Warm-up 1.5

1.5

Definitions

Construction Mathematically precise figure

Use a straightedge and a compass to draw a geometric figure

Straightedge Ruler with no markings on it.

Compass Geometric tool used to draw circles and parts of

circles called arcs.

Construct a segment congruent to a given segment.Given:

A B

Procedure:

1. Use a straightedge to draw a line. Call it l.

2. Choose any point on l and label it X.

3. Set your compass for radius AB and make a mark on the line where B lies. Then, move your compass to line l and set your pointer on X. Make a mark on the line and label it Y.

Construct: XY = AB

X Yl

Construction #1Construction #1 This is our compass.

Don’t change your radius!

Construct an angle congruent to a given angle

Given: Procedure:1) Draw a ray. Label it RY.

Construct:

E

D

2) Using B as center and any radius, draw an arc intersecting BA and BC. Label the points of intersection D and E.

4) Measure the arc from D to E.E2

Construction #2

RY

3) Using R as center and the SAME RADIUS as in Step 2, draw an arc intersecting RY. Label point E2 the point where the arc intersects RY

5) Move the pointer to E2 and make an arc that that intersects the blue arc to get point D2

D2

A

CB

6) Draw a ray from R through D2

definitions

Perpendicular lines Two lines that intersect to form right angles.

The symbol

Perpendicular bisector Is a line, segment, or ray that is perpendicular to the

segment at its midpoint, thereby bisecting the segment into two congruent segments.

definitions

Angle bisector Ray that divides an angle into two congruent

coplanar angles. Its endpoint is a the angle vertex

How do I construct a Bisector of a given angle?

Given:

C

A

B

Procedure:1. Using B as center and any radius, draw and arc that intersects BA at X and BC at point Y.

X

Y

2. Using X as center and a suitable radius, draw and arc. Using Y as center and the same radius, draw an arc that intersects the arc with center X at point Z.

Z

3. Draw BZ.

Construction #3

How do I construct a perpendicular bisector to a given segment?

Given:

Procedure:1. Using any radius greater than 1/2 AB, draw four arcs of equal radii, two with center A and two with center B. Label the points of intersection X and Y.

A B

X

Y

2. Draw XY

Construction #4

How do I construct a perpendicular bisector to a given segment at a given point?

Given: C k

Procedure: 1. Using C as center and any radius, draw arcs intersecting k at X and Y.

X Y

2. Using X as center and any radius greater than CX,draw an arc. Using Y as center and the same radius,

draw and arc intersecting the arc with center X at Z.

Z

3. Draw CZ.

Construction #5

How do I construct a perpendicular bisector to a given segment at a given point outside the line?

Given: k

P

Procedure: 1. Using P as center, draw two arcs of equal radii that intersect k at points X and Y.

X Y

2. Using X and Y as centers and a suitable radius, draw arcs that intersect at a point Z.

Z

3. Draw PZ.

Construction #6

How do I construct a line parallel to a given line through a given point?

Given: k

P

Procedure: 1. Let A and B be two points on line k. Draw PA.

A B

2. At P, construct <1 so that <1 and <PAB are congruent corresponding angles. Let l be the line containing the ray you just constructed.

1 l

Construction #7

Step 2: Open the compass to the length of KM.

Construct TW congruent to KM.

Step 1: Draw a ray with endpoint T.

Step 3: With the same compass setting, put the compass point on point T. Draw an arc that intersects the ray. Label the point of intersection W.

TW KMQuick Check

Basic ConstructionsLESSON 1-7

Additional Examples

Construct Y so that Y G.

Step 1: Draw a ray with endpoint Y.

Step 3: With the same compass setting, put the compass point on point Y. Draw an arc that intersects the ray. Label the point of intersection Z.

Step 2: With the compass point on point G, draw an arc that intersects both sides of G. Label the points of intersection E and F.

75°

Basic ConstructionsLESSON 1-7

Additional Examples

(continued)

Y G

Step 5: Draw YX to complete Y.

Step 4: Open the compass to the length EF. Keeping the same compass setting, put the compass point on Z. Draw an arc that intersects the arc you drew in Step 3. Label the point of intersection X.

Quick Check

Basic ConstructionsLESSON 1-7

Additional Examples

Start with AB.

Step 2: With the same compass setting, put the compass point on point B and draw a short arc.

Without two points of intersection, no line can be drawn, so the perpendicular bisector cannot be drawn.

Use a compass opening less than AB. Explain why the

construction of the perpendicular bisector of AB shown in the text is

not possible.

12

Step 1: Put the compass point on

point A and draw a short arc. Make

sure that the opening is less than AB.12

Basic ConstructionsLESSON 1-7

Additional Examples

Quick Check

–3x = –48 Subtract 4x from each side. x = 16 Divide each side by –3.

m AWR = m BWR Definition of angle bisector x = 4x – 48 Substitute x for m AWR and

4x – 48 for m BWR.

m AWB = m AWR + m BWR Angle Addition Postulatem AWB = 16 + 16 = 32 Substitute 16 for m AWR and

for m BWR.

Draw and label a figure to illustrate the problem

WR bisects AWB. m AWR = x and m BWR = 4x – 48. Find m AWB.

m AWR = 16 m BWR = 4(16) – 48 = 16 Substitute 16 for x.

Basic ConstructionsLESSON 1-7

Additional Examples

Quick Check

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

Step 2: Put the compass point on point B. Draw an arc in the interior of M.

Construct MX, the bisector of M.

Basic ConstructionsLESSON 1-7

Additional Examples

Step 4: Draw MX. MX is the angle bisector of M.

(continued)

Step 3: Put the compass point on point C. Using the same compass setting, draw an arc in the interior of M. Make sure that the arcs intersect. Label the point where the two arcs intersect X.

Basic ConstructionsLESSON 1-7

Additional Examples

Quick Check

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