H. Geo 1.2 & 1.3 Use Segments & Congruence & Midpoint Formula +JMJ Learning Target: Students demonstrate understanding by identifying and giving examples of undefined theorems, and inductive and deductive reasoning.

Vocabulary

Date:

Honors Geometry Lesson 2

Pair-Share: Complete the following questions using the diagram below. You need a ruler!

1) Find AB using your ruler. (Let’s  say  it  is  3  units)

2) Find the length of BC using your ruler. (Let’s  say  it  is  6.4  units)

3) Find the length of AC without using a ruler. Explain how you found the length of AC In this course, you will have to write about distances between points and lengths of segments in many, if not most, homework problems. Instead of writing 𝑑𝑖𝑠𝑡(𝐴, 𝐵) all of the time, which is a rather long and clunky notation, we will instead use the much simpler notation 𝐴𝐵 for both distance and length of segments. Even though the notation will always make the meaning of each statement clear, it is worthwhile to consider the context of the statement to ensure correct usage. Here are some examples:

� 𝐴𝐵⃖ ⃗ intersects… 𝐴𝐵⃖ ⃗ refers to a line. � 𝐴𝐵 + 𝐵𝐶 = 𝐴𝐶 Only numbers can be added and 𝐴𝐵 is a length or distance. � Find 𝐴𝐵 so that 𝐴𝐵 ∥ 𝐶𝐷. Only figures can be parallel and 𝐴𝐵 is a segment. � 𝐴𝐵 = 6. 𝐴𝐵  refers to the length of segment 𝐴𝐵 or the distance

                                                                                                       from 𝐴 to 𝐵.

Here are the standard notations for segments, lines, rays, distances, and lengths:

� A ray with vertex 𝐴 that contains the point 𝐵: 𝐴𝐵 ⃗ or  “ray  AB”

� A line that contains points 𝐴 and 𝐵: 𝐴𝐵⃖ ⃗ or  “line  AB”

� A segment with endpoints 𝐴 and 𝐵: 𝐴𝐵 or  “segment  AB”

� The length of segment 𝐴𝐵:          𝐴𝐵

� The distance from 𝐴 to 𝐵: 𝑑𝑖𝑠𝑡(𝐴, 𝐵) or 𝐴𝐵 Coordinate System on a Line: Given a line 𝑙, a coordinate system on 𝑙 is a correspondence between the points on the line and the real numbers such that (i) to every point on 𝒍 there corresponds exactly one real number, (ii) to every real number there corresponds exactly one point of 𝑙, and (iii) the distance between two distinct points on 𝑙 is equal to the absolute value of the difference of the corresponding numbers.

A

C

B

Date:

Honors Geometry Lesson 2

Pair-Share: Complete the following questions using the diagram below. You need a ruler!

1) Find AB using your ruler. (Let’s  say  it  is  3  units)

2) Find the length of BC using your ruler. (Let’s  say  it  is  6.4  units)

3) Find the length of AC without using a ruler. Explain how you found the length of AC In this course, you will have to write about distances between points and lengths of segments in many, if not most, homework problems. Instead of writing 𝑑𝑖𝑠𝑡(𝐴, 𝐵) all of the time, which is a rather long and clunky notation, we will instead use the much simpler notation 𝐴𝐵 for both distance and length of segments. Even though the notation will always make the meaning of each statement clear, it is worthwhile to consider the context of the statement to ensure correct usage. Here are some examples:

� 𝐴𝐵⃖ ⃗ intersects… 𝐴𝐵⃖ ⃗ refers to a line. � 𝐴𝐵 + 𝐵𝐶 = 𝐴𝐶 Only numbers can be added and 𝐴𝐵 is a length or distance. � Find 𝐴𝐵 so that 𝐴𝐵 ∥ 𝐶𝐷. Only figures can be parallel and 𝐴𝐵 is a segment. � 𝐴𝐵 = 6. 𝐴𝐵  refers to the length of segment 𝐴𝐵 or the distance

                                                                                                       from 𝐴 to 𝐵.

Here are the standard notations for segments, lines, rays, distances, and lengths:

� A ray with vertex 𝐴 that contains the point 𝐵: 𝐴𝐵 ⃗ or  “ray  AB”

� A line that contains points 𝐴 and 𝐵: 𝐴𝐵⃖ ⃗ or  “line  AB”

� A segment with endpoints 𝐴 and 𝐵: 𝐴𝐵 or  “segment  AB”

� The length of segment 𝐴𝐵:          𝐴𝐵

� The distance from 𝐴 to 𝐵: 𝑑𝑖𝑠𝑡(𝐴, 𝐵) or 𝐴𝐵 Coordinate System on a Line: Given a line 𝑙, a coordinate system on 𝑙 is a correspondence between the points on the line and the real numbers such that (i) to every point on 𝒍 there corresponds exactly one real number, (ii) to every real number there corresponds exactly one point of 𝑙, and (iii) the distance between two distinct points on 𝑙 is equal to the absolute value of the difference of the corresponding numbers.

A

C

B

Date:

Honors Geometry Lesson 2

Pair-Share: Complete the following questions using the diagram below. You need a ruler!

1) Find AB using your ruler. (Let’s  say  it  is  3  units)

2) Find the length of BC using your ruler. (Let’s  say  it  is  6.4  units)

3) Find the length of AC without using a ruler. Explain how you found the length of AC In this course, you will have to write about distances between points and lengths of segments in many, if not most, homework problems. Instead of writing 𝑑𝑖𝑠𝑡(𝐴, 𝐵) all of the time, which is a rather long and clunky notation, we will instead use the much simpler notation 𝐴𝐵 for both distance and length of segments. Even though the notation will always make the meaning of each statement clear, it is worthwhile to consider the context of the statement to ensure correct usage. Here are some examples:

� 𝐴𝐵⃖ ⃗ intersects… 𝐴𝐵⃖ ⃗ refers to a line. � 𝐴𝐵 + 𝐵𝐶 = 𝐴𝐶 Only numbers can be added and 𝐴𝐵 is a length or distance. � Find 𝐴𝐵 so that 𝐴𝐵 ∥ 𝐶𝐷. Only figures can be parallel and 𝐴𝐵 is a segment. � 𝐴𝐵 = 6. 𝐴𝐵  refers to the length of segment 𝐴𝐵 or the distance

                                                                                                       from 𝐴 to 𝐵.

Here are the standard notations for segments, lines, rays, distances, and lengths:

� A ray with vertex 𝐴 that contains the point 𝐵: 𝐴𝐵 ⃗ or  “ray  AB”

� A line that contains points 𝐴 and 𝐵: 𝐴𝐵⃖ ⃗ or  “line  AB”

� A segment with endpoints 𝐴 and 𝐵: 𝐴𝐵 or  “segment  AB”

� The length of segment 𝐴𝐵:          𝐴𝐵

� The distance from 𝐴 to 𝐵: 𝑑𝑖𝑠𝑡(𝐴, 𝐵) or 𝐴𝐵 Coordinate System on a Line: Given a line 𝑙, a coordinate system on 𝑙 is a correspondence between the points on the line and the real numbers such that (i) to every point on 𝒍 there corresponds exactly one real number, (ii) to every real number there corresponds exactly one point of 𝑙, and (iii) the distance between two distinct points on 𝑙 is equal to the absolute value of the difference of the corresponding numbers.

A

C

B

Date:

Honors Geometry Lesson 2

Pair-Share: Complete the following questions using the diagram below. You need a ruler!

1) Find AB using your ruler. (Let’s  say  it  is  3  units)

2) Find the length of BC using your ruler. (Let’s  say  it  is  6.4  units)

3) Find the length of AC without using a ruler. Explain how you found the length of AC In this course, you will have to write about distances between points and lengths of segments in many, if not most, homework problems. Instead of writing 𝑑𝑖𝑠𝑡(𝐴, 𝐵) all of the time, which is a rather long and clunky notation, we will instead use the much simpler notation 𝐴𝐵 for both distance and length of segments. Even though the notation will always make the meaning of each statement clear, it is worthwhile to consider the context of the statement to ensure correct usage. Here are some examples:

� 𝐴𝐵⃖ ⃗ intersects… 𝐴𝐵⃖ ⃗ refers to a line. � 𝐴𝐵 + 𝐵𝐶 = 𝐴𝐶 Only numbers can be added and 𝐴𝐵 is a length or distance. � Find 𝐴𝐵 so that 𝐴𝐵 ∥ 𝐶𝐷. Only figures can be parallel and 𝐴𝐵 is a segment. � 𝐴𝐵 = 6. 𝐴𝐵  refers to the length of segment 𝐴𝐵 or the distance

                                                                                                       from 𝐴 to 𝐵.

Here are the standard notations for segments, lines, rays, distances, and lengths:

� A ray with vertex 𝐴 that contains the point 𝐵: 𝐴𝐵 ⃗ or  “ray  AB”

� A line that contains points 𝐴 and 𝐵: 𝐴𝐵⃖ ⃗ or  “line  AB”

� A segment with endpoints 𝐴 and 𝐵: 𝐴𝐵 or  “segment  AB”

� The length of segment 𝐴𝐵:          𝐴𝐵

� The distance from 𝐴 to 𝐵: 𝑑𝑖𝑠𝑡(𝐴, 𝐵) or 𝐴𝐵 Coordinate System on a Line: Given a line 𝑙, a coordinate system on 𝑙 is a correspondence between the points on the line and the real numbers such that (i) to every point on 𝒍 there corresponds exactly one real number, (ii) to every real number there corresponds exactly one point of 𝑙, and (iii) the distance between two distinct points on 𝑙 is equal to the absolute value of the difference of the corresponding numbers.

A

C

B

A B C

Vocabulary

1. POSTULATE (Ruler Postulate): The points on a line can be matched one to one with the real numbers. The real number that corresponds to a point is the coordinate of the point. The distance between points A and B, written as AB, is the absolute value of the difference between the coordinates of A and B. AB is read as “the length of AB” AGAIN::: Æ AB is  read  as  “the  length of segment AB”Å

2. POSTULATE (Segment Addition Postulate): If B is between A and C, then AB + BC = AC. If AB + BC = AC, then B is between A and C.

AB + BC = AC

3. Congruent line segments are segments that have the same length. The notation for congruent is # .

(If AB = CD, then AB CD# .) Congruent segments ↔ equal lengths

4. The midpoint of a line segment is the point that divides the segment into two congruent segments.

Midpoint ↔ 2 # segments

5. A segment bisector is a segment, ray, line, or plane that intersects a line segment at its midpoint.

A segment bisector determines the location of the midpoint. In the diagram below, line l is a segment bisector of ABbecause line l intersects AB at its midpoint.

Segment bisector ↔ midpoint If line l is a segment bisector of AB , then C is the midpoint of AB If C is the midpoint of AB , then line l is a segment bisector of AB In geometry, we do not use the word equal for segments, we would say that 𝑴𝑩 and 𝑨𝑴 are congruent. What does the word congruent mean?

l

BA C

Review of Key Vocabulary Given Conclusion Reason

AB = CD

AB = CD

B is between A and C, ( ) AB + BC = AC

A B C

Vocabulary

1. POSTULATE (Ruler Postulate): The points on a line can be matched one to one with the real numbers. The real number that corresponds to a point is the coordinate of the point. The distance between points A and B, written as AB, is the absolute value of the difference between the coordinates of A and B. AB is read as “the length of AB” AGAIN::: Æ AB is  read  as  “the  length of segment AB”Å

2. POSTULATE (Segment Addition Postulate): If B is between A and C, then AB + BC = AC. If AB + BC = AC, then B is between A and C.

AB + BC = AC

3. Congruent line segments are segments that have the same length. The notation for congruent is # .

(If AB = CD, then AB CD# .) Congruent segments ↔ equal lengths

4. The midpoint of a line segment is the point that divides the segment into two congruent segments.

Midpoint ↔ 2 # segments

5. A segment bisector is a segment, ray, line, or plane that intersects a line segment at its midpoint.

A segment bisector determines the location of the midpoint. In the diagram below, line l is a segment bisector of ABbecause line l intersects AB at its midpoint.

Segment bisector ↔ midpoint If line l is a segment bisector of AB , then C is the midpoint of AB If C is the midpoint of AB , then line l is a segment bisector of AB In geometry, we do not use the word equal for segments, we would say that 𝑴𝑩 and 𝑨𝑴 are congruent. What does the word congruent mean?

l

BA C

A B C

Vocabulary

1. POSTULATE (Ruler Postulate): The points on a line can be matched one to one with the real numbers. The real number that corresponds to a point is the coordinate of the point. The distance between points A and B, written as AB, is the absolute value of the difference between the coordinates of A and B. AB is read as “the length of AB” AGAIN::: Æ AB is  read  as  “the  length of segment AB”Å

2. POSTULATE (Segment Addition Postulate): If B is between A and C, then AB + BC = AC. If AB + BC = AC, then B is between A and C.

AB + BC = AC

3. Congruent line segments are segments that have the same length. The notation for congruent is # .

(If AB = CD, then AB CD# .) Congruent segments ↔ equal lengths

4. The midpoint of a line segment is the point that divides the segment into two congruent segments.

Midpoint ↔ 2 # segments

5. A segment bisector is a segment, ray, line, or plane that intersects a line segment at its midpoint.

A segment bisector determines the location of the midpoint. In the diagram below, line l is a segment bisector of ABbecause line l intersects AB at its midpoint.

Segment bisector ↔ midpoint If line l is a segment bisector of AB , then C is the midpoint of AB If C is the midpoint of AB , then line l is a segment bisector of AB In geometry, we do not use the word equal for segments, we would say that 𝑴𝑩 and 𝑨𝑴 are congruent. What does the word congruent mean?

l

BA C

A B C

Vocabulary

1. POSTULATE (Ruler Postulate): The points on a line can be matched one to one with the real numbers. The real number that corresponds to a point is the coordinate of the point. The distance between points A and B, written as AB, is the absolute value of the difference between the coordinates of A and B. AB is read as “the length of AB” AGAIN::: Æ AB is  read  as  “the  length of segment AB”Å

2. POSTULATE (Segment Addition Postulate): If B is between A and C, then AB + BC = AC. If AB + BC = AC, then B is between A and C.

AB + BC = AC

3. Congruent line segments are segments that have the same length. The notation for congruent is # .

(If AB = CD, then AB CD# .) Congruent segments ↔ equal lengths

4. The midpoint of a line segment is the point that divides the segment into two congruent segments.

Midpoint ↔ 2 # segments

5. A segment bisector is a segment, ray, line, or plane that intersects a line segment at its midpoint.

A segment bisector determines the location of the midpoint. In the diagram below, line l is a segment bisector of ABbecause line l intersects AB at its midpoint.

Segment bisector ↔ midpoint If line l is a segment bisector of AB , then C is the midpoint of AB If C is the midpoint of AB , then line l is a segment bisector of AB In geometry, we do not use the word equal for segments, we would say that 𝑴𝑩 and 𝑨𝑴 are congruent. What does the word congruent mean?

l

BA C

A B C

Vocabulary

1. POSTULATE (Ruler Postulate): The points on a line can be matched one to one with the real numbers. The real number that corresponds to a point is the coordinate of the point. The distance between points A and B, written as AB, is the absolute value of the difference between the coordinates of A and B. AB is read as “the length of AB” AGAIN::: Æ AB is  read  as  “the  length of segment AB”Å

2. POSTULATE (Segment Addition Postulate): If B is between A and C, then AB + BC = AC. If AB + BC = AC, then B is between A and C.

AB + BC = AC

3. Congruent line segments are segments that have the same length. The notation for congruent is # .

(If AB = CD, then AB CD# .) Congruent segments ↔ equal lengths

4. The midpoint of a line segment is the point that divides the segment into two congruent segments.

Midpoint ↔ 2 # segments

5. A segment bisector is a segment, ray, line, or plane that intersects a line segment at its midpoint.

A segment bisector determines the location of the midpoint. In the diagram below, line l is a segment bisector of ABbecause line l intersects AB at its midpoint.

Segment bisector ↔ midpoint If line l is a segment bisector of AB , then C is the midpoint of AB If C is the midpoint of AB , then line l is a segment bisector of AB In geometry, we do not use the word equal for segments, we would say that 𝑴𝑩 and 𝑨𝑴 are congruent. What does the word congruent mean?

l

BA C

A B C

Vocabulary

1. POSTULATE (Ruler Postulate): The points on a line can be matched one to one with the real numbers. The real number that corresponds to a point is the coordinate of the point. The distance between points A and B, written as AB, is the absolute value of the difference between the coordinates of A and B. AB is read as “the length of AB” AGAIN::: Æ AB is  read  as  “the  length of segment AB”Å

2. POSTULATE (Segment Addition Postulate): If B is between A and C, then AB + BC = AC. If AB + BC = AC, then B is between A and C.

AB + BC = AC

3. Congruent line segments are segments that have the same length. The notation for congruent is # .

(If AB = CD, then AB CD# .) Congruent segments ↔ equal lengths

4. The midpoint of a line segment is the point that divides the segment into two congruent segments.

Midpoint ↔ 2 # segments

5. A segment bisector is a segment, ray, line, or plane that intersects a line segment at its midpoint.

A segment bisector determines the location of the midpoint. In the diagram below, line l is a segment bisector of ABbecause line l intersects AB at its midpoint.

Segment bisector ↔ midpoint If line l is a segment bisector of AB , then C is the midpoint of AB If C is the midpoint of AB , then line l is a segment bisector of AB In geometry, we do not use the word equal for segments, we would say that 𝑴𝑩 and 𝑨𝑴 are congruent. What does the word congruent mean?

l

BA C

A B C

Vocabulary

1. POSTULATE (Ruler Postulate): The points on a line can be matched one to one with the real numbers. The real number that corresponds to a point is the coordinate of the point. The distance between points A and B, written as AB, is the absolute value of the difference between the coordinates of A and B. AB is read as “the length of AB” AGAIN::: Æ AB is  read  as  “the  length of segment AB”Å

2. POSTULATE (Segment Addition Postulate): If B is between A and C, then AB + BC = AC. If AB + BC = AC, then B is between A and C.

AB + BC = AC

3. Congruent line segments are segments that have the same length. The notation for congruent is # .

(If AB = CD, then AB CD# .) Congruent segments ↔ equal lengths

4. The midpoint of a line segment is the point that divides the segment into two congruent segments.

Midpoint ↔ 2 # segments

5. A segment bisector is a segment, ray, line, or plane that intersects a line segment at its midpoint.

A segment bisector determines the location of the midpoint. In the diagram below, line l is a segment bisector of ABbecause line l intersects AB at its midpoint.

Segment bisector ↔ midpoint If line l is a segment bisector of AB , then C is the midpoint of AB If C is the midpoint of AB , then line l is a segment bisector of AB In geometry, we do not use the word equal for segments, we would say that 𝑴𝑩 and 𝑨𝑴 are congruent. What does the word congruent mean?

l

BA C

A B C

Vocabulary

1. POSTULATE (Ruler Postulate): The points on a line can be matched one to one with the real numbers. The real number that corresponds to a point is the coordinate of the point. The distance between points A and B, written as AB, is the absolute value of the difference between the coordinates of A and B. AB is read as “the length of AB” AGAIN::: Æ AB is  read  as  “the  length of segment AB”Å

2. POSTULATE (Segment Addition Postulate): If B is between A and C, then AB + BC = AC. If AB + BC = AC, then B is between A and C.

AB + BC = AC

3. Congruent line segments are segments that have the same length. The notation for congruent is # .

(If AB = CD, then AB CD# .) Congruent segments ↔ equal lengths

4. The midpoint of a line segment is the point that divides the segment into two congruent segments.

Midpoint ↔ 2 # segments

5. A segment bisector is a segment, ray, line, or plane that intersects a line segment at its midpoint.

A segment bisector determines the location of the midpoint. In the diagram below, line l is a segment bisector of ABbecause line l intersects AB at its midpoint.

Segment bisector ↔ midpoint If line l is a segment bisector of AB , then C is the midpoint of AB If C is the midpoint of AB , then line l is a segment bisector of AB In geometry, we do not use the word equal for segments, we would say that 𝑴𝑩 and 𝑨𝑴 are congruent. What does the word congruent mean?

l

BA C

AC – AB = BC AC – BC = AB

Given Conclusion Reason

AB CD#

AB = CD

If two segments are congruent, then they have equal length

(Short form:#segments    →  =  lengths)

AB = CD

AB CD# If two segments have equal lengths,

then they are congruent (Short form: = lengths →  #segments)

B is between A and C, (ABC)

AB + BC = AC AC – AB = BC AC – BC = AB

Segment addition postulate

CONSTRUCTION: Can you think how you might use a straightedge and compass (a construction in Geometry) to create another congruent segment if someone gives you a certain segment to copy? Video Resource Æ http://www.youtube.com/watch?v=SoIDx-u3h4Q Steps to Construct Congruent Segments 1. Draw a line and choose a point to be the endpoint of the constructed segment. 2. Open the compass to the length of the given segment. 3. Place the compass point at the point of the new segment, and strike an arc for the measured length, this intersection will be the other endpoint. Construct congruent segments: (ex 1) (ex 2)

A

C

B

We only use a straightedge and

compass to construct

Given Conclusion Reason

AB CD#

AB = CD

If two segments are congruent, then they have equal length

(Short form:#segments    →  =  lengths)

AB = CD

AB CD# If two segments have equal lengths,

then they are congruent (Short form: = lengths →  #segments)

B is between A and C, (ABC)

AB + BC = AC AC – AB = BC AC – BC = AB

Segment addition postulate

CONSTRUCTION: Can you think how you might use a straightedge and compass (a construction in Geometry) to create another congruent segment if someone gives you a certain segment to copy? Video Resource Æ http://www.youtube.com/watch?v=SoIDx-u3h4Q Steps to Construct Congruent Segments 1. Draw a line and choose a point to be the endpoint of the constructed segment. 2. Open the compass to the length of the given segment. 3. Place the compass point at the point of the new segment, and strike an arc for the measured length, this intersection will be the other endpoint. Construct congruent segments: (ex 1) (ex 2)

A

C

B

We only use a straightedge and

compass to construct

CONSTRUCTION: Video Resource Æ http://www.youtube.com/watch?v=VA_Y7GCtNGA Steps to Constructing Segment Bisector 1. Place the compass point on one endpoint and open the compass more than half the distance. 2. Strike an arc above and below the segment. 3. Move the compass point to the other endpoint, and strike the same size arc above and below the segment. 4. The intersection of the arcs above and below will be connected to construct the segment bisector. (ex 3) Construct a bisector for the line segment below. (ex 4) Construct and label midpoint B of the line segment below.

HOMEWORK

1. __________________ is the amount of space between 2 points on a line. It is always expressed as a nonnegative number.

2. Name the 2 words we discussed that have to do with dividing a segment in half. For questions 3 & 4, tell whether each statement is sometimes, always or never true. Support each of your answers with a sketch. 3. 2 segments that have the same length must be congruent. 4. If Y is between X and Z, then X, Y, and Z are collinear. 5. B is between A and C, AC = 15.8 and AB = 9.9. Find BC 6. Suppose S is between R and T. Use the Segment Addition Postulate to solve for the

variable. RS = 7y – 4 ST = y + 5 RT = 28

7. Use a compass and straightedge to construct a segment whose length is AB + CD

8. HJ is twice JK. J is between H and K. If HJ = 4x and HK = 78, find JK 9. A, D, N, and X are collinear points. D is between N and A. NA + AX = NX. Draw a

diagram that represents this information. REVIEW: Use the figure to name each of the following

10. 2 lines that contain B 11. 3 collinear points

A B

Activity

For Exercises 1-4, H is between I and J.

1. HI = 3.9 and HJ = 6.2. Find IJ. _____

2. JI = 25 and IH = 13. Find HJ. _____

3. H is the midpoint of IJ , and IH = 0.75. Find HJ. _____

4. H is the midpoint of IJ , and IJ = 9.4. Find IH. _____

Find the measurements. What theorem or definition did you use?

5.

Find LM. ______

6. Draw three collinear points where E is between D and F. Then write an equation for these points using the Segment Addition Postulate.

Homework:

x Pages  17−19: 2, 4, 6, 10, 12, 14, 18, 24, 26, 30, 32, 34, 36, 38, 40, 50, 52 x Boxes Practice Sheet x And do the following:

Divide the following segment, AB , into 4 segments of equal length.

Unit 1 Lesson 2.notebook

7

September 09, 2016

Unit 1 Lesson 2.notebook

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September 09, 2016