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We observe that triangles are classified into different types based on their side’s measurement and their angles measurement.

Based on triangle’s angles, we have 3 categories as shown below:

a) Acute-angled triangle: A triangle in which all the 3 angles measure less than 90°.

b) Obtuse-angled triangle: A triangle which has one angle greater than 90°, with other two angles less than 90°. (Since the sum of all the 3 angles in any triangle must be equal to 180°, hence there cannot be more than one obtuse angle in a triangle).

c) Right-angled triangle: A triangle in which one of the angles is equal to 90°.

Based on triangle’s sides, we have 3 categories as shown below:

a) Equilateral triangle: All the 3 sides of the triangle are equal to each other. In this triangle, all the 3 angles are also equal to each other. Each angle measure 60°.

b) Isosceles triangle: Any two sides of the triangle are equal to each other. Hence, the angles opposite to the equal sides are also equal.

c) Scalene Triangle: A triangle in which all the 3 sides (and hence 3 angles) are not equal to each other.

1) A scalene triangle has no two sides equal to each other.

2) A scalene triangle has all the 3 angles unequal to each other.

3) Sum of all the angles in any triangle = 180°

4) A scalene triangle does not have any line of symmetry.

Given that angle R = 56° and angle Q = 88°

Sum of all the angles of a triangle = 180°

Hence, angle P + angle Q + angle R = 180°

So, angle P + 88° + 56° = 180°

Angle P + 144° = 180° ==> angle P = 180° - 144°

Therefore, angle P = 36°

Sum of all angles of a triangle = 180°

Hence, 2x + 20 + x + 50 + x + 10 = 180° ==> 4x + 80 = 180° ==> 4x = 100

This gives: x = 100/4 ==> x = 25°

Now the angles are 2x + 20 = 2*25 + 20 = 70°

x + 50 = 25 + 50 = 75° and x+ 10 = 25 + 10 = 35°

Therefore, the 3 angles are 70°, 75° and 35°.

If ‘a’, ‘b’ and ‘c’ are the side lengths of the three sides of a triangle, then according to the theorem:

· a < b + c

· b < c + a

· c < a + b

This theorem can be understood by looking at two cases:

I. If one side is longer than the sum of the other two sides, then the other two sides will not meet each other to form a triangle, as shown in the figure below.

II. If one side is equal to the sum of the other two sides, then a triangle is not formed. In fact, this happens when the three points are collinear (points which lie on the same line).

Therefore for a triangle to be formed, the sum of any two sides of the triangle must always be greater than the third side!

Let the side lengths be a = 12m, b = 18m and c = 22m.

Now, 12 < 18 + 22. Hence a < b + c

18 < 22 + 12. Hence b < c + a

22 < 12 + 18. Hence c < a + b

Since the Triangle Inequality Theorem is satisfied, hence yes with the given side lengths we can form a triangle.

Perimeter of a Scalene triangle = Sum of all the side lengths of the triangle.

Perimeter = 12m + 16m + 17m = 45m

Therefore, perimeter of the given scalene triangle is 45m.

1)

If the base of a triangle is ‘b’ and height of the triangle is ‘h’, then:

2)

If a triangle has side lengths as ’a’, ‘b’ and ‘c’, then the semi-perimeter, s = (a + b+ c)/2

Then,

Given the base QR = 10m

Height is the perpendicular line drawn from the vertex to the opposite side of a triangle.

Since PN is the perpendicular line from the vertex P to the opposite base side QR, hence height is PN = 7.5m.

Area of a triangle = 1/2 * base * height = 1/2 * 10m * 7.5m = 37.5m

Hence, Area of the given triangle is 37.5 square meters.

Given the side lengths of triangle XYZ.

Since the lengths of the sides are not equal to each other, hence XYZ is a scalene triangle.

Let a = 5m, b= 7m and c = 8m.

Then s = (a + b + c)/2 = (5 + 7 + 8)/2 = 10

Now according to Heron’s formula:

Area of the triangle, A = √[s* (s-a)* (s-b)* (s-c)]

Hence, Area = √[10* (10- 5)* (10- 7)* (10- 8)] = √(10 * 5 * 3 * 2) = √300 = 17.3m

Therefore, the area of the triangle XYZ is 17.3 square meters.