Constructing Triangle

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

  1. Draw a rough sketch of the triangle having two angles as 450 and 600. And the side included between them is 3cm.

  2. Dram a segment AB of length 3cm using a ruler.

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  1. 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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  1. 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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  1. 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.

FAQ (Frequently Asked Questions)

1. How to Prove if the Triangles are Congruent or Not?

Answer: Two triangles are said to be congruent if all its sides and angles are equal. But it is not necessary that every time we know all the three sides and all the three angles. Hence we can show two triangles are congruent by different congruence rules. The five congruence rules are:

  1. SSS( side-side-side) congruence rule

  2. SAS ( side-angle-side) congruence rule

  3. ASA ( angle-side-angle) congruence rule

  4. AAS ( angle-angle-side) congruence rule

  5. RHS ( right angle- hypotenuse-side) congruence rule.

From these rules we can show that the two triangles are congruent

2. What are Similar Triangles?

Answer:

Triangles having the same shape but different sizes are known as similar triangles. Two congruent triangles are always similar but similar triangles need not be congruent. Two geometrical figures having exactly the same shape and size are said to be congruent figures. We have learned about congruent figures earlier too. Congruent figures are alike in every respect. 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.

Similar triangles have the following properties:

  • Similar triangles have the same shape but are not of the same size.

  • Each corresponding pair of angles of the two similar triangles is equal.

  • The ratio of any pair of corresponding sides of similar triangles is the same.