Triangles are the three-sided polygon geometric figure that comprises three edges and three vertices. The most significant property of a triangle is that the sum of all the interior angles of a triangle is 180°. Pythagoras theorem and trigonometry concepts are dependent on the properties of a triangle. It should be noted that the measurement of the three sides and angles of a triangle may or may not be equal in the construction of a triangle. Through ruler, compasses, protractor, and pencil, we can draw different types of a triangle. The construction of a triangle is possible if,

All the three sides of a triangle are mentioned

Two sides of a triangle one included angle is given.

Two angles of a triangle and one included side is given.

The measurement of the hypotenuse side and one other side is given.

Although different triangles have their own different properties, there are some common properties that are similar for all triangles. Some of the common properties that are similar for all the triangles are given below:

The sum of all three interior angles of a triangle is equal to 180 degrees.

The measurement of the exterior angle of a triangle is equivalent to the addition of the opposite interior angles.

The sum of the length of any two sides of a triangle is always greater than the third side of a triangle.

In a right-angle triangle, the sum of the square of the length of the hypotenuse side which is opposite to the right angle is equal to the sum of the square of the length of the other two sides of a triangle.

Triangle and the other geometrical shapes can be accurately constructed using a protractor, ruler and a pair of compasses.

SAS - Side Angle Side

ASA - Angle side angle

SSS - Side Side Side

All the dimensions and angles are not required to construct a triangle. A triangle can be constructed if any of the below-given criteria meets.

Below are the following sets of measurements that are required for the construction of a triangle.

Construction for the SSS triangle is possible when all the three sides of a triangle are given.

Criteria for the construction of the triangle when the perimeter and the two angles are given

Below, a given triangle can be constructed when the perimeter and two angles of the triangle are given.

Construction for the ASA triangle is possible when any of the two angles and one side of a triangle is given.

Construction for the SAS triangle is possible when any of the two sides and one angle of a triangle are given.

Construction for the RHS triangle is possible when the hypotenuse side and one other side of a triangle are given.

How to construct a triangle PQR with the length of PQ=5 cm, PR= 6 cm and QR = 4.5 cm.

Solution:

Given

PQ =5 cm

PR = 6 cm

QR = 4.5 cm

Steps of Construction

Draw a line of length PQ = 5cm and mark the points P and Q in it.

To draw the line segment PR, extend the two arms of the compass 6 cm away, place the one endpoint of the compass at point P and draw an arc with the pencil end.

To draw the line segment QR, fix the compass to 4.5 cm. Place the one end point of the compass at point Q and mark an arc with the pencil end. Draw an arc that intersects the previous arc drawn in step 2.Mark the intersection point of two arcs as R.

Join the line segments PR and QR.

Now , PQR is the required triangle

2. How to construct a triangle PQR with the length of PQ=4 CM, PR= 6.5 CM and ∠PQR = 60°

Solution :

Given- PQ = 4cm

PR = 6.5 cm

∠PQR = 60°

Steps Of Construction

Draw a line segment QR of length 6.5cm

Using a protractor and placing it at point Q, draw a line segment QX making an angle of 60° with QR.

Taking Q as center, draw an arc of radius 6 cm that cut the line QX at P.

Join PR.

Now, PQR is the required triangle

While constructing a triangle whose both perimeter and base angles are given, the first step is to

To draw the base of any length

To draw the base angle from any random paint

Draw base of length= 1/2* perimeter

Draw a base of any length= perimeter

If the construction of a triangle PQR in which PQ= 6cm, ∠P=70° and ∠40° is possible, then find the measurement of ∠C.

40°

70°

80°

50°

To draw the perpendicular bisector of line segment AB, we open the compass

More than ½ AB

Less than ½ AB

Equal to ½ AB

None of these

FAQ (Frequently Asked Questions)

1. How to Draw a Triangle When its Perimeter and Two Sides are Given. Mention its Steps of Construction.

For construction a triangle when its perimeter and two sides are given, you should have a ruler, a compass and protractor.

Let us take a triangle PQR whose length is 15 cm and two given angles are 55° and 60°.

Hence , P + Q+ R = 15 CM ,∠Q = 55° and ∠R = 65°

Steps of Construction

Draw a line segment XY of length AB +BC+AC = 15 cm

Using the protractor, construct an ∠LXY from the point Which is equivalent to ∠B =55°

Using the protractor, construct an ∠MYX from the point Which is equivalent to ∠C =60°

Construct the bisectors of ∠LXY and ∠MYX using the compass. Mark point A where two bisectors meet.

Draw the perpendicular bisector of AX and give a name to it as PQ.

Draw the perpendicular bisector of AY and give a name to it as RS.

Let PQ and RS touch the line segment XY at point B and C respectively.

Join the points A and B along with A and C.

2. Explain the Term Congruence of Triangle

The term congruence is used to define the object and the mirror image of any of the objects. Two shapes or objects will be considered as congruent if they overlap each other. The shapes and dimensions of the congruent triangle are similar. In Geometrical figures, the line segments within the same length and angle with the same measurement are equal.

Congruent Triangle

Congruent triangles are triangles having their sides and angles equal. The congruence is symbolized by the symbol ≅. The area and the perimeter of the congruent triangle are similar.

Two triangles are said to be congruent when their sides and angles have the same measure. Hence, two triangles can overlap side to side and angle to angle.

The triangle ABC and PQR, given below are congruent. It implies that,

Vertices- A and P , B and Q and C and R are similar.

Sides= AB=PR, QR=BC and AC=PR

Angles = ∠A = ∠P, = ∠Q AND ∠ C = ∠ R