Constructing Triangles (ASA)
When two angles and the included side (the side between those angles) are given, a unique triangle can be constructed. This is the ASA construction.
Draw the given side, construct one angle at each end, and the two rays will intersect to form the third vertex.
What is Constructing Triangles (ASA) - Grade 7 Maths (Practical Geometry)?
Steps to construct △ABC given ∠A, AB, ∠B:
- Draw AB of the given length.
- At A, construct ∠BAX = ∠A using a protractor.
- At B, construct ∠ABY = ∠B on the same side.
- Rays AX and BY intersect at point C.
△ABC is constructed.
Constructing Triangles (ASA) Formula
Two angles + Included side → Unique triangle
Note: The sum of the two given angles must be less than 180°. Otherwise, the rays will never meet.
Types and Properties
Checks before construction:
- ∠A + ∠B < 180° (otherwise rays are parallel or diverging).
- The third angle = 180° − ∠A − ∠B.
Solved Examples
Example 1: Acute Triangle
Problem: Construct △PQR: ∠P = 50°, PQ = 6 cm, ∠Q = 60°.
Solution:
- Draw PQ = 6 cm.
- At P, construct 50° above PQ.
- At Q, construct 60° above PQ on the same side.
- The rays meet at R.
Answer: △PQR constructed with ∠R = 180° − 50° − 60° = 70°.
Example 2: Right Triangle
Problem: Construct △ABC: ∠A = 90°, AB = 5 cm, ∠B = 30°.
Solution:
- Draw AB = 5 cm.
- At A, construct 90°.
- At B, construct 30°.
- Rays meet at C.
Answer: Right triangle with ∠C = 60°.
Real-World Applications
Real-world uses:
- Surveying: Triangulation — measuring two angles from known baseline.
- Navigation: Determining position using angles from known points.
Key Points to Remember
- ASA needs two angles and the side between them.
- Sum of two angles must be less than 180°.
- The triangle is unique.
- Use a protractor to construct angles at both ends.
Practice Problems
- Construct △ABC: ∠A = 45°, AB = 7 cm, ∠B = 75°.
- Construct △XYZ: ∠X = 60°, XY = 5 cm, ∠Y = 60°. What type of triangle is this?
- Can you construct a triangle with ∠A = 100° and ∠B = 90°? Why or why not?
Frequently Asked Questions
Q1. What is ASA construction?
Constructing a triangle given two angles and the side between them.
Q2. What if the two angles sum to 180° or more?
The construction is impossible. The two rays would be parallel (sum = 180°) or diverge (sum > 180°).
Q3. How is ASA different from AAS?
In ASA, the side is between the angles. In AAS, the side is not between them. Both determine a unique triangle, but the construction steps differ slightly.
Related Topics
- Constructing Triangles (SSS)
- Constructing Triangles (SAS)
- ASA 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
