Visiting for the first time? Visit Triangles Part1 for Basics of triangle. It includes Properties and Classification of triangles, Area and Perimeter of triangles with definition and formulas.

We’ve familiarized ourselves with triangle’s basic properties. We’ve classified triangles based on their sides’ length and the measure of their internal angles and understood the ways or methods of calculating any triangle’s perimeter and area. Now that we’ve gained insight about a triangle singularly, let us compare two or more triangles and understand what similarity and congruence mean.

Imagine the earth from space. You’ll see its majestic blue and green hues, its countries and its mountains. The globe that you see in Geography classroom or in people’s offices is simply a smaller version of it. The shape is the same, the countries and the oceans are exactly where they are, it’s just that the size has been reduced. It’s a scaled down form of the actual thing. The same idea can be qualified even in the case of triangles.

What are Similar Triangles?

Ideally, two triangles are said to be similar when their corresponding angles measure the same and when their corresponding sides are in proportion. Similarity is denoted by the symbol ~ .

What are Congruent Triangles?

On the other hand, two triangles are said to be congruent when all their three sides measure the same in length and the measure of their corresponding three angles too is the same. Congruence is denoted by the symbol ≅ .

Now that we’ve had a look at what similarity in triangles means, let us look at the different sorts of tests that are needed to check if two triangles are similar or congruent or both.

Tests of similarity in triangles:

• AA Test or AAA Test:

When the two of the corresponding angles of two triangles are equal, the third angle of the same pair of triangles is bound to be equal. In Such a case, the two triangles are similar.

Triangles Example:

Q.1) In triangle XYZ, segment OP is parallel to segment YZ. Find the length of XO.

Solution: We know that OP is parallel to YZ.
∴∠XOP = ∠XYZ and
∠XPO = ∠XZY (when 2 or more parallel lines exist their corresponding angles are congruent)
Since △ XYZ and △ XOP have 2 pairs of congruent angles, their third corresponding angle is bound to be congruent.
By AA Test, △ XYZ ~ △ XOP
We know that similar triangles’ corresponding sides are in proportion.
∴ OP/XO = YZ/XY
∴ 3/XO = 9/18
∴ XO = 3*18/9
∴ XO = 6 cm

• SSS Test:

The SSS Test stands for Side Side Side Test in similarity. This test states that if the ratio of corresponding measures of three sides of a triangle is the same as the ratio of the corresponding measures of three sides of any other triangle, then the two triangles are said to be similar. XY:YZ:XZ = FG:EG:EF = 5:6:10

• SAS Test:

This test is called Side Angle Side test. Two triangles are said to be similar when the ratio of 2 sides in one triangle is equal to the ratio of the 2 corresponding sides in the other triangle and the included angles are equal.

Tests of congruency in triangles:

• SSS Test:

It stands for Side Side Side test. If all the three sides of one triangle are equal to the three sides of another triangle, then the triangles are said to be congruent.

• SAS Test:

It stands for Side Angle Side test. If the two sides of a triangle and the included angle is equal to the corresponding two sides in the other triangle and its included angle, both the triangles are said to be congruent.

Triangles Example:

Q.2) In the figure below, O is the midpoint of CS and GP. Find the value of ∠ OSZ.

Solution: We know that in triangles GCO and PSO, GO = PO (O is the midpoint of GP)
∠COG = ∠SOP (vertically opposite angles)
CO = SO (O is the midpoint of CS)
∴ By the virtue of the SAS test, △ GCO ≅ △ PSO
Hence, ∠GCO = ∠PSO = 60° (corresponding angles)
Now, ∠OSZ and ∠PSO form a linear pair.
∴ ∠OSZ +∠PSO = 180°
∴∠OSZ + 60 = 180
∴ ∠OSZ = 120°

Triangles Example:

Q.3) In the given figure △ QAN and △ USL are congruent by SAS test of congruence. Find the values of x and y.

Solution: We know that both △ QAN and △ USL are congruent and hence will have congruent corresponding sides and angles.
∴ QA = US
∴ 3x + 10 = 5y + 15 …...(1)
And
∠AQN = ∠SUL
∴ 2x + 15 = 5x – 60
∴ 3x = 75
∴ x = 25
Substituting x = 25 in (1), we get
3(25) + 10 = 5y + 15
∴ 75 + 10 = 5y + 15
∴ 85 =5y + 15
∴ 85 - 15 = 5y
∴ 5y =70
∴ y = 14

• ASA Test:

It stands for Angle Side Angle Test. If in one triangle, two angles and the included side of that triangle are equal to the corresponding angles and side of another triangle, then the respective triangles are said to be congruent.

• AAS Test:

This stands for Angle Angle Side Test. According to this, If any two angles and the non-included side of a triangle are equal to the corresponding two angles and the non-included side of another triangle, then the two triangles in question are said to be congruent.

• HL Test:

It stands for Hypotenuse Leg Test. The longest side of a right-angled triangle is referred to as the hypotenuse and the other two sides are called legs. In two triangles, if the hypotenuse and one of the legs are equal, then both the triangles are said to be congruent.

Now that we’ve looked at what the variety of tests used to judge similarity and congruence in triangles are, let us study the various lines associated with the shape of a triangle.

• Angle Bisector of a Triangle:

The line that divides an angle of the triangle into two equal parts is referred to it as an angle bisector. In triangle ABC, AD, FC and BE are the angle bisectors that divide angles BAC, BCA and ABC into 2 equal parts. All triangles will have 3 angle bisectors. The point of the meeting of all the three angle bisectors of a triangle gives us the incentre T, which is the center of the largest circle that can be drawn inside triangle ABC. The incentre of all the triangles, irrespective of their shape will principally fall inside the triangle.

Triangles Example:

Q.4) In triangle APS, DS is the angle bisector of angle ASP. ∠PAS = 60°. ∠APS = 90°. Find out the measure of ∠DSP.

Solution: We know that the measure of all the angles in a triangle adds up to 180°.
Let the measure of ∠ASP be x°.
Hence, in △ APS, ∠APS +∠ASP + ∠SAP = 180°
∴ 90 + x + 60 = 180
∴ x +150 = 180
∴ x =180 – 150
∴ x = 30
∴∠ASP = 30°
Since DS is the angle bisector of △ ASP, ∠DSP = ∠DSA = ½ of ∠ASP
∴ ∠ DSP = ½ of 30 ° = 15°

Triangles Example:

Q.5) In the figure, △ACV and △GCD are right triangles. AC = 10 cm, CD = 15 cm, GD = 12 cm. Find out the measure of AV and CV.

Solution: We know that △ ACV and △ GCD are right triangles and their common angle is ∠C.
∴ The third angle is bound to be congruent.
∴△ ACV and △ GCD are similar by AA Test.
Similar triangles’ corresponding sides are in proportion.
∴ AV/AC = GD/CD
∴AV/10 = 12/15
∴ AV = 8 cm.
Since △ ACV is right angled, by Pythagoras’ theorem,
AV² + CV² = AC²
∴ 8² + CV² = 10²
∴ 64 + CV² = 100
∴ CV² = 36
∴ CV = √ 36
∴ CV = ± 6
∴ CV = 6 cm, since the length of anything cannot be negative.

• Perpendicular Bisector of a Triangle:

When a line divides the side of a triangle into 2 equal parts and is also perpendicular to it, it is termed as a perpendicular bisector. A triangle has 3 perpendicular bisectors. In triangle ABC, KO, KP and KI divide sides YX, XZ and YZ into two equal parts. The meeting of all the perpendicular bisectors of triangle XYZ gives us the Circumcenter, which is the center of the circle that can be drawn outside the triangle with all 3 of the triangle’s vertices falling on the circumference of the circle. In an acute-angled triangle, circumcenter will be inside the triangle. In an obtuse-angled triangle, it will be placed outside of the triangle. Circumcenter always is at the midpoint of the hypotenuse of a right-angled triangle.

• Altitude of a Triangle:

An altitude is a line drawn from the vertex of a triangle to the side opposite to it. The 3 altitudes of acute and right triangles will be inside the body of both the respective triangles. An obtuse angled triangle has one altitude drawn from the vertex of the obtuse angle, and the remaining two altitudes are drawn outside the body of the triangle by extending the base. The point where all three altitudes of a triangle meet is called as the orthocenter. In an acute angled triangle, the orthocenter will always be on the inside of the triangle. In an obtuse angled triangle, it will be outside. In a right triangle, the orthocenter will lie on the vertex forming the right angle. In triangle KLM, KR, QM and PL are the altitudes that meet at point O, which is the orthocenter of triangle KLM.

• Median of a Triangle:

It is the line segment which connects the vertex of a triangle to the midpoint of the side opposite to it. The point of meeting of all the three medians of a triangle is called as the Centroid. The centroid divides the median in the ratio of 2:1. It also divides the triangle into two parts of equal areas. For instance, area of triangle ENJ = area of triangle EJM. In triangle ENM, GN, JE and VM are the medians and Q is the centroid.

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