Geometric Twins Class 7 Notes Maths Part 2 Chapter 1

Easy-to-read Ganita Prakash Class 7 Notes and Part 2 Chapter 1 Geometric Twins Class 7 Notes save valuable study time during exam season.

Class 7 Maths Chapter 1 Geometric Twins Notes

Class 7 Geometric Twins Notes

Two figures or objects that have same shape and same size, and can be superimposed on each other by sliding, turning or flipping, are called geometric twins or congruent figures.

Two line segments are said to be congruent if they have the same length.

Two angles are said to be congruent if they have the same measure.

Two polygons are congruent if they are of the same size and shape, i.e. if their corresponding angles and sides are equal.

Two triangles are congruent if their corresponding sides are equal in length and their corresponding angles are equal in measure.

Criteria for Congruence of Triangles
There are five criteria for congruence of triangles as given below:

SSS (Side – Side – Side) congruence criterion
If the three sides of one triangle are equal to corresponding three sides of another triangle, then the triangles are congruent.

In general, for any two triangles, ∆ABC and ∆PQR, if B AB = PQ, BC = QR and AC = PR, then according to SSS congruence criterion, ∆ABC ≅ ∆PQR.

SAS (Side – Angle – Side) congruence criterion

If two sides and the included angle of one triangle are equal to the corresponding two sides and the included angle of another triangle, then the triangles are congruent.

In general, for any two triangles ∆ABC and ∆PQR, if AB = PQ, ∠A = ∠P and AC = PR, then according to SAS congruence criterion, ∆ABC ≅ ∆PQR.

ASA (Angle – Side – Angle) congruence criterion

If two angles and the included side of one triangle are equal to the corresponding two angles and included side of another triangley. then the triangles are congruent.

In general, for any two triangles, ∆ABC and ∆PQR, if ∠B = ∠Q BC = QR and ∠C = ∠R, then according to ASA congruence criterion, ∆ABC ≅ ∆PQR.

AAS (Angle – Angle – Side) congruence criterion

If two angles and the non-included side of one triangle are equal to the corresponding two angles and the non-included side of another triangle, then the triangles are congruent.

In general, for any two triangles, ∆ABC and ∆PQR, if ∠A = ∠P, ∠B = ∠Q and BC = QR. then according to AAS congruence criterion,
∆ABC ≅ ∆PQR.

RHS (Right Angle – Hypotenuse – Side) congruence criterion

If the hypotenuse and one side of one right-angled triangle are equal to the hypotenuse and corresponding side of another right-angled triangle, then the triangles are congruent.

In general, if ∆ABC and ∆PQR are two right triangles g such that ∠B = ∠Q= 90°, AC = PR and BC = QR, then according to RHS congruence criterion, ∆ABC ≅ ∆PQR.

Angles of Isosceles and Equilateral Triangles
Angles opposite to equal sides of an isosceles triangle are equal. Conversely, sides opposite to equal angles of a triangle are equal.

An equilateral triangle has all sides equal and all the angles in an equilateral triangle are equal, i.e. 60°.

Important Points

1. The sum of all interior angles of a triangle is 180°, and that of a quadrilateral (polygon with 4 sides) is 360°.
2. Two congruent figures have the same area and same perimeter.
3. The angle that lies between two given sides of a triangle is called the included angle.
4. An angle of a triangle that does not lie between the two given sides is called a non-included angle.
5. The side of a triangle that lies between two given angles is called the included side.
6. A non-included side of a triangle is the side that does not lie between the two given angles.
7. AAA (Angle – Angle – Angle) and SSA (Side – Side – Angle) are not congruence criteria.

Concept of congruence

1. Two figures or objects that have same shape and same size and can be superimposed on each other by sliding, turning or flipping are called congruent figures.
2. Two line segments are said to be congruent if they have the same length.
3. Two angles are said to be congruent if they hat e the same measure.
4. Two squares are said to be congruent, if they have same length of the sides.
5. Two rectangles are said to be congruent if they have same length and breadth.
6. Two circles are said to be congruent if they have the same radius.

Congruency of Triangles
SSS Congruence Rule

If measure of all three sides of triangle ABC are equal to the measure of three corresponding sides of other triangle DEE then ∆ABC ≅ ∆DEF, by SSS (Side-Side-Side) congruence rule.

SAS Congruence Rule

If any two sides and the included angle of triangle ABC are equal to the corresponding two sides and the included angle in the another triangle DEF, then ∆ABC ≅ ∆DEF, bv SAS (Side-Angle-Side) congruence rule.

RHS Congruence Rule

In right-angled ∆ABC, if the hypotenuse and any one side of triangle are equal to the hypotenuse and corresponding side of another right angled ∆DEF, then ∆ABC ≅ ∆DEF, by RHS (Right angle-Hypotenuse- Side) congruence rule.

ASA Congruence Rule

If any two angles and the included side of triangle ABC are equal to the corresponding two angles and included side of the another triangle DEF, then ∆ABC ≅ ∆DEF, by ASA (Angle-Side-Angle) congruence rule.

AAS Congruence Rule

If any two angles and ,non- included side of triangle ABC are equal to the corresponding two angles and non-included side of the another triangle DEF, then ∆ABC ≅ ∆DEF, by AAS (Angle- Angle-Side) congruence rule.