Name_________________________________ Date

UNIT 1Geometry

Page 2 Opening Exercise

Page 3 Definition of Circle

Pages 4-5 Basic Constructions: Practice using the Compass

Pages 6-10 Information of Perpendicular Bisectors and Angles

Pages 11-12 Construction of Perpendicular Bisector

Pages 13-14 Bisecting an Angle

Pages 15-16 Copying an angle

Pages 17-19 Constructing an Equilateral Triangle

Pages 20-22 Constructing an Equilateral Triangle Inscribed in a Circle

Page 23 Constructing a Hexagon Inscribed in a Circle

Pages 24-25 Constructing a 30 degree Angle and a 45 degree Angle

Pages 26-27 Constructing a line perpendicular to another line through a given point

Pages 28-30 Constructing a line parallel to another line through a given point

Pages 31-33 Constructing a Square Inscribed in a Circle

Pages 34-35 Constructing a Rectangle that is not a square inside a circle

Page 36 Constructing the Altitude of a Trapezoid

Pages 37-39 Constructing Triangle Midsegments Online Activity

Pages 40-41 Constructing the Median of a Triangle

Pages 42-43 Points of Concurrency: Circumcenter

Pages 44-45 Points of Concurrency: Incenter

Pages 46-48 Practice problems

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Name_____________________________ Date

Opening Exercise

Materials needed: Ruler, pencil

Directions:1.) There is a point on your paper labeled P2.) Plot 20 points that are exactly 3 inches away from point P

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On the previous page, you should have ended up with something that either looked like a circle or was becoming a circle. If you kept on plotting points, you would eventually have a circle.

Definition of Circle:

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Next, we are going to start CONSTRUCTIONS

What is a Construction?

Activity

Materials needed: Compass, Straight edge, pencil

Directions: Using this page and the next, construct 8 circles with each having a different radius. Label the center point of each circle with a letter. (Don’t worry if they overlap each other)

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Name________________________________ Date:

Opening Exercise

Plot 20 points that are exactly the same distance from both points A and B

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Name___________________________ Date:

PERPENDICULAR BISECTORS

LINE SEGMENT:

PERPENDICULAR:

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BISECTOR:

Line DE is bisecting line segment AC

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Practice with Angles

1.) Draw the following angles below 2.) Label the angles ABC

1.) ACUTE ANGLE

2.) OBTUSE ANGLE

3.) STRAIGHT ANGLE

4.) RIGHT ANGLE

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Perpendicular BisectorConstructing the Perpendicular Bisector of Line Segment

Activity: Using the steps below construct the perpendicular bisector of line segment AB

1. Place the compass point at one end of line segment.2. Adjust the compass to slightly longer than half the line segment length.3. Draw a circle with the center being the end point 4. Keeping the same compass width, draw a circle with the center being the other end point.5. Place ruler where the circles cross, and draw the line segment.

Do the 2 circles that you drew have the same radius? _____

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Directions: Construct the perpendicular bisector of Line segment AB.

1.)Label the point where the line segments intersect as C

2.) Label the endpoints of the line segment you drew D and E

State 2 angles that are 90 degrees __________ and __________

Do the 2 circles that you drew have the same radius? _____

Is point D the same distance from A that it is from B? _____

How do we know?

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ANGLE BISECTORActivity: Using the steps below construct the angle bisector of angle ABC

1.)Draw an circle that is centered at the vertex of the angle. This circle can have a radius of any length. However, it must intersect both sides of the angle.

** We will call these intersection points P and Q . This provides a point on each line that is an equal distance from the vertex of the angle.

2.) Draw a circle centered at point P with the same radius

3.) Draw a circle centered at point Q with the same radius

4.) Draw the bisector through the intersection point of the two circles.

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Practice Constructing the ANGLE BISECTOR

How to Copy an angle

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1.) Draw a ray that will be one of the sides of the new angle and label the end point P2.) On the original angle, draw a circle with the center being the vertex. The circle can be any size.

However, it must intersect both sides of the angle. 3.) Draw a circle centered at the point P of the ray that you drew4.) Using the compass, measure the opening of the original angle by placing the compass point and pencil

on the points where the circle intersects the sides5.) Keeping the compass radius the same, put the point of the compass on the point where the circle

intersected the ray that you drew and draw a circle.6.) Draw the other side of the angle through the intersection point of the 2 circles

Practice copying Angles

Copy the following 2 angles

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Name__________________________________ Date:

EQUILATERAL TRIANGLES16

Properties of Equilateral Triangles

1.)

2.)

Constructing an Equilateral Triangle**There are multiple ways to perform this construction. We will go over a couple of them

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Steps1.) Draw a line segment of any length and label the end points A and B2.) Place the point of the compass on point A and open up the compass so that the

other end is on point B and draw a circle3.) Repeat step 2 with the point of the compass on Point B and draw a circle4.) Draw two line segments connecting points A and B to the intersection point of the

circle

PracticeConstruct 2 Equilateral Triangles

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Name______________________________ Date:

Equilateral Triangle Inscribed in a Circle

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We already have talked about Equilateral Triangles, but what does it mean to be “Inscribed”?

When you think of INSCRIBED, think INSIDE!

INSCRIBED –

INSCRIBED NOT INSCRIBED

All three vertex points are not touching circle

Constructing an EQUILATERAL TRIANGLE Inscribed in a circle

Steps ***You may not be able to draw the entire circle for this

1.) Draw a circle of any size20

2.)Keeping the same opening of the compass, place the point anywhere on the outer edge of the circle and draw a circle

3.)Keeping the same opening of the compass, place the point of the compass on the intersection point of your 2 circles and draw another circle

4.) Continue this process all the way around the circle until there are 6 points of intersection on your original circle

5.)Label these points in order 1,2,3,4,5,66.) Use line segments to Connect points 1,3,5

Practice

Construct an Equilateral Triangle Inscribed in a Circle

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HEXAGONConstructing a Hexagon (6 sided figure) Inscribed in a circle

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Steps

1.)Follow steps 1-5 from constructing an Equilateral Triangle Inscribed in a circle2.)Connect points 1-6 in order

Constructing a 30 degree angle1.) Construct an equilateral Triangle using one of the methods that you have learned

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2.) Construct the angle bisector of one of the angles inside the triangle. This will produce two 30 degree angles

Constructing a 45 degree angle1.) Draw a line segment2.) Construct the perpendicular bisector of the line segment. This will produce four

right angles that each measure 90 degrees3.) Construct the angle bisector of one of these angles. This will produce two angles

that each measure 45 degrees.

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Name__________________________ Date:

Constructing a line Perpendicular to another line through a given point

Steps

1.) Place compass on point P and draw a circle. The circle must intersect the line segment twice. Label the two points of intersection C and D. You have created a new line segment

2.)Construct the perpendicular bisector of line segment CD25

3.)The line you draw is perpendicular to the given line segment and should go through the given point

Practice

Construct a line perpendicular to the given line through the given point

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Name______________________ Date:

Parallel Lines

Parallel Lines-

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On a coordinate grid, parallel lines have the same slope

Constructing a line parallel to another line through a given point

Steps

1.) Draw a line segment that goes through both segment AB and point P. Label point C where the line you drew intersected segment AB

2.) Draw a circle with the center being point C3.)Keeping the compass opening the same, draw a circle with the center being

point P

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4.) Next we are going to copy angle PCB with point P being the vertex of our new angle

Practice: Construct a line parallel to the given line that goes through the given point

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Name_________________________ Date:

Properties of Square

What is a Square?

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Constructing a Square Inscribed in a circleSteps:1.) Construct a circle of any size. Make sure and mark the center point

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2.)Using a straight edge, draw the diameter of the circle. Label the end points of the diameter A and B

3.)Construct the perpendicular bisector of the diameter. Label the points where the perpendicular bisector intersects the circle C and D

4.)Connect point A, B, C, D

Practice: On this page construct a square inscribed in a circle

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Challenge: Can you figure out how to construct a Rectangle inscribed in a circle. Try it on your own

NOTE: This rectangle CAN NOT be a square

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Constructing a rectangle inscribed in a circleSteps:

1.) Follow steps 1 thru 5 of constructing a Hexagon inscribed in a circle2.) Connect points 1,2,4,5

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CONSTRUCTING THE ALTITUDE OF A SHAPEDefinition of Altitude- In General, it is another word for height. It usually will connect a vertex to the base with a line that is perpendicular to the base.

Steps

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1.)Place point of compass on vertex J and draw a circle that intersects base ML twice. Label these point A and B

2.) Make two more circles with the centers being points A and B

3.) Make line through intersection of circles. (If done correctly, your line should go through vertex J

Name____________________________ Geometry Date:

Triangle Midsegment Online Assignment

1.) Go online and look up “Midsegment of Triangle”36

Below, write down the definition of a Midsegment

How many midsegments does a triangle have? _________

What are the 2 special properties of MIDSEGMENTS?

1.)

2.)

2.) Label the Triangle below with points A, B, C Research how to construct a midsegment and construct one of the midsegments of the triangle. Label the points D and E

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3.) State 2 angles that are congruent to one another

4.) Triangle ABC is shown below. Construct all three midsegments of the triangle and label the points D,E,F

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5.) After you have constructed all three midsegments, a smaller triangle is created inside the bigger one. How do the perimeters of the two triangles compare to each other?

Name_____________________ Date:

Median of a triangle

Definition of Median of Triangle-

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Construct All 3 medians of the triangle below

The three medians of the triangle intersected at one point. What is this point called?

*Below is problem #28 from the August 2016 Geometry exam

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Name ______________________________

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Points of Concurrencies

Opening Exercise

Construct the perpendicular bisectors of the three sides of the triangle below.

Discussion

When three or more lines intersect in a single point, they are _____________________, and the point of intersection is the _____________________________.

All three perpendicular bisectors pass through a common point. The point of concurrency of the three perpendicular bisectors is the _________________________________________.

The circumcenter of △ ABC is shown below as point P.

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The question that arises here is: WHY are the three perpendicular bisectors concurrent? Will these bisectors be concurrent in all triangles? To answer these questions, we must recall that all points on the perpendicular bisector are equidistant from the endpoints of the segment. This allows the following reasoning:

1. P is equidistant from A and B since it lies on the __________________________________ of AB.

2. P is also ___________________________________________ from B and C since it lies on the perpendicular

bisector of BC.

3. Therefore, P must also be equidistant from A and C. Hence, AP=BP=CP, which suggests that P is the point of

____________________________________ of all three perpendicular bisectors.

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The construction of the three angle bisectors of a triangle also results in a point of concurrency. Use the triangle below to construct the angle bisectors of each angle in the triangle.

All three angle bisectors pass through a common point. The point of concurrency of the three perpendicular bisectors is the _________________________________________.

State precisely the steps in your construction above.

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Observe the constructions below. Point A is the __________________________ of triangle △ JKL(notice that it can fall outside of the triangle). Point B is the __________________________ oftriangle △RST . The circumcenter of a triangle is the center of the circle that circumscribes that triangle. The incenter of the triangle is the center of the circle that is inscribed in that triangle.

Problem Set

1. Given line segment AB, using a compass and straightedge construct the set of points that are equidistant from A and B.

What figure did you end up constructing? Explain.45

2. For each of the following, construct a line perpendicular to segment AB that goes through point P.

3. Using a compass and straightedge, construct the angle bisector of ∠ABC shown below. What is true about every point that lies on the ray you created?

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4. The diagram below shows the construction of the center of the circle circumscribed about ∆ ABC.

This construction represents how to find the intersection of

1) the angle bisectors of ∆ ABC

2) the medians to the sides of ∆ ABC

3) the altitudes to the sides of ∆ ABC

4) the perpendicular bisectors of the sides of ∆ ABC

5. Which geometric principle is used in the construction shown below?

1) The intersection of the angle bisectors of a triangle is the center of the inscribed circle.

2) The intersection of the angle bisectors of a triangle is the center of the circumscribed circle.

3) The intersection of the perpendicular bisectors of the sides of a triangle is the center of the inscribed circle.

4) The intersection of the perpendicular bisectors of the sides of a triangle is the center of the circumscribed circle.

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6. In the diagram below of ∆ ABC , CD is the bisector of ∠BCA, AE is the bisector of ∠CAB, and BG is drawn.

Which statement must be true?

(1) DG=EG

(2) AG=BG

(3)∠ AEB≅∠ AEC

(4) ∠DBG≅∠EBG

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