Triangles come in different types based on length of sides and angle measures. They're commonly used in construction for their stability. Knowing these properties helps solve real-world problems. Triangles are important because they offer a strong base, useful in building foundations and trusses. They're seen in traffic signals, pyramids, and even the Bermuda Triangle. There are six main types of triangles.

Triangles are `2`- dimensional geometric shapes with three sides and three angles. When three lines intersect, they form a triangle. In geometry, a triangle is denoted by its vertices and sides. For example, in triangle `△ABC`, vertices are `A`, `B`, and `C`, and sides are `AB`, `BC`, and `AC`. One fundamental property of triangles is the angle sum property, which states that the total of all three interior angles of any triangle always adds up to `180^\circ`.

- A triangle has three interior angles. The total of these angles inside any triangle is always `180^\circ`. This is called the angle-sum property.

- Another important rule is the triangle inequality property, which states that the sum of the lengths of two smaller sides of a triangle is always greater than the length of the longest side. For example, in a triangle with side lengths of `5`, `7`, and `10` units, the sum of the lengths of the two smaller sides (`5` and `7`) is `12`, which is greater than the length of the longest side `(10)`.

- In any triangle, the longest side is always opposite to the largest angle, while the shortest side is opposite to the smallest angle.

In math, triangles are `2`-dimensional shapes with three sides. They come in various types, mainly determined by their sides and angles. There are `3` types of triangles based on side lengths:

- Scalene
- Isosceles
- Equilateral

Additionally, triangles are classified by their angles:

- Acute triangle if all angles are less than `90°`
- Right triangle if one angle is exactly `90°`
- Obtuse triangle if one angle is greater than `90°`

We can classify types of triangles by sides. Triangles can be classified based on their side lengths into three triangle types:

**`1`. Equilateral Triangle: **An equilateral triangle is a type of triangle where all three sides are of equal length. This means that each angle in an equilateral triangle measures `60^\circ`.

**`2`. Isosceles Triangle:** An isosceles triangle is a triangle that has at least two sides of equal length. Consequently, the angles opposite to these equal sides are also equal.

**`3`. Scalene Triangle: **A scalene triangle is a triangle in which all three sides have different lengths. As a result, the measure of the three interior angles of a scalene triangle are also different from each other.

Triangles can be classified based on their interior angles into three types:

**`1`. Acute Triangle:** An acute triangle is a type of triangle where all three angles are acute angles, meaning each angle measures less than `90^\circ`.

**`2`. Right Triangle:** Right triangles are characterized by having one angle that measures exactly `90^\circ`, which forms a right angle. The side opposite to this right angle is known as the hypotenuse, and it's the longest side in a right triangle.

**`3`. Obtuse Triangle: **Obtuse triangles have one angle that is greater than `90^\circ`, known as an obtuse angle. This means that one of the triangle's corners is wider than a right angle.

There are various other types of triangles based on their sides and angles. Some of them are as follows:

**`1`. Equiangular Triangle: **A triangle in which all three sides and angles are the same length and measure.

**`2`. Right Isosceles Triangle:** A triangle which has two sides of equal length and one angle that measures exactly `90` degrees.

**`3`. Obtuse Isosceles Triangle:** A triangle having two sides of equal length and one angle that measures more than `90` degrees.

**`4`. Acute Isosceles Triangle: **A triangle in which two sides are of equal length, and all three angles are acute, meaning they measure less than `90` degrees.

**`5`. Right Scalene Triangle:** A triangle in which one of the angles is a right angle (`90` degrees) and all three sides have different lengths.

**`6`. Obtuse Scalene Triangle: **A triangle in which one of the angles is obtuse (greater than `90` degrees) and all three sides have different lengths.

**`7`. Acute Scalene Triangle: **A triangle in which all three angles are acute (less than `90` degrees) and all three sides have different lengths.

**Example `1`: What type of triangle is it called if all the angles of a triangle are less than `90` degrees?**

**Solution: **

If all the angles of a triangle are less than `90` degrees, the triangle is called an acute triangle.

**Example `2`: In triangle ` \triangle ABC `, if angle ` \angle A ` measures ` 60^\circ ` and angle ` \angle B ` measures ` 40^\circ `, what is the measure of angle ` \angle C `?**

**Solution: **

Using the angle sum property, we know that the sum of the interior angles of a triangle is ` 180^\circ `. Therefore, to find the measure of angle ` \angle C `, we subtract the sum of angles ` \angle A ` and ` \angle B ` from ` 180^\circ `:

`\angle C = 180^\circ - (\angle A + \angle B)`

`\angle C = 180^\circ - (60^\circ + 40^\circ)`

`\angle C = 180^\circ - 100^\circ`

`\angle C = 80^\circ`

Therefore, angle ` \angle C ` measures ` 80^\circ `.

**Example `3`: If the length of all the sides of a triangle are equal, what type of triangle is it?**

**Solution: **

If the length of all the sides of a triangle are equal, the triangle can be identified as an equilateral triangle.

**Example `4`: In a right triangle, if angle `X` measures ` 60^\circ `, find angle `Z`.**

**Solution: **

Using the angle sum property:

`\angle X + \angle Y + \angle Z = 180^\circ`

`60^\circ + 90^\circ + \angle Z = 180^\circ`

`150^\circ + \angle Z = 180^\circ`

`\angle Z = 180^\circ - 150^\circ`

`\angle Z = 30^\circ`

Therefore, angle `Z` measures \(30^\circ \).

**Example `5`: Is it possible to form a triangle with its sides measuring `5` cm, `7` cm and `9` cm?**

**Solution:**

As per the properties of a triangle, sum of the lengths of two smaller sides of a triangle is always greater than the length of the longest side.

The two smaller sides are `5` cm and `7` cm.

Adding these sides we get `12` cm.

`12` cm is greater that the length of the longest side (`9` cm).

Hence, it is possible to form a triangle with its sides measuring `5` cm, `7` cm and `9` cm.

**Q`1`. In triangle `ABC`, if `AB = BC = 5` cm and `AC = 6` cm, what type of triangle is it?**

- Equilateral
- Isosceles
- Scalene
- Right

** Answer:** b

**Q`2`. Triangle `PQR` has angles measuring `50^\circ`, `35^\circ`, and `95^\circ`. What type of triangle is `PQR`?**

- Equilateral
- Isosceles
- Scalene
- Acute

**Answer:** c

**Q`3`. In triangle ` \triangle PQR `, if angle ` \angle P ` measures ` 45^\circ ` and angle ` \angle R ` measures ` 80^\circ `, what is the measure of angle ` \angle Q `?**

- ` 110^\circ `
- ` 50^\circ `
- ` 55^\circ `
- ` 95^\circ `

**Answer:** c

**Q`4`. If the length of all the sides of a triangle are different, what type of triangle is it?**

- Scalene
- Isosceles
- Equilateral
- Right

**Answer:** a

**Q`5`. Which set of measurements below is not capable of forming the sides of a scalene triangle?**

- `6` cm, `8` cm, `10` cm
- `5` m, `12` m, `13` m
- `10` cm, `12` cm, `24` cm
- `17` m, `17` m, `18` m

**Answer:** c

**Q`1`. What is an equilateral triangle?**

**Answer:** An equilateral triangle is a type of triangle in which all three sides are of equal length. Additionally, all three interior angles are also equal, each measuring `60` degrees.

**Q`2`. How can you determine if a triangle is right, obtuse, or acute?**

**Answer:** In a right triangle, one of the angles measures `90` degrees. In an acute triangle, all three angles are less than `90` degrees. In an obtuse triangle, one of the angles measures more than `90` degrees.

**Q`3`. What is the sum of angles in a triangle?**

**Answer:** The sum of the interior angles of a triangle is always `180` degrees. This property is known as the triangle angle sum theorem.

**Q`4`. What is the difference between an isosceles and a scalene triangle?**

**Answer:** An isosceles triangle has at least two sides of equal length, whereas a scalene triangle has all three sides of different lengths.

**Q`5`. How do you find the perimeter of a triangle?**

**Answer:** The perimeter of a triangle is found by adding the lengths of all three sides together.