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Lesson 1.3 Collinearity, Betweenness, and Assumptions. Objective: Recognize collinear, and non-collinear points, recognize when a point is between two others, recognize that each side of a triangle is shorter than the sum of the other two sides, and correctly interpret geometric diagrams.

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Lesson 1.3 Collinearity, Betweenness, and Assumptions

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Lesson 1 3 collinearity betweenness and assumptions

Lesson 1.3

Collinearity, Betweenness, and Assumptions

Objective:

Recognize collinear, and non-collinear points, recognize when a point is between two others, recognize that each side of a triangle is shorter than the sum of the other two sides, and correctly interpret geometric diagrams


Lesson 1 3 collinearity betweenness and assumptions

Definitions…

Def. Points that lie on the same line are called collinear.

Def. Points that do not lie on the same line are called noncollinear.

P

U

H

A

S

N

Collinear

Noncollinear


Lesson 1 3 collinearity betweenness and assumptions

Example #1

Name as many sets of points as you can that are collinear and noncollinear

M

P

O

X

Y

T

R

S


Lesson 1 3 collinearity betweenness and assumptions

Definitions…

In order for us to say that a point is between two other points, all three points MUST be collinear.

U

P

A

H

S

N

P is NOT between H and S

A is between N and U


Lesson 1 3 collinearity betweenness and assumptions

Triangle Inequality

  • For any 3 points there are only 2 possibilities:

  • They are collinear (one point is between the other two and two of the distances add up to the 3rd)

  • They are noncollinear (the 3 points determine a triangle)

C

5.5

B

12.5

A

18

B

14

11

C

A

24


Lesson 1 3 collinearity betweenness and assumptions

Triangle Inequality

Notice in this triangle, 14 + 11 > 24.

This is extra super important!

“The sum of the lengths of any 2 sides of a triangle is always greater than the length of the third”

B

14

11

C

A

24


Lesson 1 3 collinearity betweenness and assumptions

Assumptions

When given a diagram, sometimes we need to assume certain information, but you know what they say about assuming….

There are do’s and don’ts!


Lesson 1 3 collinearity betweenness and assumptions

Homework

Lesson 1.3 Worksheet


Lesson 1 3 collinearity betweenness and assumptions

Lesson 1.4

Beginning Proofs

Objective:

Write simple two-column proofs


Lesson 1 3 collinearity betweenness and assumptions

Introducing…

The Two-Column Proof!

The two-column proof is the major type of proof we use throughout our studies.

Def. A theorem is a mathematical statement that can be proved.


Lesson 1 3 collinearity betweenness and assumptions

Theorem Procedure…

  • We present a theorem(s).

  • We prove the theorem(s).

  • We use the theorems to help prove sample problems.

  • You use the theorems to prove homework problems.

Note:

The sooner you learn the theorems, the easier your homework will be! 


Lesson 1 3 collinearity betweenness and assumptions

Theorem 1

If two angles are right angles, then they are congruent.

Given: <A is a right <.

<B is a right <.

Prove:

B

A

Statement Reason

1. <A is a right <

1. Given

2. m<A = 90°

2. If an < is a right < then its measure is 90°

3. <B is a right <

3. Given

4. If an < is a right < then its measure is 90°

4. m<B = 90°

  • If 2 <‘s have the same measure then they are congruent.

5.


Lesson 1 3 collinearity betweenness and assumptions

Theorem 2

If two angles are straight angles, then they are congruent.

U

Given: <NAU is a straight <.

<PHS is a straight <.

Prove:

A

N

P

H

S

Statement Reason


Lesson 1 3 collinearity betweenness and assumptions

Practice Makes Perfect…

Now that we know the two theorems (and have proved them), we apply what we know to sample problems.

about what we can and cannot assume from a diagram! This is important with proofs!


Lesson 1 3 collinearity betweenness and assumptions

Example #1

Given: <RST = 50°

<TSV = 40°

<X is a right angle

Prove:

R

T

X

S

V

Statement Reason


Lesson 1 3 collinearity betweenness and assumptions

Example #2

X

Y

Given: <ABD = 30°

<ABC = 90°

<EFY = 50° 20’

<XFY = 9° 40’

Prove:

D

A

F

E

B

C

Statement Reason


Lesson 1 3 collinearity betweenness and assumptions

Homework

Lesson 1.4 Worksheet


Lesson 1 3 collinearity betweenness and assumptions

Lesson 1.5

Division of Segments and Angles

Objective:

Identify midpoints and bisectors of segments, trisection points and trisectors of segments, angle bisectors and trisectors.


Lesson 1 3 collinearity betweenness and assumptions

Definitions

Def. A point (or segment, ray, or line) that divides a segment into two congruent segments bisects the segment. The bisection point is called the midpoint of the segment.

Y

A

M

B

Note:

Only segments have midpoints!

X

Why can’t a ray or line have a midpoint?

X

X is not a midpoint

Y

Y is not a midpoint


Lesson 1 3 collinearity betweenness and assumptions

Example

If D is the midpoint of segment FE, what conclusions can we draw?

G

F

D

E

Conclusions:


Lesson 1 3 collinearity betweenness and assumptions

Definitions

A segment divided into three congruent parts is said to be trisected.

Def. Two points (or segments, rays, or lines) that divides a segment into 3 congruent segments trisect the segment. The 2 points at which the segment is divided are called trisections points.

Note:

One again, only segments have trisection points!


Lesson 1 3 collinearity betweenness and assumptions

Examples

If , what conclusions can we draw?

C

S

R

A

If E and F are trisection points of segment DG, what conclusions can we draw?

H

D

G

F

E


Lesson 1 3 collinearity betweenness and assumptions

Definitions

Like a segment, angles can also be bisected and trisected.

Def. A ray that divides an angle into 2 congruent angles bisects the angle. The dividing ray is called the angle bisector.

Def. Two ray that divide an angle into 3 congruent angles trisects the angle. The 2 dividing rays are called angle trisectors.


Lesson 1 3 collinearity betweenness and assumptions

Examples

A

D

40°

40°

B

C

C

A

D

35°

35°

35°

B

E


Lesson 1 3 collinearity betweenness and assumptions

Example #1

Does M bisect segment OP?

2x - 6

x + 8

O

P

M

44


Lesson 1 3 collinearity betweenness and assumptions

Example #2

A

Given: B is a midpoint of

Prove:

D

B

C

Statement Reason


Lesson 1 3 collinearity betweenness and assumptions

Example #3

Segment EH is divided by F and G in the ratio 5:3:2 from left to right. If EH = 30, find FG and name the midpoint of segment EH.

G

H

F

E


Lesson 1 3 collinearity betweenness and assumptions

Classwork

1.1-1.3 Review Worksheet


Lesson 1 3 collinearity betweenness and assumptions

Homework

Lesson 1.5 Worksheet


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