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source: ncert
topic: geometry
class_level: class_9
chapter: ch7
difficulty: intermediate
---
Class 9 | Ch7: Triangles (Congruence)
CONGRUENCE CRITERIA:
SAS: Two sides and included angle equal.
ASA: Two angles and included side equal.
AAS: Two angles and any corresponding side equal.
SSS: All three sides equal.
RHS: Right angle, hypotenuse and one side equal (right triangles).
IMPORTANT THEOREMS:
Theorem 1: Angles opposite equal sides of isosceles triangle are equal.
If AB=AC then Angle B=Angle C.
Proof: In triangles ABD and ACD (D=midpoint of BC):
AB=AC (given), BD=CD (D is midpoint), AD=AD (common). By SSS: triangles congruent. Angle B=Angle C by CPCT.
Theorem 2 (Converse): If two angles of triangle are equal, sides opposite them are equal.
Theorem 3: In right triangle, hypotenuse is the longest side.
Theorem 4: The sum of any two sides of triangle > third side.
Theorem 5: Of all line segments drawn from external point to line, perpendicular is shortest.
INEQUALITIES IN TRIANGLES:
In triangle, greater angle has greater opposite side.
If Angle A > Angle B then BC > AC.
CPCT: Corresponding Parts of Congruent Triangles.
SOLVED EXAMPLES:
Example 1: ABC is isosceles with AB=AC. D is midpoint of BC. Prove AD perpendicular to BC.
In triangles ABD and ACD: AB=AC, BD=CD, AD=AD. SSS congruence.
Angle ADB=Angle ADC (CPCT). Both on straight line, so each=90°. AD perpendicular to BC.
Example 2: Prove that triangle ABC where AB=BC=CA (equilateral) has all angles 60°.
AB=BC → Angle A=Angle C. BC=CA → Angle A=Angle B. So A=B=C. A+B+C=180°. Each=60°.
COMMON MISTAKES:
RHS works ONLY for right triangles. Cannot apply to other triangles.
CPCT used only AFTER proving congruence.
SAS: the angle must be INCLUDED (between the two sides). |