Triangles
What is a Triangle?
A triangle
is a type of polygon.
Key Features
of a Triangle is:
A Triangle
has :
- Three vertices, which are the points where the sides meet.
- Three sides, each connecting two vertices.
- Three interior angles formed by its sides. The sum of these angles is always 180 degrees.
Triangles
can be categorized based on their sides and angles:
Based on Sides:
Equilateral
Triangle: All three sides are of equal
length.
Isosceles
Triangle: At least two sides are of equal
length.
Scalene
Triangle: All three sides have different
lengths.
Based on Angles:
Acute
Triangle: All three angles are acute (less
than 90 degrees).
Right
Triangle: One angle is a right angle
(exactly 90 degrees).
Obtuse
Triangle: One angle is obtuse (greater
than 90 degrees).
Congruence
of triangles:
- Congruence is a term used to describe the relationship between two figures that have the same shape and size.
- If two triangles are congruent, their corresponding sides and angles are equal.
- Ways to show that two triangles are congruent:
- SSS
(Side-Side-Side): If the
three sides of one triangle are equal to the three sides of another triangle,
then the triangles are congruent.
- SAS
(Side-Angle-Side): If two
sides and the included angle of one triangle are equal to the corresponding two
sides and included angle of another triangle, then the triangles are congruent.
- ASA
(Angle-Side-Angle): If two
angles and the included side of one triangle are equal to the corresponding two
angles and included side of another triangle, then the triangles are congruent.
- AAS
(Angle-Angle-Side): If two
angles and a non-included side of one triangle are equal to the corresponding
two angles and non-included side of another triangle, then the triangles are
congruent.
- HL
(Hypotenuse-Leg): If the
hypotenuse and one leg of a right-angled triangle are equal to the hypotenuse
and corresponding leg of another right-angled triangle, then the triangles are
congruent.
Similar
figures:
- Similar figures are geometric figures that have the same shape but not necessarily the same size.
- In other words, if you can obtain one figure by scaling (enlarging or reducing) the other, then the figures are considered similar.
Properties
of similar figures:
- Corresponding angles in similar figures are equal.
- The ratios of the lengths of corresponding sides are equal. This ratios known as scale factor or similarity ratio.
- If triangle ABC and DEF are similar; ΔABC ~ΔDEF
- All congruent figures are similar but the similar figures need not be congruent.
- Two polygons of the same number of sides are similar, if (i) their corresponding angles are equal and (ii) their corresponding sides are in the same ratio (or proportion)
The scale factor
for the polygons:
- It is also known as Representative Fraction or same ratio or proportion.
- It is a ratio that expresses the proportional relationship between corresponding lengths in two similar polygons.
- It is used to quantify how much one polygon has been scaled (enlarged or reduced) to obtain the other. The scale factor is usually denoted by the letter k or r.
The scale factor is the same for all corresponding
sides in similar polygons.
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