Euclidean geometry has to do with shapes, lines and angles and the way they interact with each other. You have to work a lot in the beginning to learn the language of geometry. After learning the basic postulates and properties of all shapes and lines, you can begin to use this information to solve geometry problems. Unfortunately, geometry takes time, but with the effort you can understand it.
Part 1 of 3: Learn the 5 Postulates of Euclid
Step 1. Learn the first postulate
A line segment can be formed by joining any two points. If you have two points A and B, you can draw a line segment connecting them. A line segment can only be made by joining the two points.
Step 2. Know the second postulate
Any line segment can extend to infinity in any direction. After building a line segment between two points, you can extend it to form a line. To do this, you can extend either end of the segment infinitely in the same direction.
Step 3. Understand the third postulate
Given any length and any point, a circle can be drawn with one point as the center and the length as the radius. In other words, a circle can be drawn from any line segment. This postulate applies regardless of the length of the segment.
Step 4. Identify the fourth postulate
All right angles are identical. A right angle equals 90 degrees. All right angles are congruent or equal. If an angle is not 90 degrees, it is not a right angle.
Step 5. Define the fifth postulate
Given a line and a point, only one line can be drawn through the point that is parallel to the first line. Another way of expressing this postulate is to say that if two lines cross a third in such a way that the sum of their interior angles on one side equals less than two right angles, both lines will eventually cross. These two lines are not parallel to each other.
This last postulate cannot be proved as a theorem. In non-Euclidean geometry, this "parallel" postulate does not apply
Part 2 of 3: Understanding Shapes, Lines, and Angles
Step 1. Know the properties of lines
A line extends infinitely in any direction and is denoted by arrows at the ends to indicate this. A line segment is finite and exists only between two points. A ray is a hybrid between a line and a line segment: it extends infinitely in one direction from a defined point.
- A single line will always measure 180 degrees.
- Two lines are parallel if they have the same slope and never intersect.
- Perpendicular lines are two lines that meet to form a 90 degree angle.
- Intersecting lines are any two lines that intersect at one point. Parallel lines can never cross, while perpendicular lines can.
Step 2. Learn about the different types of angles
There are three types of angles: acute, obtuse, and right. An acute angle is any angle that measures less than 90 degrees. An obtuse angle is a wide angle and is defined as any angle that measures more than 90 degrees. A right angle measures 90 degrees exactly.
- Being able to identify the various types of angles is a critical part of understanding geometry.
- Two lines that form a right angle are also perpendicular to each other. They form a perfect corner.
- You may also see a straight line. This angle measures 180 degrees.
- For example, a square or rectangle has four 90-degree angles, while a circle has none.
Step 3. Identify the types of triangles
You can identify a triangle in two ways: by the size of its angles (acute, obtuse, and right) or by the number of sides and angles that are equal (equilateral, isosceles, and scalene). In an acute triangle, all the angles are less than 90 degrees. An obtuse triangle has an angle that measures more than 90 degrees, and a right triangle has an angle of 90 degrees.
- Equilateral triangles have three equal sides and three angles that measure exactly 60 degrees.
- Isosceles triangles have two equal sides and two equal angles.
- Scalene triangles do not have equal sides or angles.
Step 4. Know how to determine the perimeter and area of two-dimensional shapes
Squares, rectangles, circles, triangles, etc., are shapes whose perimeters and areas you should know how to calculate. The perimeter of an object is the measure of all its sides, while the area is the measure of the amount of space it occupies. These are the equations for the perimeter and area of the most common shapes:
- The perimeter of a circle is known as the circumference and is equal to 2πr, where "r" is the radius.
- The area of a circle is πr2, where "r" is the radius.
- The perimeter of a rectangle is 2l + 2w, where "l" is the length and "w" is the width.
- The area of a rectangle is l x w, where "l" is the length and "w" is the width.
- The perimeter of a triangle is a + b + c, where each variable denotes one side of the triangle.
- The area of a triangle is ½bh, where "b" is the base of the triangle and "h" is the vertical height (for short).
Step 5. Find the surface area and volume of three-dimensional objects
Just as it is possible to calculate the perimeter and area of a two-dimensional object, it is possible to find the total surface area and volume of a three-dimensional object. Objects like spheres, rectangular prisms, pyramids, and cylinders have special equations for this. The surface area is the total area of the entire surface of the object, while the volume constitutes the total amount of space it occupies.
- The surface area of a sphere equals 4πr2, where "r" is the radius of the sphere.
- The volume of a sphere equals (4/3) πr3, where "r" is the radius of the sphere.
- The surface area of a rectangular prism is 2lw + 2lh + 2hw, where "l" is the length, "w" is the width, and "h" is the height.
- The volume of a rectangular prism is l x w x h, where "l" is the length, "w" is the width, and "h" is the height.
Step 6. Identify pairs of angles
When a line crosses two other lines, it is known as a transversal. These lines form pairs of angles. The corresponding angles are the two corresponding corner angles against the transversal. Alternate interior angles are the two angles that lie within the two lines but on opposite sides of the transversal. Alternate exterior angles are the two angles that lie outside the two lines but on opposite sides of the transversal.
- The pairs of angles are equivalent if two of the lines are parallel.
- There are a fourth pair of angles: consecutive interior angles. These are the two angles that are inside the lines and on the same side as the transversal. When the two lines are parallel, the consecutive interior angles always add up to 180 degrees.
Step 7. Define the Pythagorean Theorem
The Pythagorean Theorem is a useful way to determine the lengths of the sides of a right triangle. It is defined as a2 + b2 = c2, where "a" and "b" are the length and height (the straight lines) of the triangle and "c" is the hypotenuse (the line at an angle). If you know two sides of a triangle, you can calculate the third using this equation.
- For example, if you have a right triangle with sides a = 3 and b = 4, you can find the hypotenuse:
- to2 + b2 = c2
- 32 + 42 = c2
- 9 + 16 = c2
- 25 = c2
- c = √25
- c = 25. The hypotenuse of the triangle is 5.
Part 3 of 3: Solve Geometry Problems
Step 1. Trace the figures
Read the problem and make a diagram to illustrate it. Label all the information given to you, including all angles, parallel or perpendicular lines, and intersecting lines. Once you have a basic sketch of the problem, you may need to draw it all over again. The second drawing can fix the scale of everything and ensure that all the angles have been drawn approximately correctly.
- Also label all the unknowns.
- The easiest way to understand the problem is with a diagram that is clearly drawn.
Step 2. Make observations based on the information given
If you are given a line segment but there are angles coming out of it, you will know that the measures of all the angles must add up to 180 degrees. Record this information on the diagram or in the margins. This is a good way to think about what you are being asked.
- For example, the angles ABC and DBE form a line, ABE. The angle ABC = 120 degrees. What is the angle DBE?
- The sum of the angles ABC and DBE must be 180, so the angle DBE = 180 - the angle ABC.
- Angle DBE = 180 - 120 = 60 degrees.
Step 3. Apply basic theorems to answer the questions
Many individual theorems that describe the properties of triangles, intersecting and parallel lines, and circles can be used to solve a problem. Identify the geometric shapes in the problem and look for the theorems that apply. Use old tests and problems as a guide to see if there are similarities between them. These are some of the general geometric theorems you will need:
- The reflective property: a variable equals itself. x = x.
- The Sum Postulate: When equal variables are added to equal variables, all sums are equivalent. A + B + C = A + C + B.
- The Subtraction Postulate: This is similar to the addition postulate. All variables that are subtracted from equal variables have the same difference. A - B - C = A - C - B.
- The substitution postulate: if two quantities are equivalent, you can substitute one for the other in any expression.
- The partition postulate: any whole is equal to the sum of all its parts. Line ABC = AB + BC.
Step 4. Learn the theorems that apply to triangles
Many geometry problems will involve triangles, and knowing their properties will help you solve them. Use these theorems to formulate geometric proofs. These are some of the most important for triangles:
- The corresponding parts of congruent triangles are congruent.
- LLL: side-side-side. If three sides of a triangle are congruent with three sides of a second triangle, they are both congruent.
- LAL: side-angle-side. If two triangles have a congruent side-angle-side, they are both congruent.
- ALA: angle-side-angle. If two triangles have a congruent angle-side-angle, they are both congruent.
- AAA: angle-angle-angle. Triangles that have congruent angles are similar but not necessarily congruent.