CHAPTER 1: Tools of Geometry Section 1-7: Basic Constructions

CHAPTER 1:

Tools of Geometry

Section 1-7:Basic Constructions

Objectives

To use a compass and a straightedge to construct congruent segments and congruent angles.

To use a compass and a straightedge to bisect segments and angles.

Vocabulary

Construction Straightedge Compass Perpendicular Lines Perpendicular Bisector Angle Bisector

Construction

In a construction you use a straightedge and a compass to draw a geometric figure.

Straightedge

A straightedge is a ruler with no markings on it.

Compass

A compass is a geometric tool used to draw circles and parts of circles called arcs.

Construction #1:Constructing Congruent Segments

Given: AB Construct: CD so that CD is congruent to AB Steps:

Draw a ray with endpoint C. Open the compass to the length of AB. With the same compass setting, put the

compass on C and draw an arc that intersects the ray. Label the intersection D.

CD is congruent to AB

Construction #2:Constructing Congruent Angles

Given: RA Construct: RS so that RS is congruent to RA Steps:

Draw a ray with endpoint S. With the compass on point A, draw an arc that intersects the sides

of RA. Label the points of intersection B and C. With the same compass setting, put the compass point on S. Draw

an arc and label its point of intersection with the ray as R. Open the compass to the length of BC. Keeping the same compass

setting, put the compass on R. Draw an arc to locate point T. Draw ST. RS is congruent to RA

Perpendicular Lines

Perpendicular lines are two lines that intersect to form a right angle.

Perpendicular Bisector

A perpendicular bisector of a segment is a line, segment, or ray that is perpendicular to a segment at its midpoint.

It bisects the segment into two congruent segments.

Construction #3:Constructing the Perpendicular Bisector

Given: AB Construct: XY so that XY is perpendicular to AB at the

midpoint M of AB. Steps:

Put the compass point on point A and draw a long arc– be sure the opening is greater than half of AB.

With the same compass setting, repeat step one, this time with the compass on point B. Label the two intersection points X and Y.

Draw XY.

Angle Bisector

An angle bisector is a ray that divides an angle into two congruent coplanar angles.

The ray “bisects” the angle.

Construction #4:Constructing the Angle Bisector

Given: RA Construct: AX, the bisector of RA. Steps:

Put the compass point on vertex A. Draw an arc that intersects both sides of the angle. Label those points B and C.

Put the compass point on C and draw an arc in the interior of the angle.

Repeat step two, this time with the compass point on B. Label the intersection point of the two arcs X. Draw AX.