make a project on the angle base triangle
Draw the page plss...
Answers
Answer:
solve with your mind
Step-by-step explanation:
In the series on the basic building blocks of geometry, after a overview of lines, rays and segments, this time we cover the types and properties of triangles.
Definition: A triangle is a closed figure made up of three line segments.
simple triangle
A triangle consists of three line segments and three angles. In the figure above, AB, BC, CA are the three line segments and ∠A, ∠B, ∠C are the three angles.
There are three types of triangles based on sides and three based on angles.
Types of triangles based on sides
Equilateral triangle: A triangle having all the three sides of equal length is an equilateral triangle.
equilateral triangle
Since all sides are equal, all angles are equal too.
Isosceles triangle: A triangle having two sides of equal length is an Isosceles triangle.
isosceles triangle
The two angles opposite to the equal sides are equal.
Scalene triangle: A triangle having three sides of different lengths is called a scalene triangle.
scalene triangle
Types of triangles based on angles
Acute-angled triangle: A triangle whose all angles are acute is called an acute-angled triangle or Acute triangle.
acute angled triangle
Obtuse-angled triangle: A triangle whose one angle is obtuse is an obtuse-angled triangle or Obtuse triangle.
obtuse angled triangle
Right-angled triangle: A triangle whose one angle is a right-angle is a Right-angled triangle or Right triangle.
right angled triangle
In the figure above, the side opposite to the right angle, BC is called the hypotenuse.
For a Right triangle ABC,
BC2 = AB2 + AC2
This is called the Pythagorean Theorem.
In the triangle above, 52 = 42 + 32. Only a triangle that satisfies this condition is a right triangle.
Hence, the Pythagorean Theorem helps to find whether a triangle is Right-angled.
Types of triangles
types of triangles
There are different types of right triangles. As of now, our focus is only on a special pair of right triangles.
45-45-90 triangle
30-60-90 triangle
45-45-90 triangle:
A 45-45-90 triangle, as the name indicates, is a right triangle in which the other two angles are 45° each.
This is an isosceles right triangle.
isosceles right triangle
In ∆ DEF, DE = DF and ∠D = 90°.
The sides in a 45-45-90 triangle are in the ratio 1 : 1 : √2.
30-60-90 triangle:
A 30-60-90 triangle, as the name indicates, is a right triangle in which the other two angles are 30° and 60°.
This is a scalene right triangle as none of the sides or angles are equal.
scalene right triangle
The sides in a 30-60-90 triangle are in the ratio 1 : √3 : 2
Like any other right triangle, these two triangles satisfy the Pythagorean Theorem.
Basic properties of triangles
The sum of the angles in a triangle is 180°. This is called the angle-sum property.
The sum of the lengths of any two sides of a triangle is greater than the length of the third side. Similarly, the difference between the lengths of any two sides of a triangle is less than the length of the third side.
The side opposite to the largest angle is the longest side of the triangle and the side opposite to the smallest angle is the shortest side of the triangle.
triangle smallest angle
In the figure above, ∠B is the largest angle and the side opposite to it (hypotenuse), is the largest side of the triangle.
triangle largest angle
In the figure above, ∠A is the largest angle and the side opposite to it, BC is the largest side of the triangle.
An exterior angle of a triangle is equal to the sum of its interior opposite angles. This is called the exterior angle property of a triangle.
triangle exterior angle
Here, ∠ACD is the exterior angle to the ∆ABC.
According to the exterior angle property, ∠ACD = ∠CAB + ∠ABC.
Similarity and Congruency in Triangles
Figures with same size and shape are congruent figures. If two shapes are congruent, they remain congruent even if they are moved or rotated. The shapes would also remain congruent if we reflect the shapes by producing mirror images. Two geometrical shapes are congruent if they cover each other exactly.
Figures with same shape but with proportional sizes are similar figures. They remain similar even if they are moved or rotated.
Similarity of triangles
Two triangles are said to be similar if the corresponding angles of two triangles are congruent and lengths of corresponding sides are proportional.
It is written as ∆ ABC ∼ ∆ XYZ and said as ∆ ABC ‘is similar to’ ∆ XYZ.
similar triangles