A triangle is a polygon with three sides, three vertices and three angles. This chapter includes the classification of triangles based on their sides and angles, a brief review of the congruence of triangles, some important properties of triangles, the mid-point theorem, the basic proportionality theorem and similarity and its criteria.
Equilateral Triangle: An equilateral triangle is a triangle whose all sides and angles are equal.
Isosceles Triangle: An isosceles triangle is a triangle whose two sides are of equal length.
Scalene Triangle: A scalene triangle is any triangle with three different side lengths.
Acute Triangle: Every angle in an acute triangle is less than 90°.
Right Triangle: One angle in a right triangle is exactly 90°.
Obtuse Triangle: An obtuse triangle has one angle measuring more than 90°.
Note: An equilateral triangle is a triangle whose all angles are 60°.
The sum of the angles of a triangle is 180°.
An exterior angle is an angle formed between one side of a polygon and the extension of an adjacent side. The measure of an exterior angle is equal to the sum of the two interior opposite angles.
When two triangles are congruent, their corresponding sides are equal in length and their corresponding angles have the same measurement.
Side-Side-Side (SSS): If the three sides of one triangle are equal in length to the three sides of another triangle, then the triangles are congruent.
Side-Angle-Side (SAS): If two sides and the angle between them in one triangle are equal to the corresponding two sides and the corresponding angle between them in another triangle, then the triangles are congruent.
Angle-Side-Angle (ASA): If two angles and the side between them in one triangle are equal to the corresponding two angles and the corresponding side between them in another triangle, then the two triangles are congruent.
Angle-Angle-Side (AAS): If two angles and a corresponding side of one triangle are equal to two angles and the corresponding side of another triangle, then the triangles are congruent.
Right Angle-Hypotenuse-Side (RHS): If in two right 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.
The sum of the angles of a triangle is 180°.
The measure of an exterior angle is equal to the sum of the measures of its interior opposite angles.
When two sides of a triangle are equal, their opposite angles are also equal.
When two angles of a triangle are equal, the sides opposite to them are also equal.
Each angle in an equilateral triangle measures 60°.
In a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.
The sum of the lengths of any two sides of a triangle is always greater than the length of the third side.
In △ABC,
⇒ AB + BC > AC
⇒ BC + AC > AB
⇒ AB + AC > BC
The difference between the lengths of any two sides of a triangle is less than the third side.
In △ABC,
⇒ (BC − AB) < AC
⇒ (AC − BC) < AB
⇒ (AC − AB) < BC
In triangle ABC, if ∠B > ∠C, then the side opposite to ∠B is longer than the side opposite to ∠C, that is, AC > AB.
In triangle ABC, if AC > BC, then the angle opposite to side AC is greater than the angle opposite to side BC, that is, ∠B > ∠A.
In a right-angled triangle, the square of the hypotenuse side is equal to the sum of the squares of the other two sides.
(Hypotenuse)^{2} = (Base)^{2} + (Perpendicular)^{2}
The midpoint theorem states that in a triangle, the line segment joining the midpoints of two sides is parallel to the third side, and it is also half of its length.
If in △ABC, D and E are the mid-points of AB and AC, respectively, then
DE || BC and DE = ^{1}⁄_{2}BC.
If a line is drawn parallel to one side of a triangle and intersects the other two sides, it divides those sides in equal ratios.
If in △ABC, PQ is drawn parallel to BC, then ^{AP}⁄_{PB }= ^{AQ}⁄_{QC}.
Example: In △ABC, D and E are points on the sides AB and AC, respectively such that DE || BC. If AD = 4 cm, AE = 2 cm and BD = 18 cm, then what is the value of CE?
a) 2.25 cm
b) 4.5 cm
c) 9 cm
d) 18 cm
Answer: c) 9 cm
Explanation: We are given in △ABC, DE || BC and AD = 4 cm, AE = 2 cm and BD = 18 cm
Using the Basic Proportionality Theorem, if DE || BC, then
⇒ ^{AD}⁄_{DB }= ^{AE}⁄_{EC}
⇒ ^{4}⁄_{18 }= ^{2}⁄_{CE}
⇒ 4 × CE = 2 × 18
⇒ CE = ^{36}⁄_{4}
∴ CE = 9 cm
Any line that divides a triangle's two sides in the same ratio is parallel to the triangle's third side.
If in △ABC, if ^{AD}⁄_{DB }= ^{AE}⁄_{EC}, then DE || BC.
Similar figures are shapes that have the same shape but can differ in size.
When two figures are similar, their corresponding angles are equal and their corresponding sides are in proportion or have the same ratio.
Example: Any two rectangles are similar to each other.
Two polygons of the same number of sides are similar if
(i) Their corresponding angles are equal.
(ii) The ratio (or proportion) of their corresponding sides is the same.
While similar figures do not necessarily need to be congruent, all congruent figures are similar.
Two triangles are similar if:
(i) Their corresponding angles are equal.
(ii) Their corresponding sides are in the same ratio (or proportion).
Note:
Two triangles are referred to as equiangular triangles if their corresponding angles are equal.
In two equiangular triangles, the ratio of any two corresponding sides is always the same.
AAA (Angle-Angle-Angle) Similarity Criterion: If in two triangles, corresponding angles are equal, then the two triangles are similar and hence their corresponding sides are in the same ratio (or proportion). It is also known as AA Similarity Criterion. AA similarity criterion states that if two angles of one triangle are respectively equal to two angles of another triangle, then the two triangles are similar.
In △ABC and △DEF,
If ∠A = ∠D, ∠B = ∠E and ∠C = ∠F, then
△ABC ~ △DEF and ^{AB}⁄_{DE} = ^{BC}⁄_{EF} = ^{AC}⁄_{DF}.
SSS (Side-Side-Side) Similarity Criterion: If in two triangles, the sides of one triangle are proportional (i.e., in the same ratio) to the sides of the other triangle, then the two triangles are similar and hence their corresponding angles are equal.
In △ABC and △DEF,
If ^{AB}⁄_{DE }= ^{BC}⁄_{EF }= ^{AC}⁄_{EDF}, then
△ABC ~ △DEF, ∠A = ∠D, ∠B = ∠E and ∠C = ∠F.
SAS (Side-Angle-Side) Similarity Criterion: If two triangles have two pairs of sides that are in the same proportion and the angles between these sides are equal, then these triangles are similar and hence the other two corresponding angles are also equal.
In △ABC and △DEF,
If ∠A = ∠D and ^{AB}⁄_{DE }= ^{AC}⁄_{DF}, then
△ABC ~ △DEF, ∠B = ∠E and ∠C = ∠F.
Example: If △ABC ~ △EDF and AB = 4 cm, AC = 6 cm, DF = 20 cm and DE = 15 cm are given, then what are the lengths of the remaining sides of the triangles?
a) EF = 22 cm and BC = 5.33 cm
b) EF = 22.5 cm and BC = 5.33 cm
c) EF = 22 cm and BC = 5.66 cm
d) EF = 22.5 cm and BC = 5.66 cm
Answer: b) EF = 22.5 cm and BC = 5.33 cm
Explanation: We are given that △ABC ~ △EDF:
AB = 4 cm, AC = 6 cm, DF = 20 cm and DE = 15 cm
Thus, EF = 22.5 cm and BC = 5.33 cm
If two triangles are similar, then the ratio of their areas is equal to the square of the ratio of their corresponding side lengths.
Example: If △ABC ~ △PQR such that the ratio between the area of △ABC and the area of △PQR is 16 : 25 and the lengths of AB and BC are 12 cm and 8 cm, respectively, then what is the value of PQ?
a) 15.2 cm
b) 15.5 cm
c) 16 cm
d) 15 cm
Answer: d) 15 cm
Explanation: Consider the figure given below:
We are given that △ABC ~ △PQR.
(area of △ABC) / (area of △PQR) = 16 / 25
AB = 12 cm and BC = 8 cm
By Area of Similar Triangles theorem,
(Area of △ABC) / (Area of △PQR) = (AB / PQ)2
⇒ 16 / 25 = (12)^{2} / (PQ)^{2}
⇒ 16 / 25 = 144 / PQ^{2}
⇒ PQ^{2} = 144 × 25 / 16
⇒ PQ = 9 × 25
⇒ PQ = √225
∴ PQ = 15 cm
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