Constructions

All geometry constructions are limited by the use of only two mathematical tools. The first is a compass and the second is an unmarked straight edge.

To make a construction successfully the user should be familiar with the use of his/her compass. One should not hold the compass by the sides (wherein the sides could inadvertently be pushed inward) but rather by the top of the compass where there is typically a short top intended for this purpose. The pencil or lead in the compass should be slightly longer than the other leg (which I shall call pin-end). Start by just familiarizing yourself with the compass. This may see juvenile but it is important that you can manage full and complete smooth circles with little effort. I should also say at this time that you should have several sheets of blank paper removed from a notebook and lying directly atop the desk. Thus place the pin-end into your paper, slanting the compass slightly down (not perpendicular to the paper but at an angle). Most novices work better with the compass at a slight angle. If it still seems uncomfortable or uneven twist the paper instead of the compass to make your circles. Make several different sizes so that you can determine the size that is best for you. I have found that circles that are approximately the size of a 50 cent piece are most comfortable. Students that tend to keep things small such as the size of a penny or a dime have much greater difficulty.

The straight edge is an unmarked solid piece that does not measure but simply can be used to produce a straight line using its edge.

We will be learning how to:

  1. Copy a line segment. (CU1)
  2. Adding Line Segments (segment addition postulate).(CU1)
  3. Perpendicular Bisector.(CU1)
  4. Perpendicular to a line through a given point not on the line. (CU1)
  5. Cutting a line Segment into specific lengths (¼ , ½, 1/8, etc). (CU1)
  6. Bisecting and Angle (students should practice bisecting acute, right and obtuse angles). (CU2)
  7. Copy and Angle (students should practice copying acute, right and obtuse angles). (CU2)
  8. Angle Addition (angle addition postulate). (CU2)
  9. Construction of Parallel Lines (there ARE more converses that we could use here but we will only spend time on the following two). (CU3)

a)Use of Alternate Interior Angles Converse.

b)Use of Corresponding Angles Converse.

  1. Cutting a line segment in to any length (specifically 1/3, 1/5, 2/3, 3/5, 1/7 and 3/7). (CU3)
  2. Construction of specific Angles (30, 45, 60, 120, 135, 75, etc). Note: the only construction still impossible to construct (using only the compass and straight edge) is the trisection of an angle.(CU3)
  3. Copy Triangles (using SSS, SAS, ASA and AAS). (CU4)

Constructions Unit Worksheet 1

  1. Copy the line segment (follow the instructions in your booklet or on p. 104 of the text).
  1. b.

c.

  1. Adding line segments (segment addition postulate). Instructions: Draw a line, copy the first segment onto the line then the next segment at the end of the first. Repeat if adding more than 2 segments.
  1. Add and from question #1 b. Add and from question #1

c. 3 times from question #1d. 2 timesfrom question # 1

  1. Perpendicular Bisector (follow the instructions in the booklet for Segment bisector and Midpoint or on p. 34 of the text).
  1. b.

c. and d. – On a separate piece of paper Construct 2 more segments and then a perpendicular bisector to each segment.

  1. Perpendicular to a line through a given point not on the line (follow the instructions in you booklet or on p. 130 of the text).
  1. b.

c. d.

e. and f. On a separate piece of paper Construct 2 more segments and then a

perpendicular line to the segment through a given point not on the segment.

  1. Cutting a line segment into specific lengths (½, ¼, 1/8, etc.) This can be done by bisection a line then bisecting it again.
  1. ½
  1. 1/4
  1. 1/8

Constructions Worksheet Unit 2

  1. Bisecting an Angle (students should practice bisecting acute, right and obtuse angles). Follow the instructions in you booklet or on p. 36 of the text.
  1. b.

c. d.

  1. Copying an Angle (students should practice copying acute, right and obtuse angles). Follow the instructions in you booklet or on p. 159 of the text.
  1. Adding Angles (angle addition postulate). Instructions: copy the first angle then copy the second angle using one side of the first angle as the base of the new angle.
  1. Add the angles in part a and b from question # 7.
  1. Add the angles in part a and c from question # 7.
  1. Add the angles in part d and e from question # 7.

Constructions Unit Worksheet 3

  1. Construction of Parallel Lines (there ARE more converses that we could use here but we will only spend time on the following two).
  1. Use of Corresponding angles Converse. Follow the instructions in the booklet or on p. 159 of the text.
  1. b.

c. d.

e. f.

  1. Use of Alternate Interior Angles Converse. Instructions: similar to corresponding angles except copy the angles in the alternate interior position.

a. b.

c. d.

e. f.

  1. Cutting a line segment into any length. Follow the instructions on p. 500 of the text.
  1. 1/3b. 1/5

c. 2/3d. 3/5

e. 1/7f. 3/7

  1. Construction of specific Angles (30, 45, 60, 120, 135, 75, etc). Note: the only construction still impossible to construct (using only the compass and straight edge) is the trisection of an angle. Instructions: First draw a good sized circle (the biggest you can get on this page). Then without changing the compass setting, walk the compass around the outside of the circle marking the circle as you go. You should have 6 marks on the circle. If you connect every other mark you will have a 60 degree angle. From this 60 degree angle you can bisect the angle to get a 30 degree angle, and then bisect the 30 degree angle to get a 15 degree angle. Now use angle addition to get the desired angle. Draw the circle below and then on a separate sheet of paper draw a:
  1. 60 degree angle
  2. 30 degree angle
  3. 45 degree angle
  4. 120 degree angle
  5. 75 degree angle
  6. 135 degree angle

Constructions Unit Worksheet 4

  1. Copying Triangles (using SSS, SAS, ASA, and AAS). Follow the instructions in the booklet or on p. 213 of the text for SSS. For SAS copy a side then and angle then the last side. For ASA copy an angle then a side and then an angle. For AAS you would really do ASA via the use of the third angle theorem.

Copy each of the following triangles on a separate piece of paper using:

  1. SSS
  2. SAS
  3. ASA
  4. Do not worry about doing AAS