How to Construct a Triangle Using SSS SAS ASA and RHS Methods
Introduction
We have studied triangles. To recall a figure formed by the intersection of three lines is said to be a triangle. A figure having three vertices, three sides, and three angles is called a triangle.
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In above figure, figure ABC is a triangle , here AB, BC, AC, are the sides of the triangle. A, B, C are the vertex and ∠ A, ∠B, ∠ C are the three angles. ASA means Angle-Side-Angle we have to construct a triangle when its two angles and included side between the angles are given. In this article we will study how to draw a triangle when measurements of two angles and a side included between these two angles are given.
ASA Triangle Construction
For constructing an ASA Triangle we must have measurements of two alternate angles and a length of a side between these two angles. If any other side rather than included side is given it does not satisfy the conditions of ASA and ASA triangle cannot be constructed.
To construct an ASA triangle we will require a ruler to measure the length of the side and a protractor to measure the angles.
Let us construct an ASA triangle with side AB = 3cm, ∠ CAB = 450 and ∠CBA = 600
Steps to Construct an ASA Triangle
Draw a rough sketch of the triangle having two angles as 450 and 600. And the side included between them is 3cm.
Dram a segment AB of length 3cm using a ruler.
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Now take a protractor and place its pointer at point B and mark a point at 600 and Make a ray joining the point and B.
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Now place the protractor point at A and measure 45p. Mark a point at 450. Join the point and the point A making a ray at point A.
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You will see both the rays intersect at a point. Name it as C.
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You will get the required triangle Δ ABC with side AB = 3cm, ∠ CAB = 450 and ∠CBA = 600.
The third angle of the triangle will be 750, by the angle sum property of a triangle.
I.e 60 + 45 + third angle = 1800
105 + third angle = 180
Third angle = 180 - 105
Third angle = 750
Congruent Triangles
Two triangles are congruent to each other if one of them is superimposed on another such that they both cover each other completely.
Two triangles are said to be congruent if the sides and angles of one triangle are exactly equal to the corresponding sides and angles of the other triangle.
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From the figure, Δ ABC is congruent to Δ DEF and it is written as Δ ABC ≅ Δ DEF
In two congruent triangles, corresponding parts of corresponding angles are equal they are:
∠ A = ∠ D,
∠ B = ∠ E,
∠ C = ∠ F
and
AB = DE ,
BC = EF ,
AC = DF.
ASA ( Angle-Side-Angle) Congruence Rule
If two angles and the side included by the two angles of one triangle are equal to the corresponding angles and side included by the angles of another triangle then the two triangles are said to be congruent.
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Here, ∠ B = ∠ Q
∠ C = ∠ R
And the side between the angles are equal
BC = QR;
therefore Δ ABC ≅ Δ PQR …….by ASA criteria
also,∠ A = ∠ P; AB = PQ ; AC = PR ………..by c.p.c.t property
Solved Examples
1. State True or False: In ΔABC, the Side Included Between ∠B and ∠C is AB.
Solution: False
Two angles form the end points of the segment included between them. Here ∠B and ∠C will form segment BC. hence segment AB is not included between ∠B and ∠C.
2. In ΔABC, BC = CA. Which of its Two Angles are Equal ?
Solution : ∠A and ∠B are equal because of the property angles opposite to equal sides are equal. And we have BC = CA, ∠A and ∠B are opposite angles
3. If m∠A=70 0 ,AB= 5cm and m∠B=600 Which Rule is Used here to Construct Triangle ABC
Solution: As measurements of two angles are given i.e m∠A=700 and m∠B=600
And the side included between the angles AB = 5cm is also given so we can construct a triangle ABC with ASA ( Angle-Side-Angle) rule.
Problems to Solve
1.If we have PQ = 5 cm, ∠PQR= 115° and ∠QRP = 30°, can we construct a ΔPQR with these measurements?
2. Construct ΔABC in which BC=6 cm, ∠B = 350 and ∠C = 100 0 .
Measure ∠A.
FAQs on Constructing a Triangle in Geometry
1. What is constructing a triangle in geometry?
Constructing a triangle means drawing a triangle accurately using given measurements with a compass and ruler. In geometric construction, no freehand drawing or scaling is allowed. The triangle is constructed using exact data such as:
- Three sides (SSS)
- Two sides and the included angle (SAS)
- One side and two angles (ASA or AAS)
2. How do you construct a triangle when three sides are given (SSS)?
To construct a triangle using SSS (Side-Side-Side), draw one side and use arcs to locate the third vertex. Steps:
- Draw base AB equal to the given first side.
- With center A, draw an arc of radius equal to the second side.
- With center B, draw an arc of radius equal to the third side.
- The intersection point is C; join AC and BC.
3. How do you construct a triangle when two sides and the included angle are given (SAS)?
To construct a triangle using SAS (Side-Angle-Side), draw the given angle between the two known sides. Steps:
- Draw base AB equal to one given side.
- At point A, construct the given angle.
- On the angle arm, mark point C equal to the second given side.
- Join C to B.
4. How do you construct a triangle when one side and two angles are given (ASA)?
To construct a triangle using ASA (Angle-Side-Angle), draw the given side and construct the two angles at its endpoints. Steps:
- Draw base AB equal to the given side.
- At A, construct the first given angle.
- At B, construct the second given angle.
- The two angle arms intersect at point C.
5. What is the triangle inequality condition for constructing a triangle?
A triangle can be constructed only if the sum of any two sides is greater than the third side. This is called the triangle inequality theorem:
- a + b > c
- b + c > a
- c + a > b
6. Can a triangle be constructed with sides 2 cm, 3 cm, and 6 cm?
No, a triangle cannot be constructed because it violates the triangle inequality rule. Check:
- 2 + 3 = 5
- 5 is not greater than 6
7. How do you construct a right-angled triangle?
A right-angled triangle is constructed by drawing a 90° angle at one vertex. Steps:
- Draw base AB.
- At point A, construct a 90° angle using a compass.
- Mark the required length on the perpendicular arm.
- Join the new point to B.
8. What tools are required for constructing a triangle?
Triangle construction requires only a compass and a straightedge (ruler without markings). These tools are used to:
- Draw exact line segments
- Construct arcs and circles
- Measure and copy angles
9. Is a triangle uniquely determined by SSS, SAS, and ASA?
Yes, a triangle is uniquely determined by the conditions SSS, SAS, and ASA. These conditions fix both the shape and size of the triangle. However:
- SSA does not always give a unique triangle (ambiguous case).
10. What are common mistakes to avoid while constructing a triangle?
Common mistakes in constructing a triangle include ignoring given measurements and violating geometric rules. Avoid the following:
- Not checking the triangle inequality condition
- Using incorrect compass radius
- Drawing angles inaccurately
- Freehand sketching instead of proper construction
