Grade 9
  1. Number Systems  1.1. Real Numbers  1.2. Rational Numbers  1.3. Irrational Numbers  1.4. Properties of Real Numbers  1.5. Laws of Exponents and Surds  1.6. Representation on the Number Line  1.7. Decimal Representation of Real Numbers
  2. Polynomials  2.1. Definition and Types of Polynomials  2.2. Degree of a Polynomial  2.3. Zeroes of a Polynomial  2.4. Remainder Theorem  2.5. Factorization of Polynomials  2.6. Algebraic Identities  2.7. Polynomial Equations and Roots
  3. Coordinate Geometry  3.1. Cartesian System  3.2. Plotting Points on the Plane  3.3. Distance Formula  3.4. Section Formula  3.5. Area of a Triangle  3.6. Coordinates of Midpoint and Centroid
  4. Linear Equations in Two Variables  4.1. Solutions of a Linear Equation  4.2. Graphical Method  4.3. Algebraic Method  4.4. Applications of Linear Equations  4.5. Linear Inequalities
  5. Introduction to Euclidean Geometry  5.1. Basic Definitions and Terms  5.2. Axioms and Postulates  5.3. Theorems on Angles and Lines  5.4. Properties of Parallel Lines  5.5. Construction of Triangles  5.6. Congruence Axioms
  6. Lines and Angles  6.1. Angle Pair Relationships  6.2. Parallel Lines and Transversal  6.3. Properties of Angles Formed by Parallel Lines  6.4. Angle Sum Property of a Triangle  6.5. Types of Angles
  7. Triangles  7.1. Congruence of Triangles  7.2. Criteria for Congruence  7.3. Properties of Triangles  7.4. Inequalities in a Triangle  7.5. Similarity of Triangles  7.6. Pythagorean Theorem
  8. Quadrilaterals  8.1. Properties of Quadrilaterals  8.2. Types of Quadrilaterals  8.3. Midpoint Theorem  8.4. Properties of Parallelograms  8.5. Trapezium and Kite Properties  8.6. Diagonals and their Properties
  9. Areas of Parallelograms and Triangles  9.1. Area of a Parallelogram  9.2. Area of a Triangle  9.3. Proofs of Area Theorems  9.4. Applications of Area Theorems
  10. Circles  10.1. Definitions and Terms  10.2. Properties of a Circle  10.3. Chord Properties  10.4. Tangents to a Circle  10.5. Cyclic Quadrilaterals  10.6. Arcs and Angles Subtended
  11. Constructions  11.1. Basic Constructions  11.2. Constructing Bisectors  11.3. Constructing Triangles  11.4. Constructing Tangents to a Circle  11.5. Constructing Angles of Given Measure
  12. Heron's Formula  12.1. Area of a Triangle Using Heron’s Formula  12.2. Application to Different Triangles and Quadrilaterals  12.3. Proof of Heron’s Formula
  13. Surface Areas and Volumes  13.1. Surface Area of a Cube Cuboid and Cylinder  13.2. Volume of a Cube Cuboid and Cylinder  13.3. Surface Area and Volume of a Cone  13.4. Surface Area and Volume of a Sphere and Hemisphere  13.5. Conversion of Units  13.6. Volume of a Prism
  14. Statistics  14.1. Collection of Data  14.2. Presentation of Data  14.3. Measures of Central Tendency  14.4. Mean Median and Mode  14.5. Graphical Representation of Data  14.6. Range and Measures of Dispersion
  15. Probability  15.1. Introduction to Probability  15.2. Experimental Probability  15.3. Theoretical Probability  15.4. Simple Problems on Probability

Grade 9 → Triangles ↓


Criteria for Congruence


Congruence is an important concept in geometry, especially when studying triangles. Understanding when triangles are congruent helps determine their properties and equivalence. Two triangles are considered congruent if they have the same shape and measurement, although their positions and orientations may differ. This concept is essential because congruent triangles have equal corresponding angles and sides.

In this detailed discussion, we will explore the various criteria used to determine the congruence of a triangle. These criteria help us understand and identify congruent triangles accurately.

What is congruence?

Congruence literally means matching in shape and size. In math, when we talk about congruent triangles, we are referring to triangles that are the same in terms of their size and shape. This means that one triangle can be superimposed on another exactly the same way. If two triangles are congruent, all corresponding sides and angles are equal.

Understanding congruence

To say that two triangles are similar, we need to verify that they satisfy certain criteria. Before diving into these criteria, let's look at some basic properties of triangles:

  • A triangle has three sides, three angles, and three vertices.
  • The sum of the interior angles of a triangle is always 180 degrees.
  • Triangles are represented by naming the vertices, such as △ABC.

Criteria for conformity

There are several established rules for checking the congruence of triangles. These are named after the parts of the triangles being compared. The main criteria are:

  1. Side-Side-Side (SSS) Criterion
  2. Side-Angle-Side (SAS) Criterion
  3. Angle-Side-Angle (ASA) Criteria
  4. Angle-Angle-Side (AAS) Criterion
  5. Right Angle-Hypotenuse-Side (RHS) Criterion

Side-Side-Side (SSS) Criterion

According to the SSS criterion, if three sides of one triangle are equal to three sides of another triangle, then the triangles are similar. This ensures that the two triangles are equal in shape and size as each aspect is compared directly.

Let us consider an example:

Suppose you have two triangles, △ABC and △DEF. Their sides are as follows:

    AB = DE, BC = EF and CA = FD

If these sides are equal respectively, then by SSS criterion, △ABC ▼ △DEF.

Visualization:

A C B D F I

Here, △ABC and △DEF are congruent as the three sides are equal in length.

Side-Angle-Side (SAS) Criterion

The SAS criterion states that if two sides and the angle between them in one triangle are equal to two sides and the angle between them in another triangle, then the triangles are similar. This means that the adjacent angles ensure a specific orientation between the sides.

Imagine △PQR and △XYZ:

    PQ = XZ, QR = YZ, and angle ∠PQR = ∠XYZ

If these conditions are valid, then according to the SAS criterion, △PQR ▼ △XYZ.

Visualization:

P R Why X Jade Y

Here, matching sides and the angle between them in both the triangles suggest SAS congruence.

Angle-Side-Angle (ASA) Criteria

According to the ASA criterion, if two angles and the side between them in a triangle are equal to two angles and the side between them in another triangle, then those triangles are congruent.

For example, in triangles △GHI and △JKL:

    ∠GHI = ∠JKL, ∠IGH = ∠LJK, and side GH = JK

If these are true, then △GHI ▼ △JKL according to the ASA criteria.

Visualization:

Yes I H J l K

In the above illustration, the two marked angles and the included side confirm congruence by ASA.

Angle-Angle-Side (AAS) Criterion

According to AAS criterion, if two angles and the corresponding nonconnected side of a triangle are equal to two angles and the corresponding nonconnected side of another triangle, then the triangles are congruent.

Consider the triangles △MNO and △PQR:

    ∠NMO = ∠QPR, ∠NOM = ∠QRP, and side NO = QR

Under these conditions, △MNO ▼ △PQR by the AAS criterion.

Visualization:

M Hey N P R Why

Given two corresponding angles and sides, the triangles are congruent through AAS.

Right Angle-Hypotenuse-Side (RHS) Criterion

The RHS criterion, sometimes called the hypotenuse-side (HL) theorem, applies specifically to right triangles. It states that if the hypotenuse and a side of one right triangle are equal to the hypotenuse and a side of another right triangle, then the triangles are right triangles.

Let's evaluate triangles △ABC and △DEF:

    Hypotenuse AC = DF and side AB = DE

If the hypotenuse and one side of the triangles coincide, then by RHS criterion △ABC ▼ △DEF.

Visualization:

A C B F D I

As indicated, given the hypotenuse and a side, the RHS criterion confirms triangle congruence.

Conclusion

Understanding congruence in triangles helps solve many geometry problems by providing a strong foundation for identifying similar figures. The SSS, SAS, ASA, AAS and RHS criteria cover the various cases and combinations of sides and angles needed to declare two triangles congruent. Although the abstract nature of geometry can be challenging initially, these criteria simplify the process by reducing the comparison to manageable rules.

Learning these concepts well lays the groundwork for more advanced topics in geometry, such as similarity and geometric transformations that rely heavily on consistency principles. Mastery of consistency extends beyond the classroom, providing critical thinking and analytical skills applicable in a variety of STEM fields.


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Congruence of Triangles
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Criteria for Congruence

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