Transcript Geometry Section 1.6 Notes

1.6 Constructions Involving Lines
and Angles
Copyright © 2014 Pearson Education, Inc.
Slide 1-1
Congruent Segments
Two segments that have the same length are called
congruent segments. The symbol ≅ means
congruent.
We mark congruent segments in a figure with
exactly the same number of tick marks.
Copyright © 2014 Pearson Education, Inc.
Slide 1-2
Example
Measuring Congruent Segments
a. Use the figure to write the
congruent segments and the equal
distances.
AD and BC are marked the same, so
AD  BC and AD  BC.
AB and DC are marked the same, so
AB  DC and AB  DC.
Copyright © 2014 Pearson Education, Inc.
Slide 1-3
Example
Measuring Congruent Segments
b. If BC = 2 feet, find AD.
BC = AD, so if BC = 2 feet, then AD = 2 feet.
c. If DC = 6 feet, find AB.
DC = AB, so if DC = 6 feet, then AB = 6 feet.
Copyright © 2014 Pearson Education, Inc.
Slide 1-4
Definitions
The midpoint of a segment is a point that divides,
or bisects, a segment into two congruent segments.
True statements:
• B is the midpoint of AC.
• Line m bisects AC.
• BD bisects AC.
A line, ray, segment, or plane
• DB bisects AC.
that intersects a segment at its
• AB  BC.
midpoint is called a segment
• AB  BC.
bisector.
Copyright © 2014 Pearson Education, Inc.
Slide 1-5
Definitions
A straight edge is a ruler with no markings on it. A
compass is a geometric tool used to draw circles
and parts of circles called arcs. A construction is a
geometric figure drawn using a straight edge and a
compass.
Copyright © 2014 Pearson Education, Inc.
Slide 1-6
Example
Constructing Congruent Segments
Construct a segment congruent to a given segment.
Given: segment AB
Construct: segment CD so that segment CD ≅
segment AB
Solution
Step 1. Draw a ray with endpoint C.
Copyright © 2014 Pearson Education, Inc.
Slide 1-7
Example
Constructing Congruent Segments
Step 2. Open the compass to the length of
segment 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.
Copyright © 2014 Pearson Education, Inc.
Slide 1-8
Example
Constructing Congruent Angles
Construct an angle congruent to a given angle.
Given: ∠A
Construct: ∠S so that ∠S ≅ ∠A
Solution
Step 1. Draw a ray with endpoint S.
Copyright © 2014 Pearson Education, Inc.
Slide 1-9
Example
Constructing Congruent Angles
Step 2. With the compass point on vertex A, draw
an arc that intersects the sides of ∠A. Label the
points of intersection B and C.
Copyright © 2014 Pearson Education, Inc.
Slide 1-10
Example
Constructing Congruent Angles
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.
Copyright © 2014 Pearson Education, Inc.
Slide 1-11
Example
Constructing Congruent Angles
Step 4. Open the compass to the length BC. Keeping
the same compass setting, put the compass point on
R. Draw an arc to locate point T.
Step 5. Draw ray ST.
The angles are congruent, or ∠S ≅ ∠A.
Copyright © 2014 Pearson Education, Inc.
Slide 1-12
Definitions
AB  CD and CD  AB.
Copyright © 2014 Pearson Education, Inc.
Slide 1-13
Constructing the Perpendicular
Example
Bisector
Copyright © 2014 Pearson Education, Inc.
Slide 1-14
Constructing the Perpendicular
Example
Bisector
Step 2. With the same compass setting, put the
compass point on point B and draw another long arc.
Label the points where the two arcs intersect as X
and Y.
Copyright © 2014 Pearson Education, Inc.
Slide 1-15
Constructing the Perpendicular
Example
Bisector
Step 3. Draw line XY. Label the point of intersection
of segment AB and line XY as M, the midpoint of
AB.
XY  AB at midpoint M, so line XY is the
perpendicular bisector of segment AB.
Copyright © 2014 Pearson Education, Inc.
Slide 1-16
Example
Constructing the Angle Bisector
Construct the bisector of an angle.
Given: ∠A
Construct: ray AD, the bisector of ∠A
Solution
Step 1. Put the compass point on vertex A. Draw an
arc that intersects the sides of ∠A. Label the points
of intersection B and C.
Copyright © 2014 Pearson Education, Inc.
Slide 1-17
Example
Constructing the Angle Bisector
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
point where the two arcs intersect as D.
Copyright © 2014 Pearson Education, Inc.
Slide 1-18
Example
Constructing the Angle Bisector
Step 3. Draw ray AD .
ray AD is the angle bisector of ∠CAB.
Copyright © 2014 Pearson Education, Inc.
Slide 1-19