Constructing Triangles (SAS)
When two sides and the included angle are given, you can construct a unique triangle. This is the SAS construction.
The included angle is the angle between the two given sides. You draw one side, construct the given angle at one end, mark the second side along the angle's ray, and join the endpoints.
What is Constructing Triangles (SAS) - Grade 7 Maths (Practical Geometry)?
Steps to construct △ABC given AB, ∠A, and AC:
- Draw side AB of the given length.
- At point A, construct ∠BAX = the given angle using a protractor.
- On ray AX, mark point C such that AC = given length.
- Join B to C.
Triangle ABC is constructed.
Constructing Triangles (SAS) Formula
Two sides + Included angle → Unique triangle
Types and Properties
Variations:
- Acute angle: The third vertex falls above the base.
- Right angle: Use perpendicular construction instead of protractor.
- Obtuse angle: The third vertex may fall beyond the base extension.
Solved Examples
Example 1: Constructing with Acute Angle
Problem: Construct △PQR with PQ = 6 cm, ∠P = 50°, PR = 5 cm.
Solution:
- Draw PQ = 6 cm.
- At P, construct ∠QPX = 50° using protractor.
- On ray PX, mark R at 5 cm from P.
- Join QR.
Answer: △PQR is constructed.
Example 2: Constructing with Right Angle
Problem: Construct △ABC with AB = 5 cm, ∠A = 90°, AC = 4 cm.
Solution:
- Draw AB = 5 cm.
- At A, construct a 90° angle (perpendicular).
- On the perpendicular ray, mark C at 4 cm from A.
- Join BC.
Answer: Right-angled △ABC is constructed.
Example 3: Constructing with Obtuse Angle
Problem: Construct △DEF with DE = 4 cm, ∠D = 120°, DF = 3 cm.
Solution:
- Draw DE = 4 cm.
- At D, construct ∠EDX = 120°.
- On ray DX, mark F at 3 cm.
- Join EF.
Answer: Obtuse-angled △DEF is constructed.
Real-World Applications
Real-world uses:
- Construction: Creating angled supports with specific measurements.
- Carpentry: Cutting triangular pieces with given side and angle.
Key Points to Remember
- SAS requires two sides and the angle between them.
- The angle MUST be the included angle.
- Use protractor to construct the angle and compass/ruler for sides.
- The constructed triangle is unique.
Practice Problems
- Construct △ABC: AB = 5 cm, ∠A = 60°, AC = 4 cm.
- Construct △XYZ: XY = 7 cm, ∠X = 45°, XZ = 6 cm.
- Construct a right triangle with legs 3 cm and 4 cm.
Frequently Asked Questions
Q1. What is SAS construction?
Constructing a triangle when two sides and the angle between them (included angle) are given.
Q2. What if the angle is not between the two sides?
Then it is not SAS. You may not get a unique triangle. SAS requires the angle to be the included angle.
Q3. What tools are needed?
A ruler (to draw sides), a protractor (to measure the angle), and optionally a compass.
Related Topics
- Constructing Triangles (SSS)
- Constructing Triangles (ASA)
- SAS Congruence Rule
- Constructing Triangles (RHS)
- Constructing a Line Segment
- Constructing Perpendicular Lines
- Constructing Angles
- Bisecting a Line Segment
- Bisecting an Angle
- Constructing Parallel Lines
- Constructing Quadrilaterals
- Constructing Special Quadrilaterals
- Triangle Construction Problems
