# UNIT 5: ANGLES

## 1. TRANSLATION OF AN ANGLE

In Geometry, “Translation” simply means moving…without rotating, resizing or anything else,just moving.
If we want to draw an angle equal to a given one with vertex at a given point V. STEPS:

1. Center the compass at vertex of the given angle and draw an arc intersecting both sides of it. Without changing the radius of the compass, center it at point V and draw another arc
2. Set the compass radius to the distance between the two intersection points of the first arc.
3. Now center the compass at the point where the second arc intersects ray V.
4. Mark the arc intersection point 1.
5. Join point V with point 1 so you get the equal angle.

## 2. ANGLE BISECTOR It 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. STEPS:

1. Draw an angle.
2. Center the compass at vertex of the given angle and draw an arc intersecting both sides of it. We get 1 and 2
3. Center the compass at point 1 and draw an arc.
4. With the same measure center it at point 2 and draw another arc.
5. Where these arcs cross we get point 3.
6. If we join point 3 with the vertex of the angle we get the angle bisector.

## 3. TRISECTION OF AN ANGLE – DIVIDE A RIGHT ANGLE IN THREE EQUAL PARTS STEPS:

1. Draw a right angle angle, to do this we use the steps of the perpendicular to a ray
2. Center the compass at vertex of the right angle (V) and draw an arc intersecting both sides of it. We get 1 and 2.
3. Without changing the radious of the compass center the compass at point 1 and draw an arc, so we get point 3.
4. Without changing the radious of the compass center the compass at point 2 and draw an arc, so we get point 4.
5. If we join points 3 and 4 with the vertex of the angle we get the three equal parts of the right angle.

## 4. 45º ANGLE STEPS:

1. Draw a right angle angle, to do this we use the steps of the perpendicular to a ray
2. Draw the angle bisector of the right angle that you have drawn

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

1. Copy angle A using translation of an angle.
2. From this new angle copy angle B.
3. The solution is angle C.

## 6. SUBTRACTION OF ANGLES

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

1. Copy angle B (the biggest one) using translation of an angle.
2. From this new angle copy angle A.
3. The solution is angle C.

## SOLUTION OF DRAWING SHEET

Here you can see how to compose the complete drawing sheet, remember that you must underline the blue lines with your 0.4 technical pen, the red ones with 0.8 and the black ones with 0.2. Here you can see one of our students’ drawing sheets:
ANGLES – DRAWING SHEET NUMBER 6 – EXAMPLE 2013

# UNIT 3: FUNDAMENTAL CONSTRUCTIONS

## EXERCISE 01: SEGMENTS

In geometry, a line segment is a part of a line that is bounded by two distinct endpoints, and contains every point on the line between its end points. Line segments are generally labeled with two capital letters corresponding to their endpoints.

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.
We use this exercise if we have two segments and we want to draw a segment whose length is the addition of the measures of those two segments. STEPS:

1. Draw a line (r).
2. Draw a point A on it.
3. Measure the given segment AB with your compass.
4. Draw an arc from A with that measure, so you get B.
5. Measure the given segment CD with your compass.
6. Draw an arc from B with that measure, so you get D.
7. The solution is the segment AD.

## 2. SUBTRACTION SEGMENTS

We use this exercise if we have two given segments and we want to draw a segment whose measure is the substraction of the measures of those two segments. STEPS:

1. Draw a line (r).
2. Measure the longest segment with your compass, in our case is the segment CD.
3. Draw a point C on r.
4. Draw an arc from dot C with the previous measure (the segment CD), so you get D.
5. Measure the smallest segment with your compass, in our case is the segment AB.
6. Draw an arc from D with that measure, so you get B.
7. The solution is the segment CB.

## 3. DIVIDE A SEGMENT IN PROPORTIONAL PARTS TO THE GIVEN SEGMENTS. STEPS:

1. Draw a segment and call it AB.
2. Draw an oblique ray (r) to the segment from A.
3. Now we are going to divide segment AB into proportional parts to the given segments CD, DE and EF. We know the measures of these segments.
4. Take the measure of the given segment CD with your compass.
5. Draw an arc from dot A with this measure and where this arc crosses the oblique ray we get a point.
6. Take the measure of the given segment DE with your compass.
7. Draw an arc from the point and where this arc crosses the ray we get another point.
8. Take the measure of the given segment EF with your compass.
9. Draw an arc from last point and where this arc crosses the ray we get another point.
10. Now we need to join this last point with dot B.
11. Using our set square draw parallel lines to that segment from the other points on the ray.
12. Where these lines cross the segment AB we get the new segments C’D’, D’E’ and E’F’.

According to Thales’s theorem the segment C’D’ is proportional to segment CD and it will be the same with segment DE and D’E’ and EF with E’F’.

## 4. DIVIDE A SEGMENT IN EQUAL PARTS

Using Thales we can divide a segment in equal parts. STEPS:

1. Draw the given segment AB. This is the segment that we want to divide.
2. From point A draw an oblique ray (r).
3. Chose a measure with your compass and from point A draw arcs on the oblique ray as many arcs as parts you need.
4. Join the last point of the oblique ray with point B.
5. Draw parallels using your set square to the segment B7 from the other points on the ray.

Here we have divided the segment in seven parts, but you can divide the segment in as many parts as you need.

# ANGLES

In geometry, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle. ## TYPES OF ANGLES

• Angles equal to 90º are called a RIGHT ANGLES. Two lines that form a right angle are said to be perpendicular.
• Angles equal to 180º are called STRAIGHT ANGLES.
• Angles equal to 360º are called ROUND OR FULL ANGLES.
• Angles smaller than a right angle (less than 90°) are called ACUTE ANGLES.
• Angles larger than a right angle and smaller than a straight angle (between 90° and 180°) are called OBTUSE ANGLES. ## 1. TRANSLATION OF AN ANGLE

In Geometry, “Translation” simply means moving…without rotating, resizing or anything else,just moving.
If we want to draw an angle equal to a given one with vertex at a given point V. STEPS:

1. Center the compass at vertex of the given angle and draw an arc intersecting both sides of it. Without changing the radius of the compass, center it at point V and draw another arc
2. Set the compass radius to the distance between the two intersection points of the first arc.
3. Now center the compass at the point where the second arc intersects ray V.
4. Mark the arc intersection point 1.
5. Join point V with point 1 so you get the equal angle.

## 2. ANGLE BISECTOR It 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. STEPS:

1. Draw an angle.
2. Center the compass at vertex of the given angle and draw an arc intersecting both sides of it. We get 1 and 2
3. Center the compass at point 1 and draw an arc.
4. With the same measure center it at point 2 and draw another arc.
5. Where these arcs cross we get point 3.
6. If we join point 3 with the vertex of the angle we get the angle bisector.

## 3. TRISECTION OF AN ANGLE – DIVIDE A RIGHT ANGLE IN THREE EQUAL PARTS STEPS:

1. Draw a right angle angle, to do this we use the steps of the perpendicular to a ray
2. Center the compass at vertex of the right angle (V) and draw an arc intersecting both sides of it. We get 1 and 2.
3. Without changing the radious of the compass center the compass at point 1 and draw an arc, so we get point 3.
4. Without changing the radious of the compass center the compass at point 2 and draw an arc, so we get point 4.
5. If we join points 3 and 4 with the vertex of the angle we get the three equal parts of the right angle.

## 4. 45º ANGLE STEPS:

1. Draw a right angle angle, to do this we use the steps of the perpendicular to a ray
2. Draw the angle bisector of the right angle that you have drawn

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

1. Copy angle A using translation of an angle.
2. From this new angle copy angle B.
3. The solution is angle C.

## 6. SUBTRACTION OF ANGLES

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

1. Copy angle B (the biggest one) using translation of an angle.
2. From this new angle copy angle A.
3. The solution is angle C.

## SOLUTION OF DRAWING SHEET

Here you can see how to compose the complete drawing sheet, remember that you must underline the blue lines with your 0.4 technical pen, the red ones with 0.8 and the black ones with 0.2. Here you can see one of our students’ drawing sheets:
ANGLES – DRAWING SHEET NUMBER 6 – EXAMPLE 2013

# SEGMENTS

In geometry, a line segment is a part of a line that is bounded by two distinct endpoints, and contains every point on the line between its end points. Line segments are generally labeled with two capital letters corresponding to their endpoints.

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.
We use this exercise if we have two segments and we want to draw a segment whose length is the addition of the measures of those two segments. STEPS:

1. Draw a line (r).
2. Draw a point A on it.
3. Measure the given segment AB with your compass.
4. Draw an arc from A with that measure, so you get B.
5. Measure the given segment CD with your compass.
6. Draw an arc from B with that measure, so you get D.
7. The solution is the segment AD.

## 2. SUBTRACTION SEGMENTS

We use this exercise if we have two given segments and we want to draw a segment whose measure is the substraction of the measures of those two segments. STEPS:

1. Draw a line (r).
2. Measure the longest segment with your compass, in our case is the segment CD.
3. Draw a point C on r.
4. Draw an arc from dot C with the previous measure (the segment CD), so you get D.
5. Measure the smallest segment with your compass, in our case is the segment AB.
6. Draw an arc from D with that measure, so you get B.
7. The solution is the segment CB.

## 3. THALES THEOREM

If two lines which are in the same plane are cut by parallel lines, segments determined in one line are proportional to the segments determined in the other line. We use the Thales theorem to divide segments into equal or proportional parts.

## 4. DIVIDE A SEGMENT IN PROPORTIONAL PARTS TO THE GIVEN SEGMENTS. STEPS:

1. Draw a segment and call it AB.
2. Draw an oblique ray (r) to the segment from A.
3. Now we are going to divide segment AB into proportional parts to the given segments CD, DE and EF. We know the measures of these segments.
4. Take the measure of the given segment CD with your compass.
5. Draw an arc from dot A with this measure and where this arc crosses the oblique ray we get a point.
6. Take the measure of the given segment DE with your compass.
7. Draw an arc from the point and where this arc crosses the ray we get another point.
8. Take the measure of the given segment EF with your compass.
9. Draw an arc from last point and where this arc crosses the ray we get another point.
10. Now we need to join this last point with dot B.
11. Using our set square draw parallel lines to that segment from the other points on the ray.
12. Where these lines cross the segment AB we get the new segments C’D’, D’E’ and E’F’.

According to Thales’s theorem the segment C’D’ is proportional to segment CD and it will be the same with segment DE and D’E’ and EF with E’F’.

## 5. DIVIDE A SEGMENT IN EQUAL PARTS

Using Thales we can divide a segment in equal parts. STEPS:

1. Draw the given segment AB. This is the segment that we want to divide.
2. From point A draw an oblique ray (r).
3. Chose a measure with your compass and from point A draw arcs on the oblique ray as many arcs as parts you need.
4. Join the last point of the oblique ray with point B.
5. Draw parallels using your set square to the segment B7 from the other points on the ray.

Here we have divided the segment in seven parts, but you can divide the segment in as many parts as you need.