Geometric Constructions with Understanding **This is meant as a resource to the teacher! It is NOT intended to replace teaching in the classroom OR the discourse that should take place!!

Disclaimers! **This is meant as a resource to the teacher! **It is NOT intended to replace teaching in the classroom OR the discourse that should take place!! **The understanding can only take place if you have reviewed properties of special quadrilaterals & triangles (square, rhombus, parallelogram, kite, equilateral & isosceles triangles, etc.). **Triangle congruence is also a prerequisite.

Copy a Segment Start with the segment that you want to copy.

Copy a Segment Create a ray that you will copy the segment on.

Copy a Segment Draw a circle with the center at point A and radius on segment AB.

Copy a Segment Without changing the width of your compass, draw a second circle centered at point C on the ray.

Copy a Segment Segment AB is congruent to segment CD because they are both radii to congruent circles. Therefore, we have copied segment AB.

Bisect a Segment Start with the segment that you want to bisect.

Bisect a Segment Draw a circle centered at Point A and radius AB.

Bisect a Segment Draw a circle centered at Point B and radius AB.

Bisect a Segment Find the points of intersection of the circles.

Bisect a Segment Connect point A with both points of intersection. What do you know about those segments?

Bisect a Segment The segments are ALL radii of circle A so they are all the same length.

Bisect a Segment Connect point B with both points of intersection. What do you know about those segments?

Bisect a Segment The segments are ALL radii of circle B, which are the same length as the radii of circle A, so they are all the same length.

Bisect a Segment Connect the points of intersection (C & D). This segment bisects segment AB. Why?

Bisect an Angle Start with the angle you want to bisect.

Bisect an Angle Construct a circle centered at the vertex, Point A.

Bisect an Angle Find the points of intersection.

Bisect an Angle Segment AB is congruent to segment AC because both are radii of circle A.

Bisect an Angle Construct a circle centered at point C.

Bisect an Angle Without changing your compass setting, construct another circle about point B.

Bisect an Angle Find the points of intersection between the two new circles, B and C. These are points D and E.

Bisect an Angle Segment CE is a radius of circle C and BE is a radius of circle B. Circle C and circle B are congruent, so their radii are congruent.

Bisect an Angle Construct a ray that passes through points A & E. Notice that the ray also passes through point D. Why?

Bisect an Angle In hiding the circles and point D, we have this.

Bisect an Angle Segment AE is congruent to itself by the reflexive property.

Bisect an Angle Triangle ABE is congruent to Triangle ACE due to SSS congruence.

Bisect an Angle By CPCTC, angle BAE is congruent to angle CAE.

Bisect an Angle Since angle BAE is congruent to angle CAE, we know that angle BAC was bisected.

Copy an Angle Start with the angle you want to copy.

Copy an Angle Draw a ray on which you will copy the angle.

Copy an Angle Construct circle A & find points of intersection on each ray of the angle. What do you know about segments AC & AD?

Copy an Angle Without changing the width of your compass, construct circle B.

Copy an Angle Find the point of intersection of circle B with the given ray. What do you know about segment BE?

Copy an Angle Construct circle D so that it passes through point C.

Copy an Angle Draw segment CD. What do you know about it?

Copy an Angle Without changing the width of your compass, draw circle E.

Copy an Angle Find the points of intersection between circles B & E calling them F & G.

Copy an Angle We are going to continue with point F. We can use point G instead – we are simply making a choice here!!

Copy an Angle Draw segments BF & EF. What do you know about them?

Copy an Angle

Hide all circles. What do you notice?

Copy an Angle By SSS congruence, triangle ACD is congruent to triangle BFE.

Copy an Angle By CPCTC, angle CAD is congruent to angle FBE. So, we have copied that angle!

Construct a Parallel Line Start with a line and a point not on the line. We will construct a line parallel to the given line that passes through the point not on the line.

Construct a Parallel Line We know that if corresponding angles of two lines cut by a transversal are congruent, then we have parallel lines. So, we are going to create an angle on line j that passes through point A that we can copy on point A.

Construct a Parallel Line Draw a line that passes through point A and the line. This will be the transversal of our parallel lines.

Construct a Parallel Line Construct circle B and find the points of intersection with the given line and transversal.

Construct a Parallel Line Segments BC and BE are congruent because they are both radii of circle B.

Construct a Parallel Line Without changing the width of your compass, construct circle A.

Construct a Parallel Line Find the point of intersection of circle A & the transversal that corresponds with point C.

Construct a Parallel Line Segments BC and AD are congruent because they are both radii of congruent circles.

Construct a Parallel Line Construct circle C so that it passes through point E. Segment CE is a radius of circle C.

Construct a Parallel Line Without changing the width of your compass, construct circle D.

Construct a Parallel Line Find the point of intersection of circle D and circle A that corresponds to point E. DF is a radius of circle D.

Construct a Parallel Line Segments DF and CE are congruent because they are both radii of congruent circles.

Construct a Parallel Line Segment AF is a radius of circle A.

Construct a Parallel Line Segments AF and AD are congruent because they are both radii of circle A.

Construct a Parallel Line Hide all of the circles. What do you notice?

Construct a Parallel Line By SSS congruence, triangles BCE & ADF are congruent.

Construct a Parallel Line By CPCTC, angles CBE & DAF are congruent.

Construct a Parallel Line Because these corresponding angles are congruent, we know that segment AF is parallel to line BE.

Construct a Parallel Line Extend segment AF into line AF so that line AF is parallel to line BE.

Construct a Perpendicular Line Start with a line and a point not on the line. You will construct a line perpendicular to the given line that passes through the point not on the line.

Construct a Perpendicular Line Place two arbitrary points on the line.

Construct a Perpendicular Line Construct a circle centered at each arbitrary point that passes through point A.

Construct a Perpendicular Line Circles B and C both pass through point A. Find the second point of intersection of circles B and C.

Construct a Perpendicular Line Draw segments BA and BD. What do you know about those segments?

Construct a Perpendicular Line Draw segments CA and CD. What do you know about those segments?

Construct a Perpendicular Line Construct the line that passes through points A & D.

Construct a Perpendicular Line Hide circles B and C. What shape are we left with?

Construct a Perpendicular Line We have a kite. The diagonals of a kite are perpendicular. Therefore, our constructed line AD is perpendicular to given line BC.