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Measuring
and
1-3
1-3 Measuring and Constructing Angles
Constructing Angles
Warm Up
Lesson Presentation
Lesson Quiz
Holt
Holt
McDougal
Geometry
Geometry
1-3 Measuring and Constructing Angles
Warm Up
1. Draw AB and AC, where A, B, and C are
noncollinear.
Possible answer: A
B
C
2. Draw opposite rays DE and DF.
F
Solve each equation.
3. 2x + 3 + x – 4 + 3x – 5 = 180 31
4. 5x + 2 = 8x – 10 4
Holt McDougal Geometry
D
E
1-3 Measuring and Constructing Angles
Objectives
Name and classify angles.
Measure and construct angles and angle
bisectors.
Holt McDougal Geometry
1-3 Measuring and Constructing Angles
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.
There are four ways to name this angle.
1.
Holt McDougal Geometry
<R
< SRT
<1
< TRS
1-3 Measuring and Constructing Angles
2. 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.
E
B
<A=33
Which angle am I talking about?
A
Hmmm…
C
D
Holt McDougal Geometry
We have no clue! This is why you
CAN’T name an angle by 1 letter if
there’s a lot going on
1-3 Measuring and Constructing Angles
3. How many different ways can we name all of the
following angles?
Holt McDougal Geometry
<1
<2
< MAH
< HAT
< MAT
< TAM
< TAH
< HAM
1-3 Measuring and Constructing Angles
4. 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.
Holt McDougal Geometry
1-3 Measuring and Constructing Angles
5. The measure of an angle is usually given
in degrees.
Since there are 360° in a circle, one
degree
is
of a circle.
Holt McDougal Geometry
1-3 Measuring and Constructing Angles
Holt McDougal Geometry
1-3 Measuring and Constructing Angles
6) •
Congruent angles are angles that have the
same measure.
• In the diagram, mÐABC = mÐDEF so you can
write ÐABC @ ÐDEF .
• This is read as “angle ABC is congruent to angle
DEF.”
• Arc marks are used to show that the two angles are
congruent. I call them “swoops.”
Holt McDougal Geometry
1-3 Measuring and Constructing Angles
Holt McDougal Geometry
1-3 Measuring and Constructing Angles
7. An angle bisector is a ray that divides an angle
into two congruent angles.
JK bisects
ÐLJM ; thus. ÐLJK @ ÐMJK
*K is in the interior
of <LJM
Holt McDougal Geometry
1-3 Measuring and Constructing Angles
Example 8
KM bisects ÐJKL ,
mÐMKL = (7x -12)
Find mÐJKM .
mÐJKM = (4x + 6)
and
m<JKM = m<MKL
(4x + 6)° = (7x – 12)°
+12
+12
4x + 18
–4x
= 7x
–4x
18 = 3x
6=x
Holt McDougal Geometry
4(6) + 6
mÐJKM = 30
1-3 Measuring and Constructing 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.
Step 1 Find y.
Step 2 Find m∠ PQS.
m∠ PQS = 5y – 1
= 5(7) – 1
5y – 1 = 4y + 6
y–1=6
y=7
Holt McDougal Geometry
= 34