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Students consider the definition of midpoint and the congruence of two line segments.
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Congruent SegmentsCongruent Segments
You will learn to identify congruent segments and find the midpoints of segments.
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.
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.
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
________________________
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
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.
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.
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.
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
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
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 ________.
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
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.
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.
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
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
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
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
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
Congruent SegmentsCongruent Segments
There is a unique point on every segment called the _______.
Congruent SegmentsCongruent Segments
There is a unique point on every segment called the _______.midpoint
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?
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?
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
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
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
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_____ _____.
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
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 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
Congruent SegmentsCongruent Segments
In the figure, B is the midpoint of .AC Find the value of x.
5x - 6 2x
CBA
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
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
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
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
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
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!
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!
Congruent SegmentsCongruent Segments
To bisect something means to separate it into ___ congruent parts.
Congruent SegmentsCongruent Segments
To bisect something means to separate it into ___ congruent parts.two
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.
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
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
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
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
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
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
Congruent SegmentsCongruent Segments
