midpoints and congruence (geometry 2_3)
DESCRIPTION
Students consider the definition of midpoint and the congruence of two line segments.TRANSCRIPT
![Page 1: Midpoints and Congruence (Geometry 2_3)](https://reader035.vdocuments.mx/reader035/viewer/2022081516/54be3bd94a795907688b45d3/html5/thumbnails/1.jpg)
Congruent SegmentsCongruent Segments
You will learn to identify congruent segments and find the midpoints of segments.
![Page 2: Midpoints and Congruence (Geometry 2_3)](https://reader035.vdocuments.mx/reader035/viewer/2022081516/54be3bd94a795907688b45d3/html5/thumbnails/2.jpg)
Congruent SegmentsCongruent Segments
In geometry, two segments with the same length are called ________ _________
You will learn to identify congruent segments and find the midpoints of segments.
![Page 3: Midpoints and Congruence (Geometry 2_3)](https://reader035.vdocuments.mx/reader035/viewer/2022081516/54be3bd94a795907688b45d3/html5/thumbnails/3.jpg)
Congruent SegmentsCongruent Segments
In geometry, two segments with the same length are called ________ _________congruent segments
You will learn to identify congruent segments and find the midpoints of segments.
![Page 4: Midpoints and Congruence (Geometry 2_3)](https://reader035.vdocuments.mx/reader035/viewer/2022081516/54be3bd94a795907688b45d3/html5/thumbnails/4.jpg)
Congruent SegmentsCongruent Segments
In geometry, two segments with the same length are called ________ _________congruent segments
You will learn to identify congruent segments and find the midpoints of segments.
Definition of
Congruent
Segments
Two segments are congruent if and only if
________________________
![Page 5: Midpoints and Congruence (Geometry 2_3)](https://reader035.vdocuments.mx/reader035/viewer/2022081516/54be3bd94a795907688b45d3/html5/thumbnails/5.jpg)
Congruent SegmentsCongruent Segments
In geometry, two segments with the same length are called ________ _________congruent segments
You will learn to identify congruent segments and find the midpoints of segments.
Definition of
Congruent
Segments
Two segments are congruent if and only if
________________________they have the same length
![Page 6: Midpoints and Congruence (Geometry 2_3)](https://reader035.vdocuments.mx/reader035/viewer/2022081516/54be3bd94a795907688b45d3/html5/thumbnails/6.jpg)
Congruent SegmentsCongruent Segments
BA C
R
S
P
Q
In the figures at the right, AB is
congruent to BC, and PQ is
congruent to RS.
![Page 7: Midpoints and Congruence (Geometry 2_3)](https://reader035.vdocuments.mx/reader035/viewer/2022081516/54be3bd94a795907688b45d3/html5/thumbnails/7.jpg)
Congruent SegmentsCongruent Segments
BA C
R
S
P
Q
In the figures at the right, AB is
congruent to BC, and PQ is
congruent to RS.
The symbol is used to
represent congruence.
![Page 8: Midpoints and Congruence (Geometry 2_3)](https://reader035.vdocuments.mx/reader035/viewer/2022081516/54be3bd94a795907688b45d3/html5/thumbnails/8.jpg)
Congruent SegmentsCongruent Segments
BA C
R
S
P
Q
In the figures at the right, AB is
congruent to BC, and PQ is
congruent to RS.
The symbol is used to
represent congruence.
AB BC, and PQ RS.
![Page 9: Midpoints and Congruence (Geometry 2_3)](https://reader035.vdocuments.mx/reader035/viewer/2022081516/54be3bd94a795907688b45d3/html5/thumbnails/9.jpg)
Congruent SegmentsCongruent Segments
x
11-5 -3 -1 1 3 5 7 9-6 -4 0 4 82 106-2
R S T Y
Use the number line to determine if the statement is True or False.Explain you reasoning.
RS TY
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Congruent SegmentsCongruent Segments
x
11-5 -3 -1 1 3 5 7 9-6 -4 0 4 82 106-2
R S T Y
Use the number line to determine if the statement is True or False.Explain you reasoning.
RS TY
So, RS is not congruent to TY,
and the statement is false.
Because RS = 4 and TY = 5, TYRS
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Congruent SegmentsCongruent Segments
Since congruence is related to the equality of segment measures, there areproperties of congruence that are similar to the corresponding propertiesof equality.
These statements are called ________.
![Page 12: Midpoints and Congruence (Geometry 2_3)](https://reader035.vdocuments.mx/reader035/viewer/2022081516/54be3bd94a795907688b45d3/html5/thumbnails/12.jpg)
Congruent SegmentsCongruent Segments
Since congruence is related to the equality of segment measures, there areproperties of congruence that are similar to the corresponding propertiesof equality.
These statements are called ________.theorems
![Page 13: Midpoints and Congruence (Geometry 2_3)](https://reader035.vdocuments.mx/reader035/viewer/2022081516/54be3bd94a795907688b45d3/html5/thumbnails/13.jpg)
Congruent SegmentsCongruent Segments
Since congruence is related to the equality of segment measures, there areproperties of congruence that are similar to the corresponding propertiesof equality.
These statements are called ________.theorems
Theorems are statements that can be justified by using logical reasoning.
![Page 14: Midpoints and Congruence (Geometry 2_3)](https://reader035.vdocuments.mx/reader035/viewer/2022081516/54be3bd94a795907688b45d3/html5/thumbnails/14.jpg)
Congruent SegmentsCongruent Segments
Since congruence is related to the equality of segment measures, there areproperties of congruence that are similar to the corresponding propertiesof equality.
These statements are called ________.theorems
Theorems are statements that can be justified by using logical reasoning.
2 – 1 Congruence of segments is reflexive.
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Congruent SegmentsCongruent Segments
Since congruence is related to the equality of segment measures, there areproperties of congruence that are similar to the corresponding propertiesof equality.
These statements are called ________.theorems
Theorems are statements that can be justified by using logical reasoning.
2 – 1 Congruence of segments is reflexive. AB AB
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Congruent SegmentsCongruent Segments
Since congruence is related to the equality of segment measures, there areproperties of congruence that are similar to the corresponding propertiesof equality.
These statements are called ________.theorems
Theorems are statements that can be justified by using logical reasoning.
2 – 1 Congruence of segments is reflexive. AB AB
2 – 2 Congruence of segments is symmetric. If , then AB CD
![Page 17: Midpoints and Congruence (Geometry 2_3)](https://reader035.vdocuments.mx/reader035/viewer/2022081516/54be3bd94a795907688b45d3/html5/thumbnails/17.jpg)
Congruent SegmentsCongruent Segments
Since congruence is related to the equality of segment measures, there areproperties of congruence that are similar to the corresponding propertiesof equality.
These statements are called ________.theorems
Theorems are statements that can be justified by using logical reasoning.
2 – 1 Congruence of segments is reflexive. AB AB
2 – 2 Congruence of segments is symmetric. If , then AB CD CD AB
![Page 18: Midpoints and Congruence (Geometry 2_3)](https://reader035.vdocuments.mx/reader035/viewer/2022081516/54be3bd94a795907688b45d3/html5/thumbnails/18.jpg)
Congruent SegmentsCongruent Segments
Since congruence is related to the equality of segment measures, there areproperties of congruence that are similar to the corresponding propertiesof equality.
These statements are called ________.theorems
Theorems are statements that can be justified by using logical reasoning.
2 – 1 Congruence of segments is reflexive. AB AB
2 – 2 Congruence of segments is symmetric. If , then AB CD CD AB
2 – 3 Congruence of segments is transitive.
If , and
then
AB CD CD EF
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Congruent SegmentsCongruent Segments
Since congruence is related to the equality of segment measures, there areproperties of congruence that are similar to the corresponding propertiesof equality.
These statements are called ________.theorems
Theorems are statements that can be justified by using logical reasoning.
2 – 1 Congruence of segments is reflexive. AB AB
2 – 2 Congruence of segments is symmetric. If , then AB CD CD AB
2 – 3 Congruence of segments is transitive.
If , and
then
AB CD CD EF AB EF
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Congruent SegmentsCongruent Segments
There is a unique point on every segment called the _______.
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Congruent SegmentsCongruent Segments
There is a unique point on every segment called the _______.midpoint
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Congruent SegmentsCongruent Segments
There is a unique point on every segment called the _______.midpoint
On the number line below, M is the midpoint of . ST
x
11-5 -3 -1 1 3 5 7 9-6 -4 0 4 82 106-2
S M T
What do you notice about SM and MT?
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Congruent SegmentsCongruent Segments
There is a unique point on every segment called the _______.midpoint
On the number line below, M is the midpoint of . ST
x
11-5 -3 -1 1 3 5 7 9-6 -4 0 4 82 106-2
S M T
What do you notice about SM and MT?
![Page 24: Midpoints and Congruence (Geometry 2_3)](https://reader035.vdocuments.mx/reader035/viewer/2022081516/54be3bd94a795907688b45d3/html5/thumbnails/24.jpg)
Congruent SegmentsCongruent Segments
There is a unique point on every segment called the _______.midpoint
On the number line below, M is the midpoint of . ST
x
11-5 -3 -1 1 3 5 7 9-6 -4 0 4 82 106-2
S M T
What do you notice about SM and MT?
SM = MT
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Congruent SegmentsCongruent Segments
Definition ofMidpoint
A point M is the midpoint of a segment if and only if M is between S and T and SM = MT
ST
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Congruent SegmentsCongruent Segments
Definition ofMidpoint
A point M is the midpoint of a segment if and only if M is between S and T and SM = MT
ST
MS T
SM = MT
![Page 27: Midpoints and Congruence (Geometry 2_3)](https://reader035.vdocuments.mx/reader035/viewer/2022081516/54be3bd94a795907688b45d3/html5/thumbnails/27.jpg)
Congruent SegmentsCongruent Segments
Definition ofMidpoint
A point M is the midpoint of a segment if and only if M is between S and T and SM = MT
ST
MS T
SM = MT
The midpoint of a segment separates the segment into two segments of_____ _____.
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Congruent SegmentsCongruent Segments
Definition ofMidpoint
A point M is the midpoint of a segment if and only if M is between S and T and SM = MT
ST
MS T
SM = MT
The midpoint of a segment separates the segment into two segments of_____ _____.equal length
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Congruent SegmentsCongruent Segments
Definition ofMidpoint
A point M is the midpoint of a segment if and only if M is between S and T and SM = MT
ST
MS T
SM = MT
The midpoint of a segment separates the segment into two segments of_____ _____.equal length
So, by the definition of congruent segments, the two segments are _________.
![Page 30: Midpoints and Congruence (Geometry 2_3)](https://reader035.vdocuments.mx/reader035/viewer/2022081516/54be3bd94a795907688b45d3/html5/thumbnails/30.jpg)
Congruent SegmentsCongruent Segments
Definition ofMidpoint
A point M is the midpoint of a segment if and only if M is between S and T and SM = MT
ST
MS T
SM = MT
The midpoint of a segment separates the segment into two segments of_____ _____.equal length
So, by the definition of congruent segments, the two segments are _________.congruent
![Page 31: Midpoints and Congruence (Geometry 2_3)](https://reader035.vdocuments.mx/reader035/viewer/2022081516/54be3bd94a795907688b45d3/html5/thumbnails/31.jpg)
Congruent SegmentsCongruent Segments
In the figure, B is the midpoint of .AC Find the value of x.
5x - 6 2x
CBA
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Congruent SegmentsCongruent Segments
In the figure, B is the midpoint of .AC Find the value of x.
5x - 6 2x
CBA
Since B is the midpoint: AB = BC
![Page 33: Midpoints and Congruence (Geometry 2_3)](https://reader035.vdocuments.mx/reader035/viewer/2022081516/54be3bd94a795907688b45d3/html5/thumbnails/33.jpg)
Congruent SegmentsCongruent Segments
In the figure, B is the midpoint of .AC Find the value of x.
5x - 6 2x
CBA
Since B is the midpoint: AB = BC
Write the equation involving x: 5x – 6 = 2x
![Page 34: Midpoints and Congruence (Geometry 2_3)](https://reader035.vdocuments.mx/reader035/viewer/2022081516/54be3bd94a795907688b45d3/html5/thumbnails/34.jpg)
Congruent SegmentsCongruent Segments
In the figure, B is the midpoint of .AC Find the value of x.
5x - 6 2x
CBA
Since B is the midpoint: AB = BC
Write the equation involving x: 5x – 6 = 2x
Solve for x: 5x – 2x – 6 = 2x – 2x
![Page 35: Midpoints and Congruence (Geometry 2_3)](https://reader035.vdocuments.mx/reader035/viewer/2022081516/54be3bd94a795907688b45d3/html5/thumbnails/35.jpg)
Congruent SegmentsCongruent Segments
In the figure, B is the midpoint of .AC Find the value of x.
5x - 6 2x
CBA
Since B is the midpoint: AB = BC
Write the equation involving x: 5x – 6 = 2x
Solve for x: 5x – 2x – 6 = 2x – 2x
3x – 6 + 6 = 0 + 6
3x = 6
![Page 36: Midpoints and Congruence (Geometry 2_3)](https://reader035.vdocuments.mx/reader035/viewer/2022081516/54be3bd94a795907688b45d3/html5/thumbnails/36.jpg)
Congruent SegmentsCongruent Segments
In the figure, B is the midpoint of .AC Find the value of x.
5x - 6 2x
CBA
Since B is the midpoint: AB = BC
Write the equation involving x: 5x – 6 = 2x
Solve for x: 5x – 2x – 6 = 2x – 2x
3x – 6 + 6 = 0 + 6
3x = 6
x = 2
![Page 37: Midpoints and Congruence (Geometry 2_3)](https://reader035.vdocuments.mx/reader035/viewer/2022081516/54be3bd94a795907688b45d3/html5/thumbnails/37.jpg)
Congruent SegmentsCongruent Segments
In the figure, B is the midpoint of .AC Find the value of x.
5x - 6 2x
CBA
Since B is the midpoint: AB = BC
Write the equation involving x: 5x – 6 = 2x
Solve for x: 5x – 2x – 6 = 2x – 2x
3x – 6 + 6 = 0 + 6
3x = 6
x = 2
AB = 5x – 6 = 5(2) – 6 = 10 – 6 = 4
Check!
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Congruent SegmentsCongruent Segments
In the figure, B is the midpoint of .AC Find the value of x.
5x - 6 2x
CBA
Since B is the midpoint: AB = BC
Write the equation involving x: 5x – 6 = 2x
Solve for x: 5x – 2x – 6 = 2x – 2x
3x – 6 + 6 = 0 + 6
3x = 6
x = 2
AB = 5x – 6 = 5(2) – 6 = 10 – 6 = 4
BC = 2x = 2(2) = 4
Check!
![Page 39: Midpoints and Congruence (Geometry 2_3)](https://reader035.vdocuments.mx/reader035/viewer/2022081516/54be3bd94a795907688b45d3/html5/thumbnails/39.jpg)
Congruent SegmentsCongruent Segments
To bisect something means to separate it into ___ congruent parts.
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Congruent SegmentsCongruent Segments
To bisect something means to separate it into ___ congruent parts.two
![Page 41: Midpoints and Congruence (Geometry 2_3)](https://reader035.vdocuments.mx/reader035/viewer/2022081516/54be3bd94a795907688b45d3/html5/thumbnails/41.jpg)
Congruent SegmentsCongruent Segments
To bisect something means to separate it into ___ congruent parts.two
The ________ of a segment bisects the segment because it separates thesegment into two congruent segments.
![Page 42: Midpoints and Congruence (Geometry 2_3)](https://reader035.vdocuments.mx/reader035/viewer/2022081516/54be3bd94a795907688b45d3/html5/thumbnails/42.jpg)
Congruent SegmentsCongruent Segments
To bisect something means to separate it into ___ congruent parts.two
The ________ of a segment bisects the segment because it separates thesegment into two congruent segments.
midpoint
![Page 43: Midpoints and Congruence (Geometry 2_3)](https://reader035.vdocuments.mx/reader035/viewer/2022081516/54be3bd94a795907688b45d3/html5/thumbnails/43.jpg)
Congruent SegmentsCongruent Segments
To bisect something means to separate it into ___ congruent parts.two
The ________ of a segment bisects the segment because it separates thesegment into two congruent segments.
A point, line, ray, or plane can also bisect a segment.
midpoint
B
A
E
G
CD
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Congruent SegmentsCongruent Segments
To bisect something means to separate it into ___ congruent parts.two
The ________ of a segment bisects the segment because it separates thesegment into two congruent segments.
A point, line, ray, or plane can also bisect a segment.
Point C bisects AB
midpoint
B
A
E
G
CD
![Page 45: Midpoints and Congruence (Geometry 2_3)](https://reader035.vdocuments.mx/reader035/viewer/2022081516/54be3bd94a795907688b45d3/html5/thumbnails/45.jpg)
Congruent SegmentsCongruent Segments
To bisect something means to separate it into ___ congruent parts.two
The ________ of a segment bisects the segment because it separates thesegment into two congruent segments.
A point, line, ray, or plane can also bisect a segment.
Point C bisects AB
bisects ABDC��������������
midpoint
B
A
E
G
CD
![Page 46: Midpoints and Congruence (Geometry 2_3)](https://reader035.vdocuments.mx/reader035/viewer/2022081516/54be3bd94a795907688b45d3/html5/thumbnails/46.jpg)
Congruent SegmentsCongruent Segments
To bisect something means to separate it into ___ congruent parts.two
The ________ of a segment bisects the segment because it separates thesegment into two congruent segments.
A point, line, ray, or plane can also bisect a segment.
Point C bisects AB
bisects ABDC��������������
bisects ABEC�������������� �
midpoint
B
A
E
G
CD
![Page 47: Midpoints and Congruence (Geometry 2_3)](https://reader035.vdocuments.mx/reader035/viewer/2022081516/54be3bd94a795907688b45d3/html5/thumbnails/47.jpg)
Congruent SegmentsCongruent Segments
To bisect something means to separate it into ___ congruent parts.two
The ________ of a segment bisects the segment because it separates thesegment into two congruent segments.
A point, line, ray, or plane can also bisect a segment.
Point C bisects AB
bisects ABDC��������������
bisects ABEC�������������� �
Plane GCD bisects AB
midpoint
B
A
E
G
CD
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Congruent SegmentsCongruent Segments