Midpoints and Congruence (Geometry 2_3)

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