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Triangles - Congruence of Triangles

Grade 9CBSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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Congruence of Triangles: Two triangles are congruent if they are identical in shape and size. Visually, if one triangle is placed over the other, they will coincide perfectly. This implies that all corresponding sides and corresponding angles are equal. Symbolically, we write △ABC≅△PQR\triangle ABC \cong \triangle PQR.

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SAS (Side-Angle-Side) Congruence Rule: Two triangles are congruent if two sides and the included angle of one triangle are equal to the corresponding two sides and the included angle of the other triangle. Visually, if you know the length of two sides and the exact 'opening' or angle between them, the length of the third side is automatically fixed, making the triangle unique.

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ASA (Angle-Side-Angle) Congruence Rule: Two triangles are congruent if two angles and the included side of one triangle are equal to the corresponding two angles and the included side of the other triangle. Visually, having a fixed base length with two specific angles at either end forces the remaining two sides to meet at a single, specific point in space.

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AAS (Angle-Angle-Side) Rule: Two triangles are congruent if any two pairs of angles and one pair of corresponding sides are equal. Since the sum of angles in a triangle is always 180∘180^\circ, if two angles are equal, the third must also be equal, making AAS a variation of the ASA rule.

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SSS (Side-Side-Side) Congruence Rule: If three sides of one triangle are equal to the three sides of another triangle, then the two triangles are congruent. Visually, this represents the 'rigidity' of a triangle; unlike a quadrilateral, a triangle's shape cannot be deformed if its side lengths are fixed.

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RHS (Right Angle-Hypotenuse-Side) Congruence Rule: If in two right-angled triangles the hypotenuse and one side of one triangle are equal to the hypotenuse and one side of the other triangle, then the two triangles are congruent. Visually, in a right triangle, the relationship defined by the Pythagorean theorem ensures that if the longest side (hypotenuse) and one other side are fixed, the third side must also be identical.

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CPCT (Corresponding Parts of Congruent Triangles): Once two triangles are proved congruent, we can conclude that all their remaining corresponding parts (which were not part of the criteria) are equal. For example, if △ABC≅△PQR\triangle ABC \cong \triangle PQR by SAS, we can state ∠C=∠R\angle C = \angle R by CPCT.

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Properties of Isosceles Triangles: Angles opposite to equal sides of an isosceles triangle are equal. Conversely, sides opposite to equal angles of a triangle are equal. Visually, an isosceles triangle possesses a line of symmetry that bisects the vertex angle and the base perpendicularly.

📐Formulae

△ABC≅△PQR  ⟹  AB=PQ,BC=QR,AC=PR\triangle ABC \cong \triangle PQR \implies AB=PQ, BC=QR, AC=PR

△ABC≅△PQR  ⟹  ∠A=∠P,∠B=∠Q,∠C=∠R\triangle ABC \cong \triangle PQR \implies \angle A = \angle P, \angle B = \angle Q, \angle C = \angle R

\text{Angle Sum Property: } \angle A + \angle B + \angle C = 180^\circ$

Triangle Inequality: AB+BC>AC\text{Triangle Inequality: } AB + BC > AC

\text{In } \triangle ABC, \text{ if } AB = AC \iff \angle C = \angle B$

💡Examples

Problem 1:

In △ABC\triangle ABC, the bisector ADAD of ∠A\angle A is perpendicular to side BCBC. Show that AB=ACAB = AC and △ABC\triangle ABC is isosceles.

Solution:

  1. In △ABD\triangle ABD and △ACD\triangle ACD:
  2. ∠BAD=∠CAD\angle BAD = \angle CAD (Given that ADAD bisects ∠A\angle A)
  3. AD=ADAD = AD (Common side to both triangles)
  4. ∠ADB=∠ADC=90∘\angle ADB = \angle ADC = 90^\circ (Given AD⊥BCAD \perp BC)
  5. Therefore, △ABD≅△ACD\triangle ABD \cong \triangle ACD by the ASA congruence rule.
  6. So, AB=ACAB = AC by CPCT.
  7. Since two sides of △ABC\triangle ABC are equal, it is an isosceles triangle.

Explanation:

We use the properties of the angle bisector and the perpendicularity to establish two angles and a shared side, satisfying the ASA criteria. Once congruence is proved, CPCT allows us to equate the main sides of the triangle.

Problem 2:

Line segment ABAB is parallel to another line segment CDCD. OO is the mid-point of ADAD. Show that △AOB≅△DOC\triangle AOB \cong \triangle DOC and OO is also the mid-point of BCBC.

Solution:

  1. Consider △AOB\triangle AOB and △DOC\triangle DOC:
  2. ∠OAB=∠ODC\angle OAB = \angle ODC (Alternate interior angles as AB∥CDAB \parallel CD and ADAD is the transversal)
  3. OA=ODOA = OD (Given OO is the mid-point of ADAD)
  4. ∠AOB=∠DOC\angle AOB = \angle DOC (Vertically opposite angles)
  5. Thus, △AOB≅△DOC\triangle AOB \cong \triangle DOC by the ASA congruence rule.
  6. Consequently, OB=OCOB = OC by CPCT.
  7. Since OB=OCOB = OC, OO is the mid-point of BCBC.

Explanation:

The parallel lines provide equal alternate interior angles. Combined with the midpoint definition and vertically opposite angles, we satisfy the ASA rule. CPCT is then used to prove the second part of the problem regarding the other midpoint.