Students can go through AP 9th Class Maths Notes Chapter 7 Triangles to understand and remember the concepts easily.
Class 9 Maths Chapter 7 Notes Triangles
→ Triangle : A closed figure formed by three intersecting lines is called a triangle.
Ex:
l, m, n are 3 intersecting lines giving closed figure. Hence it is a triangle.
Three lines are there but not intersecting and not formed a closed figure. Hence, above is not a triangle
→ Triangle has 3 sides (AB, BC, CA), three angles (∠A, ∠B, ∠C) and three vertices (A, B, C). Triangle ABC is denoted by Δ ABC
→ Angle sum property of a triangle : Sum of angles (interior) of a triangle is 180°.
∠A + ∠B + ∠C = 180°
→ If a side (AB / BC / CA) is produced, the exterior angle so formed is equal to the sum of the two interior opposite angles.
If \(\overline{\mathrm{AB}}\) is produced to ‘E’, then the exterior angle formed at ‘B’ is
∠CBE = ∠BCA + ∠CAB → (1)
If \(\overline{\mathrm{BC}}\) is produced to ’F’, then the exterior angle at ‘C’ is
∠FCA = ∠CAB + ∠ABC → (2)
If \(\overline{\mathrm{CA}}\) is produced to ‘D’, then the exterior angle at ‘A’ is
∠BAD = ∠BCA + ∠CBA → (3)
→ Congruence of triangles: When two or more triangles have same shape and same and, then they are said to be congruent. They are identical.
→ To determine if two triangles are congruent, we need to check if they have same shape and same size. Though there are several ways to do this, one common method is to use the congruence criteria.
These criteria are a set of rules that we can use to determine, if two triangles are congruent.
→ Some of the most commonly used criteria are
i) SSS
ii) SAS
iii) ASA
iv) AAS
v) HL or RHS
Where S – Stands for side
A – Stands for angle
→ We use the symbol ≅ to denote congruency.
→ If ΔABC is congruent to ΔDEF it means the corresponding sides are equal.
⇒ AB = DE, BC = EF, AC = DF and the corresponding angles are equal.
⇒ ∠A = ∠D, ∠B = ∠E, ∠C = ∠F
→ It means vertex ‘A’ overlaps vertex ‘D’.
It means vertex ‘B’ overlaps vertex ‘E’.
It means vertex ‘C’ overlaps vertex ‘F.
Then ΔABC fits on ΔDEF.
→ We can write ‘A’ corresponds ‘D’, A ↔ D
We can write ‘B’ corresponds ‘E’, B ↔ E
We can write ‘C’ corresponds ‘F, C ↔ F
→ We can write ΔABC ≅ ΔDEF (or)
ΔDEF ≅ ΔABC
but we cannot say
ΔABC is congruent to ΔEDF.
→ We can write ΔABC ≅ ΔDEF as follows
ΔBCA ≅ ΔEFD (or)
ΔCBA ≅ ΔFED (or)
ΔCAB ≅ ΔFDE
all above 3 are same.
→ It is necessary to write the correspondence of vertices correctly for writing of congruency of triangles in symbolic form.
→ CPCT : It states that corresponding parts of congruent – triangles are equal. It means,
in congruent triangles, corresponding parts are always equal.
→ We observe’
i) equality of one pair of sides (or)
ii) one pair of sides and one pair of angles is not sufficient to state them as congruent triangles.
→ Criteria for congruence of triangles :
Two triangles are said to be congruent if one of the following 5 criteria (criteria).
i) SAS (Side – Angle – Side)
(Note : SAS is not to say ASS or SSA because the pattern is very important)
ii) ASA (Angle – Side – Angle)
iii) SSS (Side – Side – Side)
iv) AAS (Angle – Angle – Side)
v) RHS (Right angle – hypotenuse – side) The above 5 rules are said to be congruence rules.
→ We can say that,
i) two figures are congruent, if they are of the same shape and same size.
ii) two circles are congruent, if and only if they are of same radii,
iii) two equilateral triangles are congruent, if they are of same sided.
iv) If ΔABC and ΔDEF are congruent, then we write
ΔABC ≅ ΔDEF which means (or) we can infer
∠A = ∠D, ∠B = ∠E, ∠C = ∠F
also AB = DE, BC = EF and AC = DF and also
perimeter of ΔABC = perimeter of ΔDEF and also area of ΔABC = area of ΔDEF.
v) In any triangle,
a) angles opposite to equal sides are equal.
b) sides opposite to equal angles are equal.
c) side opposite to a larger angle is always larger.
d) angle oppsite to a larger side is always greater (If it is not an Isosceles)
vi) Basing on lengths of sides, triangles are 3 types:
a) If all 3 sides are equal, then it is “equilateral triangle.
b) If two sides only are equal, then it is an isosceles triangle.
c) If all 3 sides have different lengths, then it is called scalene triangle.
vii) Basing on the measurements of angles of a triangle, triangles are 3 types :
a) Obtuse triangle : If one of 3 angles of a triangle is more than 90°, then it is called obtuse angled triangle.
b) Right angled triangle : If one angle of a triangle is a right angle (= 90°), then it is called right angled triangle.
∠B = 90°
The side (AC) opposite to right angle
(B) is called hypotenuse.
c) Acute angled triangle : A triangle is called acute angled triangle, if all 3 interior angles of it are acute angles (i.e. < 90° each)
→ Axiom 7.1 (AS congruence rule) : Two triangles are congruent if two sides and the included angle of one triangle are equal to the two sides and the included angle of the other triangle.
This result cannot be proved with the help of previously known results and so it is accepted true as an axiom (see Appendix 1).
→ Some additional properties of triangles:
- Angles opposite to equal sides of an isosceles triangle are equal.
- The sides opposite to equal angles of a triangle are equal.
→ Two line segments are congruent if they have equal length.
→ Two squares are congruent if they have same side.
→ Squares that have same measure of diagonals are also congruent.
→ Two triangles are congruent if they cover each other exactly.
→ If two triangles are congruent then Corresponding Parts of Congruent Triangles (CPCT) are equal.
→ Two triangles are congruent if two sides and the included angle of one triangle are equal to the two sides and included angle of the other triangle (S.A.S. congruence rule).
→ Two triangles are congruent, if two angles and the included side of one triangle are equal to two angles and the included side of the other triangle (A.S.A. congruence rule).
→ Two triangles are congruent if any two pairs of angles and one pair of corresponding sides are equal (A.A.S.).
→ Angles opposite equal sides of a triangle are equal.
→ The sides opposite to equal angles of a triangle are equal.
→ If three sides of one triangle are respectively equal to the corresponding three sides of another triangle, then the two triangles are congruent (SSS).
→ If in two right triangles the hypotenuse and one side of one triangle are equal to the hypotenuse and one side of the another triangle, then the two triangles are congruent (RHS).
→ In a triangle, of the two sides, side opposite to greater angle is greater.
→ Sum of any two sides of a triangle is greater than the third side.