geometry. Always use a sharp pencil for clear, thin lines. A blunt pencil can lead to thick lines and inaccurate intersections, making your construction appear messy and imprecise. Use a well-calibrated ruler and a precise protractor or compass. When drawing angles, align the protractor's baseline exactly with the line segment and its center with the vertex. For compass constructions, ensure your compass is firm and doesn't slip. Double-check your measurements at each step. Drawing a rough sketch first is not just a suggestion; it's a critical planning step that helps avoid common errors and ensures you apply the correct criterion.

Practice Questions with Solutions

• Q: Construct triangle LMN where LM = 6 cm, MN = 4 cm and ∠LMN = 80°. A: Step 1: Draw a rough sketch of triangle LMN. Step 2: Draw a line segment LM of length 6 cm. Step 3: At point M, use a protractor to construct an angle of 80°. Draw a ray MX such that ∠LMX = 80°. Step 4: With M as the center and radius 4 cm (length of MN), draw an arc intersecting the ray MX at point N. Step 5: Join N to L. Triangle LMN is the required triangle. Final answer: The triangle LMN is constructed according to the given measurements.
• Q: Construct triangle DEF where DE = 5 cm, EF = 7 cm and ∠DEF = 45°. A: Step 1: Draw a rough sketch of triangle DEF. Step 2: Draw a line segment DE of length 5 cm. Step 3: At point E, use a protractor to construct an angle of 45°. Draw a ray EY such that ∠DEY = 45°. Step 4: With E as the center and radius 7 cm (length of EF), draw an arc intersecting the ray EY at point F. Step 5: Join F to D. Triangle DEF is the required triangle. Final answer: The triangle DEF is constructed.
• Q: Construct triangle PQR where PQ = 4.5 cm, PR = 6 cm and ∠QPR = 110°. A: Step 1: Draw a rough sketch of triangle PQR, ensuring the included angle is correctly identified. Step 2: Draw a line segment PQ of length 4.5 cm. Step 3: At point P, use a protractor to construct an angle of 110°. Draw a ray PZ such that ∠QPZ = 110°. Step 4: With P as the center and radius 6 cm (length of PR), draw an arc intersecting the ray PZ at point R. Step 5: Join R to Q. Triangle PQR is the required triangle. Final answer: The triangle PQR is constructed.
• Q: Construct triangle XYZ such that XY = 6.5 cm, YZ = 5.5 cm and ∠XYZ = 90°. A: Step 1: Draw a rough sketch of triangle XYZ. Step 2: Draw a line segment XY of length 6.5 cm. Step 3: At point Y, construct a right angle (90°) using a compass or protractor. Draw a ray YW such that ∠XYW = 90°. Step 4: With Y as the center and radius 5.5 cm (length of YZ), draw an arc intersecting the ray YW at point Z. Step 5: Join Z to X. Triangle XYZ is the required right-angled triangle. Final answer: The triangle XYZ is constructed.

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