ch. 1
DESCRIPTION
Ch. 1. B. 3. 3. P. A. Midpoint of a segment. The point that divides the segment into two congruent segments. B. 3. 3. P. A. Bisector of a segment. A line, segment, ray or plane that intersects the segment at its midpoint. Bisector of an Angle. - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: Ch. 1](https://reader035.vdocument.in/reader035/viewer/2022062718/56812c7d550346895d9126ad/html5/thumbnails/1.jpg)
Ch. 1
![Page 2: Ch. 1](https://reader035.vdocument.in/reader035/viewer/2022062718/56812c7d550346895d9126ad/html5/thumbnails/2.jpg)
Midpoint of a segment
• The point that divides the segment into two congruent segments.
A
B
P
3
3
![Page 3: Ch. 1](https://reader035.vdocument.in/reader035/viewer/2022062718/56812c7d550346895d9126ad/html5/thumbnails/3.jpg)
Bisector of a segment
• A line, segment, ray or plane that intersects the segment at its midpoint.
A
B
P
3
3
![Page 4: Ch. 1](https://reader035.vdocument.in/reader035/viewer/2022062718/56812c7d550346895d9126ad/html5/thumbnails/4.jpg)
Bisector of an Angle
• The ray that divides the angle into two congruent adjacent angles (pg 19)
![Page 5: Ch. 1](https://reader035.vdocument.in/reader035/viewer/2022062718/56812c7d550346895d9126ad/html5/thumbnails/5.jpg)
More DefinitionsMore Definitions• IntersectIntersect – –
Two or more figures intersect if they Two or more figures intersect if they have one or more points in common.have one or more points in common.
• IntersectionIntersection – – All points or sets of points the All points or sets of points the
figures have in common.figures have in common.
![Page 6: Ch. 1](https://reader035.vdocument.in/reader035/viewer/2022062718/56812c7d550346895d9126ad/html5/thumbnails/6.jpg)
When a line and a point intersect, When a line and a point intersect, their intersection is a point.their intersection is a point.
BB
ll
![Page 7: Ch. 1](https://reader035.vdocument.in/reader035/viewer/2022062718/56812c7d550346895d9126ad/html5/thumbnails/7.jpg)
When 2 lines intersect, their When 2 lines intersect, their intersection is a point.intersection is a point.
![Page 8: Ch. 1](https://reader035.vdocument.in/reader035/viewer/2022062718/56812c7d550346895d9126ad/html5/thumbnails/8.jpg)
When 2 planes intersect, their When 2 planes intersect, their intersection is a line.intersection is a line.
![Page 9: Ch. 1](https://reader035.vdocument.in/reader035/viewer/2022062718/56812c7d550346895d9126ad/html5/thumbnails/9.jpg)
When a line and plane intersect, When a line and plane intersect, their intersection is a point.their intersection is a point.
![Page 10: Ch. 1](https://reader035.vdocument.in/reader035/viewer/2022062718/56812c7d550346895d9126ad/html5/thumbnails/10.jpg)
Segment Addition Postulate
• If B is between A and C, then AB + BC = AC. A
C
B
![Page 11: Ch. 1](https://reader035.vdocument.in/reader035/viewer/2022062718/56812c7d550346895d9126ad/html5/thumbnails/11.jpg)
Angle Addition Postulate
• If point B lies in the interior of AOC, – then m AOB + m BOC = m AOC.– What is the interior of an angle?
If AOC is a straight angleand B is any point not on AC, then m AOB + m
BOC = 180. Why does it add up to 180?
![Page 12: Ch. 1](https://reader035.vdocument.in/reader035/viewer/2022062718/56812c7d550346895d9126ad/html5/thumbnails/12.jpg)
Ch. 2
![Page 13: Ch. 1](https://reader035.vdocument.in/reader035/viewer/2022062718/56812c7d550346895d9126ad/html5/thumbnails/13.jpg)
The If-Then Statement
Conditional: is a two part statement with an actual or implied if-then.
If p, then q. p ---> q
hypothesis conclusion
If the sun is shining, then it is daytime.
![Page 14: Ch. 1](https://reader035.vdocument.in/reader035/viewer/2022062718/56812c7d550346895d9126ad/html5/thumbnails/14.jpg)
• Circle the hypothesis and underline the conclusion
If a = b, then a + c = b + c
![Page 15: Ch. 1](https://reader035.vdocument.in/reader035/viewer/2022062718/56812c7d550346895d9126ad/html5/thumbnails/15.jpg)
Other Forms
• If p, then q• p implies q• p only if q• q if p
What do you notice?
Conditional statements are not always written with the “if” clause first.
All of these conditionals mean the same thing.
![Page 16: Ch. 1](https://reader035.vdocument.in/reader035/viewer/2022062718/56812c7d550346895d9126ad/html5/thumbnails/16.jpg)
Properties of Equality
Addition Property
if x = y, then x + z = y + z.
Subtraction Property
if x = y, then x – z = y – z.
Multiplication Property
if x = y, then xz = yz.
Division Property
if x = y, and z ≠ 0, then x/z = y/z.
Numbers, variables, lengths, and angle measures
![Page 17: Ch. 1](https://reader035.vdocument.in/reader035/viewer/2022062718/56812c7d550346895d9126ad/html5/thumbnails/17.jpg)
Substitution Property
if x = y, then either x or y may be substituted for the other in any equation.
Reflexive
Property
x = x.
A number equals itself.
Symmetric Property
if x = y, then y = x.
Order of equality does not matter.
Transitive
Property
if x = y and y = z, then x = z.
Two numbers equal to the same number are equal to each other.
![Page 18: Ch. 1](https://reader035.vdocument.in/reader035/viewer/2022062718/56812c7d550346895d9126ad/html5/thumbnails/18.jpg)
Properties of Congruence
Reflexive
Property
AB AB≅
A segment (or angle) is congruent to itself
Symmetric Property
If AB CD, then CD AB≅ ≅
Order of equality does not matter.
Transitive
Property
If AB CD and CD EF, then AB ≅ ≅ EF≅
Two segments (or angles) congruent to the same segment (or angle) are congruent to each other.
Segments, angles and polygons
![Page 19: Ch. 1](https://reader035.vdocument.in/reader035/viewer/2022062718/56812c7d550346895d9126ad/html5/thumbnails/19.jpg)
Complimentary AnglesAny two angles whose measures add up to 90.If mABC + m SXT = 90, then ABC and SXT are complimentary.
S
X
T
A
B C
See It!
ABC is the complement of SXT SXT is the
complement of ABC
![Page 20: Ch. 1](https://reader035.vdocument.in/reader035/viewer/2022062718/56812c7d550346895d9126ad/html5/thumbnails/20.jpg)
Supplementary AnglesAny two angles whose measures sum to 180.If mABC + m SXT = 180, then ABC and SXT are supplementary.
S
X
T
A
BC
See It!
ABC is the supplement of SXT SXT is the
supplement of ABC
![Page 21: Ch. 1](https://reader035.vdocument.in/reader035/viewer/2022062718/56812c7d550346895d9126ad/html5/thumbnails/21.jpg)
Theorem
If two angles are supplementary to congruent angles (the same angle) then they are congruent.
If 1 suppl 2 and 2 suppl 3, then
1 3.1
2
3
![Page 22: Ch. 1](https://reader035.vdocument.in/reader035/viewer/2022062718/56812c7d550346895d9126ad/html5/thumbnails/22.jpg)
Theorem
If two angles are complimentary to congruent angles (or to the same angle) then they are congruent.
If 1 compl 2 and 2 compl 3, then
1 3.
1
2
3
![Page 23: Ch. 1](https://reader035.vdocument.in/reader035/viewer/2022062718/56812c7d550346895d9126ad/html5/thumbnails/23.jpg)
TheoremVertical angles are congruent (The definition of Vert.
angles does not tell us anything about congruency… this theorem proves that they are.)
1
2
3
4
![Page 24: Ch. 1](https://reader035.vdocument.in/reader035/viewer/2022062718/56812c7d550346895d9126ad/html5/thumbnails/24.jpg)
Perpendicular Lines ()
Two lines that intersect to form right angles.
If l m, then
angles are right.l
m
See It!
![Page 25: Ch. 1](https://reader035.vdocument.in/reader035/viewer/2022062718/56812c7d550346895d9126ad/html5/thumbnails/25.jpg)
Theorem
If two lines are perpendicular, then they form congruent, adjacent angles.
l
m1 2
If l m, then
1 2.
![Page 26: Ch. 1](https://reader035.vdocument.in/reader035/viewer/2022062718/56812c7d550346895d9126ad/html5/thumbnails/26.jpg)
Theorem
If two lines intersect to form congruent, adjacent angles, then the lines are perpendicular.
l
m1 2
If 1 2, then
l m.
![Page 27: Ch. 1](https://reader035.vdocument.in/reader035/viewer/2022062718/56812c7d550346895d9126ad/html5/thumbnails/27.jpg)
Ch. 3
![Page 28: Ch. 1](https://reader035.vdocument.in/reader035/viewer/2022062718/56812c7d550346895d9126ad/html5/thumbnails/28.jpg)
Parallel Lines ( or )
The way that we mark that two lines are parallel is by putting arrows on the lines.
m
n
m || n
![Page 29: Ch. 1](https://reader035.vdocument.in/reader035/viewer/2022062718/56812c7d550346895d9126ad/html5/thumbnails/29.jpg)
Skew Lines ( no symbol )Non-coplanar, non-intersecting lines.
p
q
What is the difference between the definition of parallel and skew lines?
![Page 30: Ch. 1](https://reader035.vdocument.in/reader035/viewer/2022062718/56812c7d550346895d9126ad/html5/thumbnails/30.jpg)
Parallel Planes
Planes that do not intersect.
Q
P
Can a plane and a line be parallel?
![Page 31: Ch. 1](https://reader035.vdocument.in/reader035/viewer/2022062718/56812c7d550346895d9126ad/html5/thumbnails/31.jpg)
Postulate
If two parallel lines are cut by a transversal, then corresponding angles are congruent. r
s
t
1 2
3 4
5 6
7 8
Can you name the corresponding angles?
![Page 32: Ch. 1](https://reader035.vdocument.in/reader035/viewer/2022062718/56812c7d550346895d9126ad/html5/thumbnails/32.jpg)
Theorem
If two parallel lines are cut by a transversal, then alternate interior angles are congruent. r
s
t
1 2
3 4
5 6
7 8
![Page 33: Ch. 1](https://reader035.vdocument.in/reader035/viewer/2022062718/56812c7d550346895d9126ad/html5/thumbnails/33.jpg)
Theorem
If two parallel lines are cut by a transversal, then same side interior angles are supplementary. r
s
t
1 2
3 4
5 6
7 8
![Page 34: Ch. 1](https://reader035.vdocument.in/reader035/viewer/2022062718/56812c7d550346895d9126ad/html5/thumbnails/34.jpg)
Ways to Prove Lines are Parallel (pg. 85)
1. Show that corresponding angles are congruent
2. Show that alternate interior angles are congruent
3. Show that same side interior angles are supplementary
4. In a plane, show that two lines are perpendicular to the same line
5. Show that two lines are parallel to a third line
![Page 35: Ch. 1](https://reader035.vdocument.in/reader035/viewer/2022062718/56812c7d550346895d9126ad/html5/thumbnails/35.jpg)
Types of Triangles(by sides)
Isosceles
2 congruen
t sides
Equilateral
All sides congruent
Scalene
No congruent
sides
![Page 36: Ch. 1](https://reader035.vdocument.in/reader035/viewer/2022062718/56812c7d550346895d9126ad/html5/thumbnails/36.jpg)
Types of Triangles(by angles)
Acute
3 acute angels
Right
1 right angle
Obtuse
1 obtuse angle
Equiangular
![Page 37: Ch. 1](https://reader035.vdocument.in/reader035/viewer/2022062718/56812c7d550346895d9126ad/html5/thumbnails/37.jpg)
Theorem
The sum of the measures of the angles of a triangle is 180
A
B
C
mA + mB + mC = 180
See It!
![Page 38: Ch. 1](https://reader035.vdocument.in/reader035/viewer/2022062718/56812c7d550346895d9126ad/html5/thumbnails/38.jpg)
Corollary
3. In a triangle, there can be at most one _right____ or obtuse angle.
![Page 39: Ch. 1](https://reader035.vdocument.in/reader035/viewer/2022062718/56812c7d550346895d9126ad/html5/thumbnails/39.jpg)
Theorem
The measure of an exterior angle of a triangle equals the sum of the measures of the two remote interior angles
12
3
4
m1 = m3 + m4
See It!
![Page 40: Ch. 1](https://reader035.vdocument.in/reader035/viewer/2022062718/56812c7d550346895d9126ad/html5/thumbnails/40.jpg)
Regular Polygon
All angles congruent
All side congruent
![Page 41: Ch. 1](https://reader035.vdocument.in/reader035/viewer/2022062718/56812c7d550346895d9126ad/html5/thumbnails/41.jpg)
Theorem (pg 102)
The sum of the measures of the interior angles of a convex polygon with n sides is
(n-2)180.
![Page 42: Ch. 1](https://reader035.vdocument.in/reader035/viewer/2022062718/56812c7d550346895d9126ad/html5/thumbnails/42.jpg)
Theorem
The sum of the measures of the exterior angles, one at each vertex, of a convex polygon is 360.
1
23
1
2
3
4
1 + 2 + 3 = 360 1 + 2 + 3 + 4 = 360
![Page 43: Ch. 1](https://reader035.vdocument.in/reader035/viewer/2022062718/56812c7d550346895d9126ad/html5/thumbnails/43.jpg)
REGULAR POLYGONS
• All the interior angles are congruent • All of the exterior angles are congruent
(n-2)180
n= the measure of each interior angle
360
n= the measure of each exterior angle
![Page 44: Ch. 1](https://reader035.vdocument.in/reader035/viewer/2022062718/56812c7d550346895d9126ad/html5/thumbnails/44.jpg)
Problems for Ch. 1 - 3
• 1 – 7 - 8 – 10
• 15 – 16 – 17
• 22 – 24 – 25 – 26 – 28
• 31 – 34 – 36
![Page 45: Ch. 1](https://reader035.vdocument.in/reader035/viewer/2022062718/56812c7d550346895d9126ad/html5/thumbnails/45.jpg)
Ch. 4
![Page 46: Ch. 1](https://reader035.vdocument.in/reader035/viewer/2022062718/56812c7d550346895d9126ad/html5/thumbnails/46.jpg)
Definition of Congruency
Two polygons are congruent if corresponding vertices can be matched up so that:
1. All corresponding sides are congruent2. All corresponding angles are congruent.
![Page 47: Ch. 1](https://reader035.vdocument.in/reader035/viewer/2022062718/56812c7d550346895d9126ad/html5/thumbnails/47.jpg)
The order in which you name the triangles mattersmatters !
ABC DEF
A
BC
F
E
D
![Page 48: Ch. 1](https://reader035.vdocument.in/reader035/viewer/2022062718/56812c7d550346895d9126ad/html5/thumbnails/48.jpg)
ABC XYZ• Based off this information with or without a
diagram, we can conclude…• Letters X and A, which appear first, name
corresponding vertices and that– X A.
• The letters Y and B come next, so – Y B and–XY AB
![Page 49: Ch. 1](https://reader035.vdocument.in/reader035/viewer/2022062718/56812c7d550346895d9126ad/html5/thumbnails/49.jpg)
Five Ways to Prove ’s
All Triangles:ASA SSS SAS AASRight Triangles Only:HL
![Page 50: Ch. 1](https://reader035.vdocument.in/reader035/viewer/2022062718/56812c7d550346895d9126ad/html5/thumbnails/50.jpg)
Isosceles Triangle
By definition, it is a triangle with two congruent sides called legs.
X
Y Z
Base
Base Angles
Legs Vertex Angle
![Page 51: Ch. 1](https://reader035.vdocument.in/reader035/viewer/2022062718/56812c7d550346895d9126ad/html5/thumbnails/51.jpg)
Theorem
If two sides of a triangle are congruent, then the angles opposite those sides are congruent.
A
B
C
![Page 52: Ch. 1](https://reader035.vdocument.in/reader035/viewer/2022062718/56812c7d550346895d9126ad/html5/thumbnails/52.jpg)
Conclusions
P
Q
SR
1. R S
2. PQ bisects RPS
3. PQ bisects RS
4. PQ RS at Q
5. PR PS
![Page 53: Ch. 1](https://reader035.vdocument.in/reader035/viewer/2022062718/56812c7d550346895d9126ad/html5/thumbnails/53.jpg)
Given: MK OK;KJ bisects MKO;
Prove: JK bisects MJO
Statements Reasons
1. MK OK; KJ bisects MKO
1. Given
2. 3 4 2. Def of bisector
3. JK JK 3. Reflexive Property
4. MKJ OKJ 4. SAS Postulate
5. 1 2 5. CPCTC
6. JK bisects MJO 6. Def of bisector
K
O
J
M
1
2
34
K
O
J
M
1
2
34
![Page 54: Ch. 1](https://reader035.vdocument.in/reader035/viewer/2022062718/56812c7d550346895d9126ad/html5/thumbnails/54.jpg)
Ch. 5
![Page 55: Ch. 1](https://reader035.vdocument.in/reader035/viewer/2022062718/56812c7d550346895d9126ad/html5/thumbnails/55.jpg)
Parallelograms: What we now know…
• From the definition..1. Both pairs of opposite sides are parallel
• From theorems…1. Both pairs of opposite sides are congruent 2. Both pairs of opposite angles are congruent 3. The diagonals of a parallelogram bisect each
other
![Page 56: Ch. 1](https://reader035.vdocument.in/reader035/viewer/2022062718/56812c7d550346895d9126ad/html5/thumbnails/56.jpg)
Five ways to prove a Quadrilateral is a Parallelogram
1. Show that both pairs of opposite sides parallel2. Show that both pairs of opposite sides congruent3. Show that one pair of opposite sides are both
congruent and parallel4. Show that both pairs of opposite angles congruent5. Show that diagonals that bisect each other
![Page 57: Ch. 1](https://reader035.vdocument.in/reader035/viewer/2022062718/56812c7d550346895d9126ad/html5/thumbnails/57.jpg)
Rectangle
By definition, it is a quadrilateral with four right angles.
R
S T
V
![Page 58: Ch. 1](https://reader035.vdocument.in/reader035/viewer/2022062718/56812c7d550346895d9126ad/html5/thumbnails/58.jpg)
TheoremThe diagonals of a rectangle are congruent.WY XZ
W
X Y
Z
P
![Page 59: Ch. 1](https://reader035.vdocument.in/reader035/viewer/2022062718/56812c7d550346895d9126ad/html5/thumbnails/59.jpg)
Rhombus
By definition, it is a quadrilateral with four congruent sides.
A
B C
D
![Page 60: Ch. 1](https://reader035.vdocument.in/reader035/viewer/2022062718/56812c7d550346895d9126ad/html5/thumbnails/60.jpg)
Theorem
The diagonals of a rhombus are perpendicular.
J
K
L
M
X
What does the definition of perpendicular lines tell us?
![Page 61: Ch. 1](https://reader035.vdocument.in/reader035/viewer/2022062718/56812c7d550346895d9126ad/html5/thumbnails/61.jpg)
Theorem
Each diagonal of a rhombus bisects the opposite angles.
J
K
L
M
X
![Page 62: Ch. 1](https://reader035.vdocument.in/reader035/viewer/2022062718/56812c7d550346895d9126ad/html5/thumbnails/62.jpg)
Square
By definition, it is a quadrilateral with four right angles and four congruent sides.
A
B C
D
The square is the most specific type of quadrilateral.
What do you notice about the definition compared to the previous two?
![Page 63: Ch. 1](https://reader035.vdocument.in/reader035/viewer/2022062718/56812c7d550346895d9126ad/html5/thumbnails/63.jpg)
Trapezoid
A quadrilateral with exactly one pair of parallel sides.
A
B C
D
Trap. ABCD
How does this definition differ from that of a parallelogram?
![Page 64: Ch. 1](https://reader035.vdocument.in/reader035/viewer/2022062718/56812c7d550346895d9126ad/html5/thumbnails/64.jpg)
The Median of a Trapezoid
A segment that joins the midpoints of the legs.
A
B C
D
X Y
Note: this applies to any trapezoid
![Page 65: Ch. 1](https://reader035.vdocument.in/reader035/viewer/2022062718/56812c7d550346895d9126ad/html5/thumbnails/65.jpg)
Theorem
The median of a trapezoid is parallel to the bases and its length is the average of the bases.
B C
D
X Y
AA
B C
D
X Y
How do we find an average of the bases ?
Note: this applies to any trapezoid
![Page 66: Ch. 1](https://reader035.vdocument.in/reader035/viewer/2022062718/56812c7d550346895d9126ad/html5/thumbnails/66.jpg)
Problems For Ch. 4, 5
• 5 • 11- 19 • 20 – 29 • 30 – 33 – 35 – 37 – 38 – 39 - 40
![Page 67: Ch. 1](https://reader035.vdocument.in/reader035/viewer/2022062718/56812c7d550346895d9126ad/html5/thumbnails/67.jpg)
Ch. 6
![Page 68: Ch. 1](https://reader035.vdocument.in/reader035/viewer/2022062718/56812c7d550346895d9126ad/html5/thumbnails/68.jpg)
Indirect Proof
• Are used when you can’t use a direct proof.• BUT, people use indirect proofs everyday to
figure out things in their everyday lives.• 3 steps EVERYTIME (p. 214 purple box)
![Page 69: Ch. 1](https://reader035.vdocument.in/reader035/viewer/2022062718/56812c7d550346895d9126ad/html5/thumbnails/69.jpg)
Step 1
• “Assume temporarily that….” (the conclusion is false). I know I always tell you not to ASSume, but here you can. You want to believe that the opposite of the conclusion is true (the prove statement).
![Page 70: Ch. 1](https://reader035.vdocument.in/reader035/viewer/2022062718/56812c7d550346895d9126ad/html5/thumbnails/70.jpg)
Step 2
• Using the given information of anything else that you already know for sure…..(like postulates, theorems, and definitions), try and show that the temporary assumption that you made can’t be true. You are looking for a contradiction* to the GIVEN information.
• “This contradicts the given information.”• Use pictures and write in a paragraph.
![Page 71: Ch. 1](https://reader035.vdocument.in/reader035/viewer/2022062718/56812c7d550346895d9126ad/html5/thumbnails/71.jpg)
Step 3
• Point out that the temporary assumption must be false, and that the conclusion must then be true.
• “My temporary assumption is false and…” ( the original conclusion must be true). Restate the original conclusion.
![Page 72: Ch. 1](https://reader035.vdocument.in/reader035/viewer/2022062718/56812c7d550346895d9126ad/html5/thumbnails/72.jpg)
Given: XYZW; m X = 80ºProve: XYZW is not a rectangle
Assume temporarily that XYZW is a rectangle. Then XYZW have four right angles because this is the definition of a rectangle. This contradicts the given information that m X = 80º.* My temporary assumption is false and XYZW is not a rectangle.