Section 1-5: Constructions SPI 32A: Identify properties of plane figuresTPI 42A: Construct bisectors of angles and line segments

Objective:• Use a compass and straight edge to construct and bisect congruent segments and angles

Perpendicular Lines

Two lines that intersect to form right angles.

Perpendicular Bisector

A line, segment, or ray that is perpendicular to the segment at its midpoint, thereby bisecting the segment into two congruent segments. 2 congruent segments

midpoint

Angle Bisector Ray that divides an angle into two congruent coplanar angles.

• Its endpoint is the angle vertex

• Say… Ray or Segment bisects the angle

Straight EdgeRuler with no markings

CompassTool used to draw circlesor arcs

Next, you will learn to construct:1. Congruent Segments2. Congruent Angles3. Perpendicular Bisectors4. Angle Bisectors

Tools used for Geometric Constructions

Construct Congruent Segments with Straight Edge and Compass

Construct a segment congruent to a given segment.

Given: ABConstruct: CD so that CD AB A B

Step 1: Draw a ray with endpoint CC

Step 2: Open a compass to the length AB

Step 3: With the same compass setting, put the compass point on point C. Draw an arc that intersects the ray. Label the point of intersection D. CD AB

Construct Congruent Angles using Straight Edge and Compass

Given: AConstruct: S so that S A

Step 1: Draw a ray with endpoint S.

Construct an angle congruent to a given angle.

A

S

Step 2: With the compass point on point A, draw an arc that intersects the sides of A. Label the points of intersection B and C.

Step 3: With the same compass setting, put the compass point on point S. Draw an arc and label its point of intersection with the ray as R.

Step 4: Open the compass to length BC. Keeping the same compass setting, put the compass point on R. Draw an arc to locate point T.

Step 5: Draw ST.

Construct Perpendicular Bisector

Construct the perpendicular bisector of a segment.

Given: ABConstruct: XY so that XY AB at the midpoint M of AB

A B

Step 1: Put the compass of point A and draw a long arc as shown. Be sure the opening is greater than ½ AB.

Step 2: With the same compass setting, put the compass point on point B and draw another long arc. Labe the points where the two arcs intersect as X and Y.

Step 3: Draw XY. The point of intersection of AB and XY is M, the midpoint of AB.

XY AB at the midpoint of AB, so XY is the Perpendicular bisector of AB.

Construct Angle Bisector

Construct the bisector of an angle

Given: AConstruct: AX, the bisector of A

Step 1: Put the compass of vertex A. Draw an arc that intersects the sides of A. Label the points of intersection B and C.

Step 2: Put the compass point on point C and draw an arc. With the same compass setting, draw an arc using point B. Be sure the arcs intersect. Label the intersection as X.

Step 3: Draw AX.

AX is the bisector of CAB.

Construction and Vocabulary Activity

1. Use your compass to draw a circle.

2. Locate three points A, B, and C on the circle.

3. Construct perpendicular bisectors of AB and BC.

4. Label the intersection of the two perpendicular bisectors as point O.

5. Make a conjecture about point O.