Download Geometry 1.5 Measuring and Constructing Angles Day 1

Geometry
1.5 Measuring and Constructing Angles
Day 1
1.5 Measuring and Constructing Angles
September 2, 2015
1.5 Essential Question
How can I measure and classify an angle?
1.4 Perimeter and Area in the Coordinate Plane
September 2, 2015
What You Will Learn
•
•
•
•
Name, measure, and classify angles.
Identify congruent angles.
Use Angle Addition Postulate.
Bisect angles.
1.5 Measuring and Constructing Angles
September 2, 2015
What is an Angle?
An ANGLE consists of
two different rays that
have the same endpoint.
Sides
Endpoint
Vertex
1.5 Measuring and Constructing Angles
The rays are the sides
of the angle.
We call the intersecting
endpoints the VERTEX of the
angle.
September 2, 2015
Interior/Exterior
R
R is in the interior of A
A
S S is in the exterior of A
1.5 Measuring and Constructing Angles
September 2, 2015
Naming Angles
You can name an angle several
different ways.
A
ABC or CBA
1
B
or justB
C
or 1.
Important point: when using three letters in the name
of an angle, the vertex must be the middle one.
1.5 Measuring and Constructing Angles
September 2, 2015
Example 1
A lighthouse keeper
measures the angles
formed by the lighthouse
at point M and three
boats. Name three angles
shown in the diagram.
1.5 Measuring and Constructing Angles
September 2, 2015
Let’s See What You Know
What is NOT a
name for the angle
below?
1.5 Measuring and Constructing Angles
Write 3 names for
the angle below?
September 2, 2015
Let’s See What You Know
What is NOT a
name for the angle
below?
1.5 Measuring and Constructing Angles
Write 3 names for
the angle below?
September 2, 2015
Let’s See What You Know
What is NOT a
name for the angle
below?
1.5 Measuring and Constructing Angles
Write 3 names
for the angle
below.
September 2, 2015
Angle Measure
• The measure of ABC is denoted mABC.
• Angles are measured in degrees, using the symbol °.
• For example, if the measure of D is 24 degrees, we
would write m D = 24°.
• Angles are measured using a tool called a protractor.
1.5 Measuring and Constructing Angles
September 2, 2015
Angle Classification
A
0  mA  90
1.5 Measuring and Constructing Angles
Acute
September 2, 2015
Angle Classification
Use a square at
the vertex to
indicate this is a
right angle.
A
mA  90
1.5 Measuring and Constructing Angles
Right
September 2, 2015
Angle Classification
A
90  mA  180
1.5 Measuring and Constructing Angles
Obtuse
September 2, 2015
Angle Classification
A
mA  180
1.5 Measuring and Constructing Angles
Straight
September 2, 2015
Congruent Angles
• Angles that have the same measure are congruent.
• The symbol for congruent is

If TUF and LUK each measure 50°, then they are
congruent.
Angles are congruent
TUF  LUK
1.5 Measuring and Constructing Angles
Measures are equal
mTUF  mLUK
September 2, 2015
Postulate 1.3
Protractor Postulate
The measure of an angle is a real
number from 0 to 180.
(This is hugely simplified from the book’s
version.)
1.5 Measuring and Constructing Angles
September 2, 2015
Example 2
Find the measure of each angle.
Then classify each angle.
1.5 Measuring and Constructing Angles
September 2, 2015
Postulate 1.4
Angle Addition Postulate
A
B
D
C
If B is in the interior of ADC, then
mADB + mBDC = mADC
1.5 Measuring and Constructing Angles
September 2, 2015
Example 3
70°
40°
30°
1.5 Measuring and Constructing Angles
September 2, 2015
Example 4
1.5 Measuring and Constructing Angles
September 2, 2015
Essential Question
How can you measure and classify an
angle?
Day 1 Homework:
Pg. 43 – 44
#1, 2, 3 – 13 Odd, 14, 21 – 29 Odd
1.4 Perimeter and Area in the Coordinate Plane
September 2, 2015
Geometry
1.5 Measuring and Constructing Angles
Day 2
Essential Question
When is a ray an angle bisector?
1.4 Perimeter and Area in the Coordinate Plane
September 2, 2015
Angle Bisector
A
A ray that divides an
angle into two angles
that are congruent is
an angle bisector.
D
B
C
ABD  CBD
Use congruence marks to
indicate congruent angles.
1.5 Measuring and Constructing Angles
September 2, 2015
Example 5
mBUM = 110°
G
Ray UG is the angle
bisector of BUM.
110°
B
55° 55°
M
U
Find mGUM
and mBUG
Think: Ray UG divides BUM
into two equal parts, so each is 55°.
1.5 Measuring and Constructing Angles
September 2, 2015
Example 6
mRUG = 43°
G
86°
Ray UG is the angle
bisector of RUT.
R
43° 43°
U
T
Find mGUT
and mRUT
Think: Ray UG divides RUT into two
equal parts, so each is 43°. The total is 86.
1.5 Measuring and Constructing Angles
September 2, 2015
Example 7
RT bisects ARC.
mART = (7x + 12)°
mTRC = (4x + 30)°
Find the measure of each angle.
1.5 Measuring and Constructing Angles
September 2, 2015
Your Turn
Ray AF bisects CAB
F
B
C
mFAB = 8x + 10
mFAC = 3x + 25
A
1.5 Measuring and Constructing Angles
Find mBAF and mCAB.
September 2, 2015
Your Turn Solution
F
B
8x + 10
3x + 25
C
A
8x + 10 = 3x + 25
5x = 15
x=3
mBAF = 8(3) + 10 = 34°
mCAB = 2(34) = 68
1.5 Measuring and Constructing Angles
September 2, 2015
Essential Question
When is a ray an angle bisector?
1.4 Perimeter and Area in the Coordinate Plane
September 2, 2015
Assignment
Day 2 Homework:
Pg. 44 – 45
17 – 20 ALL, 33 – 39 Odd, 44, 45
1.5 Measuring and Constructing Angles
September 2, 2015