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1-2&3 Measuring and Constructing Segments and Angles
Warm Up
Solve.
1.
2 x  9 x  18  0
2
3
x   or x  6
2
2.
6 x  54 x  84  0
2
x  7 or x  2
Holt Geometry
1-2&3 Measuring and Constructing Segments and Angles
Objectives
Use length and midpoint of a segment.
Construct midpoints and congruent segments.
Holt Geometry
1-2&3 Measuring and Constructing Segments and Angles
Vocabulary
coordinate
midpoint
distance
bisect
length
segment bisector
construction
between
congruent segments
Holt Geometry
1-2&3 Measuring and Constructing Segments and Angles
A ruler can be used to measure the distance
between two points. A point corresponds to one
and only one number on a ruler. The number is
called a coordinate. The following postulate
summarizes this concept.
Holt Geometry
1-2&3 Measuring and Constructing Segments and Angles
Holt Geometry
1-2&3 Measuring and Constructing Segments and Angles
The distance between any two points is the absolute
value of the difference of the coordinates. If the
coordinates of points A and B are a and b, then the
distance between A and B is |a – b| or |b – a|.
The distance between A and B is also called the length
of AB, or AB.
A
B
AB = |a – b| or |b - a|
a
Holt Geometry
b
1-2&3 Measuring and Constructing Segments and Angles
Congruent segments are segments that have the
same length. In the diagram, PQ = RS, so you
can write PQ  RS. This is read as “segment PQ
is congruent to segment RS.” Tick marks are
used in a figure to show congruent segments.
Holt Geometry
1-2&3 Measuring and Constructing Segments and Angles
You can make a sketch or measure and
draw a segment. These may not be exact. A
construction is a way of creating a figure that
is more precise. One way to make a
geometric construction is to use a compass
and straightedge.
Holt Geometry
1-2&3 Measuring and Constructing Segments and Angles
Check It Out! Example 2 Continued
Sketch, draw, and construct a segment
congruent to JK.
Holt Geometry
1-2&3 Measuring and Constructing Segments and Angles
In order for you to say that a point B is between two points A and C, all three
points must lie on the same line, and AB + BC = AC.
Holt Geometry
1-2&3 Measuring and Constructing Segments and Angles
Example 3A: Using the Segment Addition Postulate
G is between F and H, FG = 6, and FH = 11.
Find GH.
Holt Geometry
1-2&3 Measuring and Constructing Segments and Angles
Example 3B: Using the Segment Addition Postulate
M is between N and O. Find NO.
Holt Geometry
1-2&3 Measuring and Constructing Segments and Angles
The midpoint M of AB is the point that bisects, or
divides, the segment into two congruent segments.
If M is the midpoint of AB, then AM = MB.
So if AB = 6, then AM = 3 and MB = 3.
Holt Geometry
1-2&3 Measuring and Constructing Segments and Angles
Example 5: Using Midpoints to Find Lengths
D is the midpoint of EF, ED = 4x + 6,
and DF = 7x – 9. Find ED, DF, and EF.
F
E
4x + 6
Holt Geometry
D
7x – 9
1-2&3 Measuring and Constructing Segments and Angles
Objectives
Name and classify angles.
Measure and construct angles and angle
bisectors.
Holt Geometry
1-2&3 Measuring and Constructing Segments and Angles
Vocabulary
angle
vertex
interior of an angle
exterior of an angle
measure
degree
acute angle
Holt Geometry
right angle
obtuse angle
straight angle
congruent angles
angle bisector
1-2&3 Measuring and Constructing Segments and Angles
A transit is a tool for measuring angles. It consists
of a telescope that swivels horizontally and
vertically. Using a transit, a survey or can measure
the angle formed by his or her location and two
distant points.
An angle is a figure formed by two rays, or sides,
with a common endpoint called the vertex (plural:
vertices). You can name an angle several ways: by
its vertex, by a point on each ray and the vertex,
or by a number.
Holt Geometry
1-2&3 Measuring and Constructing Segments and Angles
The set of all points between the sides of the
angle is the interior of an angle. The exterior
of an angle is the set of all points outside the
angle.
Angle Name
R, SRT, TRS, or 1
You cannot name an angle just by its vertex if the
point is the vertex of more than one angle. In this
case, you must use all three points to name the
angle, and the middle point is always the vertex.
Holt Geometry
1-2&3 Measuring and Constructing Segments and Angles
The measure of an angle is usually given
in degrees. Since there are 360° in a circle,
one degree is
of a circle. When you use
a protractor to measure angles, you are
applying the following postulate.
Holt Geometry
1-2&3 Measuring and Constructing Segments and Angles
You can use the Protractor Postulate to help you
classify angles by their measure. The measure of
an angle is the absolute value of the difference of
the real numbers that the rays correspond with on
a protractor.
If OC corresponds with c
and OD corresponds with d,
mDOC = |d – c| or |c – d|.
Holt Geometry
1-2&3 Measuring and Constructing Segments and Angles
Holt Geometry
1-2&3 Measuring and Constructing Segments and Angles
Example 2: Measuring and Classifying Angles
Find the measure of each angle. Then classify
each as acute, right, or obtuse.
A. WXV
mWXV = 30°
WXV is acute.
B. ZXW
mZXW = |130° - 30°| = 100°
ZXW = is obtuse.
Holt Geometry
1-2&3 Measuring and Constructing Segments and Angles
Congruent angles are angles that have the same
measure. In the diagram, mABC = mDEF, so you
can write ABC  DEF. This is read as “angle ABC
is congruent to angle DEF.” Arc marks are used to
show that the two angles are congruent.
The Angle Addition Postulate is
very similar to the Segment
Addition Postulate that you
learned in the previous lesson.
Holt Geometry
1-2&3 Measuring and Constructing Segments and Angles
Holt Geometry
1-2&3 Measuring and Constructing Segments and Angles
An angle bisector is a ray that divides an angle
into two congruent angles.
JK bisects LJM; thus LJK  KJM.
Holt Geometry
1-2&3 Measuring and Constructing Segments and Angles
Example 4: Finding the Measure of an Angle
KM bisects JKL, mJKM = (4x + 6)°, and
mMKL = (7x – 12)°. Find mJKM.
Holt Geometry
1-2&3 Measuring and Constructing Segments and Angles
Check It Out! Example 4a
Find the measure of each angle.
QS bisects PQR, mPQS = (5y – 1)°, and
mPQR = (8y + 12)°. Find mPQS.
Holt Geometry
1-2&3 Measuring and Constructing Segments and Angles
Homework:
Pg 18 #21, 23, 27, 32, 40-44
Pg 25 #18, 29-32, 37, 38, 46-50
Holt Geometry