# Lesson1

```					                                            Lesson Plan Guide
Georgia Learning Connections

(*Fields with an asterisk are required)

*TITLE:
Congruent Constructions

*Annotation
Geometer Sketchpad will be used to construct basic geometric congruent drawings that could also be shown using a
straightedge and a compass.

*Primary Learning Outcome:
Students should know what steps are necessary to construct geometric congruent drawings with a straightedge and
a compass.
Sample Assessment:
Write the steps necessary to construct and construct congruent segments, angles, triangles, and circles.
Write the steps necessary to construct and construct an angle bisector and a perpendicular bisector.
Write the steps necessary to construct and construct a perpendicular line from a point on a line, parallel lines, and
proportional segments.
Write the steps necessary to construct and construct a tangent line, inscribed polygons, and circumscribed
polygons.

Students should understand the definition of congruence and associated geometric drawings.
Sample Assessment:
What is congruence?
What is a bisector (angle, perpendicular)?
What is perpendicular, parallel, and proportional?
What are a tangent line and the definition of inscribed and circumscribed?

*Assessed QCC:
Euclidean Geometry – GA (9-12) http://www.glc.k12.ga.us
16.     Topic: Congruence
Standard: Uses tools such as compass and straightedge, paper folding, tracing
paper, mira, or computer to construct congruent segments, angles, triangles, and
circles; an angle bisector; a perpendicular bisector; a perpendicular line from a point
on a line; parallel lines; proportional segments; tangents; and inscribed and
circumscribed polygons.

Non-Assessed QCC:
Technology Integration – GA (10) http://www.glc.k12.ga.us
2.      Topic: Basic Skills
Standard: Demonstrates the ability to transfer technology skills into a significantly
new situation.
5.      Topic: Basic Skills
Standard: Communicates and disseminates technology skills and knowledge in
multiple ways.
6.      Topic: Basic Skills
Standard: Demonstrates appropriate and effective care and use of technology tools.

Georgia Learning Connections 2000                                     Georgia Department of Education
*Total Duration:
1 hour 50 minutes

*Materials and Equipment:
1. Geometer’s Sketchpad Construction Directions Sheets (below-copy class set)
2. Straightedge
3. Compass

Technology Connection:
Computer Lab – individual computer with Geometer’s Sketchpad for each student

Procedures:
Step One
Introduce the ruler and compass

Ruler and compass (taken from http://en.wikipedia.org/wiki/Ruler_and_compass_constructions)
A number of ancient problems in geometry involve the construction of lengths or angles using
only an idealized ruler and compass.

The "ruler" and "compass" of ruler-and-compass constructions is an idealization of rulers and
compasses in the real world:
    The ruler is infinitely long, but it has no markings on it (thus making it a straightedge instead of what we
usually think of as a ruler). The only thing you can use it for is to draw a line segment between two points.

    The compass can be opened arbitrarily wide, but (unlike most real compasses) it also has no markings it. It
can only be opened to widths you have already constructed.

Each construction must be exact. Eyeballing it and getting close does not count as a solution.
Estimated Time:
5 minutes

Step Two
Teacher demonstrates the following construction with a ruler and compass to the class:

Taken from http://mathforum.org/library/drmath/view/54688.html
Find the angle bisector of an angle.
Draw angle ABC using the straightedge.
Draw a circle centered at B.
Label the intersections of the rays BA and BC with the circle as points G and E.
Draw circles through B centered at E and G.
Label the intersection as point H.
BH is the angle bisector.
Estimated Time:
5 minutes

Step Three

Georgia Learning Connections 2000                                      Georgia Department of Education
Review any necessary computer guidelines and handout the introduction and assignment sheets.

Students should be familiar with Geometers Sketchpad.

Estimated Time:
5 minutes

Step Four
Students should work individually while the teacher monitors and assists as needed.

Estimated Time:
1 hour

Lesson Materials to be Attached:
Annotation: Step by step procedures that the student should perform on Geometer’s Sketchpad

Title: The Math Forum @ Drexel
URL: http://mathforum.org/library/drmath/view/54688.html
Annotation: Step by step directions for some constructions.

Title: Geometric Constructions Using a Compass and Straightedge
Annotation: A Geometry unit planned by Colorado teachers for 6-8 graders with much detail.

*Assessment:
The student should print a copy of the constructions performed on GSP.

Georgia Learning Connections 2000                                  Georgia Department of Education
CONSTRUCTIONS

Write the steps necessary to construct and construct congruent segments, angles, triangles, and circles.
Write the steps necessary to construct and construct a perpendicular line from a point on a line, parallel lines, and
proportional segments.
Write the steps necessary to construct and construct a tangent line, inscribed polygons, and circumscribed
polygons.
Angle Bisector
1. Draw and label angle ABC.
2. Draw a circle centered at B.
3. Label the intersections of the rays BA and BC with the circle as points G and E.
4. Draw circles through B centered at E and G.
5. Label the intersection as point H.
6. Construct a segment between points B and H.
7. Type: BH is the angle bisector.
8. Verify this statement by measuring the angles.
(This should look similar to the construction performed by hand in class.)

Perpendicular Bisector
1. Draw segment AB.
2. Draw a circle centered at A where the radius is greater than half the segment.
3. Highlight A and intersection of circle and segment to create a new segment.
4. Highlight new segment and B to construct a circle by center + radius.
5. Label the intersections of the circles as point C and D.
6. Construct a line between C and D.
7. Type: CD is the perpendicular bisector of AB.
8. Verify this statement by measuring the angle and the segment lengths.

Congruent Angles
1. Draw an angle with vertex A with rays as the sides.
2. Plot point B on one ray of the angle.
3. Draw a ray next to the side, but not touching with endpoint O.
4. Draw a circle centered at A that intersects point B. (Use Circle by Center + Point)
5. Label the intersection point of the circle and other ray as C.
6. Construct a segment between A and B.
7. Highlight segment and O.
8. Construct a circle using Circle by Center + Radius.
9. Plot point P on intersection of new circle and ray.
10. Construct a segment between A and C.
11. Highlight AC and P.
12. Construct a circle using Circle by Center + Radius.
13. Plot point Q on intersection of the two circles.
14. Construct a ray between O and Q.
15. Type: Angle POQ is congruent to Angle BAC.
16. Verify this statement by measuring the angles.

Georgia Learning Connections 2000                                     Georgia Department of Education
Congruent Segments
1. Draw segment AB.
2. Draw an arbitrary line.
3. Highlight segment and point on line to construct a circle by center + radius.
4. Type: The radius of the circle is congruent to AB.
5. Verify by measuring segments.

Congruent Triangles
1. Draw triangle ABC.
2. Copy segment AB.
3. Label DE.
4. Highlight AC and D to construct a circle by center + radius.
5. Highlight BC and E to construct a circle by center + radius.
6. Label intersection of circle as F.
7. Type: Triangle ABC is congruent to Triangle DEF.
8. Verify by measuring sides and angles.

Perpendicular line at a point on the line
1. Draw segment AB with point P on it.
2. Construct a circle with the center at P.
3. Mark intersections of circle and segment as R and S.
4. Construct two congruent circles, centered at points R and S, with radii large enough so
that the two circles intersect.
5. Label the intersection of the circles as Q.
6. Construct segment PQ.
7. Type: PQ is perpendicular to AB.
8. Verify this by measuring the angles.

Parallel Lines
1. Draw segment AB and arbitrary point C.
2. Construct segment AC and highlight.
3. Construct two circles centered at A and C by using the center plus radius (AC) tool.
4. Label intersection of circles on segment as F.
5. Construct a circle of radius AC at the point F.
6. Label intersection of circle C and circle F as I.
7. Construct segment CI.
8. Type: CI is parallel to AB and AC is parallel to FI.
9. Verify this by measuring the slope.

Georgia Learning Connections 2000                        Georgia Department of Education

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