GEOMETRY REVIEW FOR 3rd ESO STUDENTS

• Point (punto)                    

−      A point is the place where two lines intersect.

−      In geometry, a point has no dimension

−      We represent a point with a dot.

−      We identify this point with a CAPITAL letter.

• Line (línea)

−        A line is made up of infinite points.

−        It has not endpoints and continues endlessly on a plane. 

−        These lines are named with small letters such r ,s, t...

• Ray (semirrecta)

−      A ray is a "straight" line that begins at a certain point and extends forever in one direction.

−      The point where the ray begins is known as its endpoint.

• Line segment (segmento)

−        A line segment is a portion of a "straight" line.

−        A line segment does not extend forever, but has two endpoints.

−      The point halfway between the endpoints of a line segment is called the midpoint.  A midpoint divides a line segment into two equal segments.

• Intersection

−        The term intersect is used when lines, ray lines or segments share a common point.

−        The point they share is called the point of intersection.

 

2. Operations with Line Segments

2.1. Adding and subtracting line segments

       a) Adding

•  The addition of two segments is another segment that begins at the origin of the first segment and ends  at   the end of the second segment. 
• The length of the new segment is the addition of the measures of those two segments.

      b) Subtrating

• The subtration of two segments is another segment that takes as the origin, the end of the smaller segment and as the end, the end of the biggest segment.
• The length of the segment difference is equal to the subtraction of the lengths of two segments.

2.2. Perpendicular Bisector

• The perpendicular bisector is a perpendicular line that passes through the midpoint of the segment, so it divides the segment in two equal parts.
• All the points which belong to the perpendicular bisector are at the same distance from the endpoints of the segment. 

To draw a line bisector we use the compass and the rulers. Let's check how to draw it with this video:

 


2.3. How to divide a line segment into equal parts (Thales theorem)

1. Draw a ray that shares the origin of point A with the line segment segment AB.

2. Mark in the ray as many equal units as you want to obtain starting from point A. In this case, we are goint to divide the segment into three equal parts.

 
3. Join the point B with the end of the ray. For each of the divisions of the ray, draw parallel lines to the segment joining B. The points obtained in the segment AB represent 3 equal parts.

 

If you have any doubt about the process, you can watch this video:

3. Angles

3.1 DEFINITION

In geometry, an angle is the figure formed by two rays, called the arms of the angle, sharing a common endpoint, called the vertex of the angle.

3.2. TYPES OF ANGLES

• ACUTE ANGLES (Ángulos agudos): angles smaller than right angles (less than 90°)

• RIGHT ANGLES (ángulos rectos): angles equal to 90º. 

• STRAIGHT ANGLES (ángulos llanos): angles equal to 180º. 

• OBTUSE ANGLES (ángulos obtusos): angles larger than a right angle and smaller than a straight angle (between 90° and 180°) 

• REFLEX ANGLE (ángulo cóncavo): angles between 180º and 360º

• ROUND OR FULL ANGLES (ángulos completos) : angles equal to 360º. 

3.2.COPYING AN ANGLE

If we want to draw an angle equal to a given one with vertex at a given point V, we must follow the next steps:

3.3.ANGLE BISECTOR

The angle bisector is a line which divides the angle in two equal parts. Each point of an angle bisector is equidistant from the sides of the angle. 

 

3.4. DRAWING ANGLES WITH THE SET SQUARE RULERS

3.5.  ADDING AND SUBTRACTING ANGLES WITH THE COMPASS

 The addition of two angles is another angle whose measure is the addition of the measures of those two angles.

The subtraction of two angles is another angle whose measure is the subtraction of the measures of those two angles.

To practice all these concepts you have to print the first worksheet: