Post on 29-Dec-2015
transcript
CONTENTSCONTENTSNaming Figures
Describing Figures
Distance on a number line
Distance on a grid
Segment Addition Postulate
Angles and Their Measures
Measuring Angles
Angle Addition Postulate
Classify Angles
Segment Bisectors and Midpoints
Angle Bisectors
NAMING FIGURESNAMING FIGURESFIGURE DESCRIPTION NAME IT
A POINT A
BD
C3 POINTS B, C, D
A line containing 3 known points FE FG
EF GE
OR
OR
OR.....
H
J
A segment with 2 end points HJ JHOR
EF GG
NAMING FIGURESNAMING FIGURESFIGURE DESCRIPTION NAME IT
Ray with endpoint K
A plane containing 3 known points NOP
Collinear points
KM
L
ORKL KM
NO
P
QOR Q
R, S, & TR
TS
U Noncollinear points U, R, S, & T
NO
P
Q
R
yDescribe the figure:
Plane Q contains Line NP, Line PR,and Points N, P, R, and O.
Line NP and Line PR intersect at Point P.
Line y intersects plane Q at point O.
DESCRIBING FIGURESDESCRIBING FIGURES
DISTANCEDISTANCEOn a Number LineOn a Number Line
EF G
-15 -2 7
Finding the length of a segment isthe same as finding the distance between its endpoints. When we measure a segment and attach a number to it we drop the bar in the symbol:
Since the length of AB is 12, we write AB = 12.
The length of FG is | F – G |.FG = | F – G |
YOU TRY: Find GE and FE.
= | -15 – - 2 |
= | -15 + 2 |
= | -13|
= 13
DISTANCEDISTANCEOn a Number LineOn a Number Line
EF G
-15 -2 7
The length of GE is | G – E |.GE = | G – E | = | - 2 – 7 | = | - 9|
= 9
Find GE and FE.
The length of FE is | F – E |.FE = | F – E | = | - 15 – 7 | = | - 22|
= 22
DISTANCEDISTANCEOn a Number LineOn a Number Line
The length of PQ is | P – Q |.PQ = | P – Q | = | 16 – - 4 | = | 20|
= 20
Find the length of the segment that has
endpoints with coordinates P(16) and Q(- 4).
DISTANCEDISTANCEOn a GridOn a Grid
Subtract x-values
To find the distance between two points on a
Grid, use the Distance Formula:
2122
12 yyxx Subtract y-values
Square the result Square the result
Add the results
Take the SQUARE ROOT and simplify
DISTANCEDISTANCEOn a GridOn a Grid
Example: Find the distance between
A( - 10, 4) and B( - 6, 1)
2122
12 yyxxAB
22 41106
22 34
525916 AB = 5
DISTANCEDISTANCEOn a GridOn a Grid
Find the distance between C( 7, - 3) and D( - 5, 2)
2CD2
CD yyxxCD
223275
22512
1316925144
CD = 13
Segment Addition PostulateSegment Addition Postulate
If B is between A and C,If B is between A and C,
A
C
B
Then Then ABAB + + BC BC = = ACAC
Segment Addition PostulateSegment Addition PostulateIf W is between X and Z, If W is between X and Z,
X
Z
W XWXW + + WZWZ = = XZXZ
2424
2424
+ + 5353
5353
= = XZXZ
7777 = = XZXZ
XW = 24 ,XW = 24 , WZ = 53 ,WZ = 53 ,
Find XZ .Find XZ .
Segment Addition PostulateSegment Addition PostulateIf W is between X and Z, If W is between X and Z,
X
Z
W XWXW + + WZWZ = = XZXZ
6969
6969
+ + WZWZ
142142
= = 142142
7373WZ WZ ==
XW = 69 ,XW = 69 ,
Find WZ .Find WZ .
XZ = 142,XZ = 142,
– – 6969 – – 6969
Segment Addition PostulateSegment Addition Postulate
== MPMP
Find all three segment measures .Find all three segment measures .
PGPG ++ MGMG
P MG4x + 64x + 6
9x + 129x + 12
3x + 263x + 26
4x + 64x + 6 3x + 263x + 26 9x + 129x + 12++ ==
If G is between P and M, If G is between P and M, PG = 4x + 6 ,PG = 4x + 6 ,MP = 9x + 12,MP = 9x + 12, and MG = 3x + 26,and MG = 3x + 26,
Segment Addition PostulateSegment Addition Postulate
4x + 64x + 6 3x + 263x + 26 9x + 129x + 12++ ==
7x7x
9x + 129x + 12==
9x + 129x + 12==
2x + 122x + 12==3232- 7x- 7x - 7x- 7x
2020 2x2x==xx==1010
PG = 4x + 6 = 4PG = 4x + 6 = 4(10)(10) + 6 = 46 + 6 = 46MG = 3x + 26 = 3MG = 3x + 26 = 3(10)(10) +26 = 56 +26 = 56
MP = 9x + 12 = 9MP = 9x + 12 = 9(10)(10) + 12 = 102+ 12 = 102
+ 32+ 32
- 12- 12 - 12- 12
PG = 46PG = 46
MG = 56MG = 56
MP = 102MP = 102
Angles and Their MeasuresAngles and Their Measures
L
J
KJ and KL form JKL
sides
Angle symbol
Naming angles (4 ways)1) JKL2) LKJ3) K (only if 1 angle)4) 1
1
Vertex is K
K
KK
Measuring AnglesMeasuring Angles
Congruent angles are angles with the same measure.
If m ABC = 50 and m JKL = 50 Then ABC JKL
Angles are congruentAngle Measures are equal!!
Angle Addition PostulateAngle Addition Postulate
If P is in the interior of RST, then m RSP + m PST = m RST .
TS
RP
Angle Addition PostulateAngle Addition Postulate
Suppose that the angle at the right measures 60° and that there is a point K in the interior of the angle such that m GHK = 25 . Find m KHI .
m GHK + m KHI = m GHI
K 60°?°
25°
25 + x = 60X = 60 – 25 = 35 m KHI = 35
Classify AnglesClassify Angles
Right Angle90
Obtuse Angle90 < x < 180Straight Angle
180
Acute Angle0 < x < 90
Segment BisectorSegment Bisector
Bisect means to cut into 2 congruent pieces.
The midpoint of a segment is the point that bisects the segment.
A segment bisector is a segment, ray, line or plane that intersects the segment at its
midpoint.
MidpointsMidpoints
If X is the midpoint of AB,If X is the midpoint of AB,
A
X
B
Then, Then, AXAX = = XBXB..
Midpoints on number linesMidpoints on number lines
To find the midpoint of a segment To find the midpoint of a segment on a number line, just on a number line, just averageaverage
the coordinates of the endpoints.the coordinates of the endpoints.
- 23- 23 4747
-23 + 4722
2424 22
= 12= 12
1212
==
Endpoints on number linesEndpoints on number linesTo find the To find the endpointendpoint of a segment on of a segment on
a number line with one endpoint a number line with one endpoint and the midpoint: Midpoint x 2, and the midpoint: Midpoint x 2,
then subtract the known endpoint.then subtract the known endpoint.
- 14- 14 2323
46 - -14 =46 - -14 = 23 x 2 - -14 =
6060
46 +14 =46 +14 = 6060
Midpoint FormulaMidpoint FormulaMidpoints on a GridMidpoints on a Grid
The Midpoint Formula: The midpoint of a segment
with endpoints (x1 , y1) and (x2 , y2)
has coordinates
( )
Midpoint on a GridMidpoint on a Grid
A is (-3, 4)
B is ( 2, 1)
Midpoint is -3 +2, 4 + 1 2 2
(- .5, 2.5)
Endpoint on a GridEndpoint on a Grid
A segment has endpoint J(-7, 8) and midpoint P(2, -1). Find the other endpoint.
Double the midpoint P(2, -1). Then subtract the endpoint you know J(-7, 8).
P(2, -1) x 2 gives (4, -2). (4, -2) - (-7, 8)(4 - -7, -2 – 8)
(11, -10)
The other endpoint is (11, -10).
Angle BisectorsAngle Bisectors
An angle bisector is a ray that dividesan angle into two adjacent congruent angles.
Angle bisector