Download 3.1 Angles and Their Properties -- the union of 2 rays that have the

3.1 Angles and Their Properties
_________________ -- the union of 2 rays that have the same endpoint (symbol:
)
What is the name of this angle?
________________
Which point is the vertex of this angle?
________________
Name the sides of this angle.
_______ & _______
_________________ -- is another name for a line. Draw an example below!
_________________ -- is another name for a ray. Draw an example below!
Give three different names to the angle at
right. Why is the “A” in middle of each
name?
Draw a bisector to the angle at left. How
many angles are there in the drawing now?
_______.
An ____________ _____________ must always divide an
angle _____ ____________.
Draw an angle and label the Interior and the Exterior. The exterior of an angle is always _________________________.
When mentioning the angle , we write _________________. When mentioning the _______________ of an angle, we
write __________________.
Is
o
the angle bisector to XWZ?
,
∠XWY = 79o, and m∠YWZ = 38o
Find m∠ZWV.
In the picture below, m∠YWZ = 5x – 11 and m∠YWX = 2x + 25. What is the value of x? What is the measure of each
angle?
What CAN you assume about the
picture below?
What CAN’T you assume about the
picture below?
*
*
*
*
*
*
3.2 Arcs and Rotations
Identify a minor arc.
Identify a major arc.
Name the circle.
Name a central angle.
The measure of a minor arc is ____________________ to the measure of the ____________________________
Using the circle at the top of the page, if
measure of
is 300O, what is the measure of ∠ABC?______________ What is the
There are ____________ degrees in a circle.
Draw a circle with a straight angle cutting through the center of the circle. This line ___________________ the circle.
Each half of the picture you drew is called a ________________________________________.
Draw four concentric circles below.
The amount of degrees an object rotates is called its _________________________________.
A rotation can be turned either ______________________________________ or
____________________________________________________
A positive magnitude is turned ___________________________________________________
A negative magnitude is turned __________________________________________________
Examples 3-2
1.
m  RWD = 113. m
2.
Then what is m
3.
A counterclockwise rotation (turn) has a _______________________ magnitude.
4.
m  RWD = 121. m
6.
A rotation of -170° is the same as a rotation of _________________°.
7.
When the hour hand of a clock moves from 3 to 6, what is the magnitude of the rotation?
= ________
? _____________
= ________
3.3 Properties of Angles
Name of Angle:
Zero Angle
Acute Angle
Right Angle
Obtuse Angle
Straight Angle
Complementary Angles
Supplementary Angles
Adjacent Angles
Linear Pair
Vertical Angles
Picture:
Symbols:
Examples 3-3
1.
Complementary angles ___________________________ have equal measures. (always, sometimes, never)
2.
Two acute angles are ______________________ supplementary. (always, sometimes, never)
3.
Vertical angles _______________________ have equal measure. (always, sometimes, never)
4.
Supplementary angles are ___________________________ linear pairs. (Always, sometimes, never)
5.
Right angles are __________________________ complementary. (Always, sometimes, never)
6.
m  2 = 3x –6
m  3 = 17 – x
x = __________
7.
m  2 = 3x – 6
m  4 = 7x + 8
m  2 = _______
8.
Two angles are supplementary. One measures 2x + 7, the other measures
3x – 2. Find the measure of each angle.
9.
A support wire makes a 57° angle with the ground. What is the measure of the other angle formed by the wire and the
support?
3.4 Properties
For any real numbers a, b, c:
**Reflexive Property of Equality: ____________________________
Example:
**Symmetric Property of Equality:_____________________________
Example:
**Transitive Property of Equality: _____________________________
Example:
Which is which??
1) If 1  2, and 2  3 , then 1  3
________________________
2) A  A
___________________
3) If N  K , then K  N
__________________
**Addition Property of Equality: _______________________________
Example:
**Multiplication Property of Equality:____________________________
Example:
**Equation to Inequality:
If a + b = c; then a< c and b < c
Example:
**Substitution Property:
If a = b, then a may be substituted for b
Example:
3.5 One Step Proofs
A proof is a sequence of __________________________________ ________________________________________
that start with the
_________________________________________ and end with the __________________________________________.
A proof must always have _________________ information, and we are always trying to justify statements to come to a
logical conclusion.
Example:
 Given: 4r – 3 = 11
 Prove: r = 3.5
Statement
Reason
What is a midpoint? _________________________________________________________
Draw line segment AB. Put C where you think the midpoint of the segment should be. Use "tick marks" to show
equality.
Given C is a midpoint above, what do we know? _____________________________________
Why do we know this is true? ____________________________________________________
Draw two lines that intersect and identify the four angles as angles 1, 2, 3,& 4 in a clockwise manner.
Why is the measure of angle 1 = to the measure of angle 3? __________________________________________.
Examples 3-5
State the justification in the following 1-step proof:
1.
Conclusion
Justification
0) Wis the midpoint of CR
1)
CW = CR
2.
Conclusion
1)
Justification
0) ∠7 is a right angle
0) Given
m∠7 = 90O
1)
3.
Conclusion
1)
Conclusion
1)
Justification
0) ∠3 and ∠4 are complementary
4.
0) Given
0) Given
m∠ 3 + m∠4 = 90
1)
Justification
0) m∠5 = 35O
0) Given
1) ∠5 is acute
1)
5.
State a conclusion and give its justification:
6. State a conclusion and give its justification:
7. State a conclusion and give its justification:
8. What kinds of statements can be used as justifications in proofs?
3.6 Parallel Lines
What is a Transversal? _________________________________________________________.
Which one is the transversal in the picture?
How many angles were formed above? __________
Label the corresponding angles in the drawing above.
Identify all pairs of corresponding angles in the picture:
___________________________________________________________________________
**If lines are parallel then all corresponding angles are _________________________.
**If we know all corresponding angles are congruent, then we know what must be true about the lines that create
them? _________________________________________________________.
The two marked lines above are the _______________________________________________________
The formula to find the slope of two ordered pairs is: _________________________________.
A horizontal line has a slope of ___________________________.
A vertical line has a slope of _____________________________.
All parallel lines slopes are _____________________________________.
Examples:
1.
What is the slope of the line through points (-2, 6) and (8, 11)?
2.
What is the slope of the line through points (-23, -27) and (16, -46)?
3.
If a line has an equation of 2x + 3y = 12, what is the slope of the line? (Solve
for y)
4.
If line one is parallel to line two, and line one has a slope of 2/3 what is the
slope of line two?
5.
If two lines are parallel to the same line, then the two lines must be
___________________ to each other.
6. Corresponding angles are _______________________ equal in measure. (Always, sometimes, never)
7. a // b. m∠3 = 134. m∠5 = ___________
8. a // b. Which angle(s) have the same measure as angle 6?
3.7 Perpendicular Lines
Define Perpendicular Lines:
Draw two lines that are perpendicular and identify the four angles as 1, 2,3 & 4 in a clockwise manner.
True or False: Angles 1 & 2 are complimentary.
True or False: Angles 1 & 2 are supplementary.
True or False: Angles 1 & 2 form a linear pair.
Draw two lines, a & b, that are perpendicular to a third line, c. Put in the appropriate markings.
What is true about lines a & b? Why is this true?
If lines a & b from the previous drawing are parallel what is true about their slopes?
Lines that are perpendicular have slopes that are the _________________ ___________________ to each other.
If line x has a slope of 2/3 and line y is perpendicular to line x, then the slope of line y is? __________.
To be perpendicular lines, the slopes of the two lines must multiply to give you _________________
Examples:
1.
What is the slope of a line perpendicular to y = -
2.
What is the slope of a line parallel to the line with the equation 5x + 4y = 12
3.
What is the slope of a line perpendicular to 3x – 2y = 6?
4.
If two lines are perpendicular, then their slopes would multiply to give you
what number?
?