Constructing Triangles, ASA

Introduction

Constructing ASA triangles explains to construct a triangle where the measure of two angles and one of the side lengths is given to us. Geometry is the branch of mathematics that deals with properties & relations of points, lines, surfaces & solids. Also, it deals with geometrical construction.

Geometrical construction is constructing or drawing geometrical figures like lines, line segments, triangles, circles & quadrilaterals etc. There are several methods of constructing geometrical figures. We can construct geometrical figures by using geometrical instruments like ruler, protractor, compass, divider & set squares. Triangles are mostly constructed by using a ruler, protractor & compass. There are several criteria for constructing different types of triangles like SSS criteria, ASA criteria, SAS criteria & RHS criteria. Let’s study the construction of triangle by ASA criteria.

Construction of triangle when a side & two angles adjacent to it are given:

If side & two angles adjacent to a side are given, then a triangle is constructed by using the following steps −

• Step 1 − Before constructing a triangle, draw a rough figure using the given measurements.

• Step 2 − Draw the segment as a base of a triangle using the ruler.

• Step 3 − Place the centre of the protractor on the endpoint of the segement. Measure one of the given angles & mark it then, draw a ray through the first point.

• Step 4 − Place the centre of the protractor on another endpoint of the base segment measure the other angle & draw a ray.

• Step 5 − The intersection of two rays gives the required triangle.

For example, Construct ΔXYZ such that l(YX) = 6 cm, $\mathrm{m\angle\:ZXY\:=\:30°,\:m\angle\:ZXY\:=\:100°}$.

Answer − Step 1 − Before constructing a triangle, draw a rough figure as shown below.

Step 2 − Draw seg YX as a base having a length of 6 cm.

Step 3 − Place the centre of the protractor at Y, and measure the angle of 1000, draw a ray from point Y.

Step 4 − Place the centre of the protractor at X, measure the angle of 300, draw a ray from point X. Name the point of intersection as Z. This is required ΔXYZ.

Solved examples

1) In ΔABC, BC = 6cm $\mathrm{m\angle\:A\:=\:65°\:n\angle\:C\:=\:55°}$

Answer −

• Before constructing a triangle draw a rough figure as shown below.

• Draw base segment BC having a length of 6 cm.

• By using protractor measure $\mathrm{\angle\:A\:=\:65°}$ & mark it then draw a ray.

• On another endpoint of line segment i.e., at C measure $\mathrm{\angle\:C\:=\:55°}$ & mark it then draw a ray .

• Intersection of two rays will give the third vertice of a triangle. Name the point of intersection as A. This is required ΔABC.

2) In ΔLMN, MN = 5.2 cm $\mathrm{m\angle\:M\:=\:70°\:,\:m\angle\:N\:=\:40°}$

Answer − Before constructing a triangle, draw a rough figure as shown below.

• Draw base segment MN having a length of 5.2cm.

• By using a protractor, measure $\mathrm{\angle\:L\:=\:45°}$ & draw a ray.

• On another endpoint of line segment i.e., at T measure $\mathrm{\angle\:N\:=\:40°}$ & draw a ray.

• Intersection of two rays will give the third vertice of a triangle. Name the point of intersection as M. This is required ΔLMN.

3) In ΔDEF, EF = 7.3 cm, $\mathrm{m\angle\:E\:=\:34°,m\angle\:=\:95°}$

Answer −

• Before constructing a triangle draw a rough figure as shown below.

• Draw base segment EF having a length of 7.3 cm.

• By using protractor measure $\mathrm{\angle\:E\:=\:34°}$ & draw a ray.

• On another endpoint of line segment i.e., at F measure $\mathrm{\angle\:F\:=\:95°}$ & draw a ray.

• Intersection of two rays will give the third vertice of a triangle. Name the point of intersection as D. This is required ΔDEF.

4) In ΔXYZ, XZ = 6 cm, $\mathrm{m\angle\:Y\:=\:60°,m\angle\:Z\:=\:40°}$

Answer − We have the length of another side of a triangle, which is not adjacent to one angle, hence for finding adjacent, we will use the property of the sum of angles of a triangle,

By the property of the sum of angles of a triangle,

$\mathrm{\angle\:X\:\angle\:Y\:\angle\:Z\:=\:180°}$

$\mathrm{\angle\:X\:\angle\:60°\:\angle\:40°\:=\:180°}$

$\mathrm{\angle\:X\:\angle\:100°\:=\:180°}$

$\mathrm{\angle\:X\:=\:180°\:100°}$

$\mathrm{\angle\:X\:=\:80°}$

• Before constructing a triangle, draw a rough figure as shown below.

• Draw base segment XZ having a length of 6 cm.

• By using a protractor, measure $\mathrm{\angle\:X\:=\:80°}$ & draw a ray.

• On another endpoint of the line segment i.e., at T, measure $\mathrm{\angle\:T\:=\:40°}$ & draw a ray TD.

• Intersection of two rays will give the third vertice of a triangle. Name the point of intersection as X. This is required ΔXYZ.

Conclusion

• This tutorial covers the topic of constructing triangles, ASA in brief.

• In this tutorial, we have learned steps to constructing triangles if two adjacent angles & one side (ASA) are given with solved examples.

• Geometric construction is one of the important topic in geometry.

• Geometrical construction is constructing or drawing geometrical figures like lines, line segments, triangles, circles, quadrilaterals etc.

• We can draw these figures by using geometrical instruments like a ruler, compass, protractor, divider & set square.

• There are several criteria for constructing a triangle.

• These criteria are SSS, SAS, ASA & RHS

• Triangles can be constructed using a protractor, ruler & compass.

FAQs

1. What are the types of triangles?

The equilateral triangle, isosceles triangle, scalene triangle, acute-angled triangle, right-angled triangle & obtuse-angled triangle are the types of triangles.

2. What is the criterion for constructing triangles?

There are four criteria for constructing triangles, they are as follows −

• SSS criteria

• SAS criteria

• ASA criteria

• RHS criteria

3. State whether the following statement is true or false. The sum of the angles of triangles is 𝟏𝟖𝟎°.

True, by the angle sum property, we know that the sum of all angles of a triangle is 180°.

4. How do you construct a triangle if three sides of a triangle are given?

First, draw a line with the length of the longest side. Then draw arcs from each endpoint of the line segment such that they intersect each other. The intersection of arcs gives the third vertex of a triangle. Join the intersecting point with the endpoints of the line segment