Geometry facts
Hello and welcome to all my readers! The exams are over for all of you and it is time for your big annual vacation. I know you all have had a hectic term 3 with lots of revision followed by your annual examination. Though you will be away from schools for the next 7-8 weeks, it is always better to keep yourself occupied with something that would upgrade your skills and prepare you better for the next academic year.
We would just go over some basic topics and hone our mathematical brain! This week we look at the different types of triangles and their properties in depth
Angles are formed when two or more rays or lines meet each other at a point. When the angles are closed on all sides without leaving any gap in the figure, we get a polygon.
A polygon is any closed figure with three or more straight sides. "Closed" means that there are no gaping holes in it and that all sides connect together.
Triangle :
Triangles are very common shapes. We see them on road signs, on houses, and in food (think of Doritos or a slice of pizza). When you think of a triangle, you might imagine three equal sides because we see this type of triangle a lot in our daily lives, but did you know that there are lots of different types of triangles? Triangles can be stretched out and tall like the Eiffel Tower or short and squished like a hanger. What are some other examples of different types of triangles that you can think of?
All types of triangles can be grouped or categorized. Triangles are categorized by the length of their sides and the size of their angles. Before we get into the different types of triangles, take a minute and think about what a triangle is. (Hint: 'tri' means three). Some important facts on triangles : They have 3 sides
They have 3 angles
The sum of all the 3 angles must be equal to 180
The side opposite to the smallest angle is the shortest side and vice versa.
A triangle with all 3 equal sides is an equilateral triangle
A triangle with 2 equal sides is an isosceles triangle
The angles opposite to the equal sides in an isosceles triangle are equal.
A triangle with a right angle is a right triangle
The side opposite to the right angle is called the "HYPOTENUSE" in a right triangle.
A triangle with 3 unequal sides is a scalene triangle
A triangle with all 3 acute angles is an acute triangle ( Remember an acute triangle need not necessarily be an equilateral triangle)
A triangle with one of its angles greater than 90, i.e., obtuse angle is called an obtuse triangle
An equilateral triangle has 3 lines of symmetry, an isosceles triangle has one line of symmetry and a scalene triangle has none.
Example 1: A right triangle has one other angle that is 35°. What is the size of the third angle?
Solution:
A right triangle has one angle = 90°. Sum of known angles is 90° + 35° = 125°. The sum of all the angles in any triangle is 180°.
Subtract sum of known angles from 180°. 180° − 125° = 55°
So the size of the third angle is 55°.
Example 2: Is it possible for a triangle to have more than one obtuse angle?
Solution:
Let the angles of the triangle be a, b and c. Let a be the obtuse angle.
The sum of all the angles in any triangle is 180°. a + b + c = 180°.
If a > 90° then b + c must be less than 90°.
Therefore, b and c must be acute angles. No, a triangle can only have one obtuse angle.
Example 3: An isosceles triangle has one angle of 96°. What are the sizes of the other two angles?
Solution:
Since it is an isosceles triangle it will have two equal angles.
The given 96° angle cannot be one of the equal pair because a triangle cannot have two obtuse angles. (Refer to obtuse triangle example above).
So the other 2 angles must be equal and their sum should be 180 - 96 = 84
Let x be one of the two equal angles. The sum of all the angles in any triangle is 180°.
So, x + x + 96° = 180° 2x = 84° x = 42° The sizes of the other two angles are 42° each.