Lesson 1.3Collinearity, 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

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

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

Definitions…

U

A

N

S

H

P

NoncollinearCollinear

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

noncollinear

Example #1

YX

SR

O

M

P

T

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

Definitions…

U

A

NS

H

P

P is NOT between H and SA is between N and U

For any 3 points there are only 2 possibilities:

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

2.They are noncollinear (the 3 points determine a triangle)

Triangle Inequality

5.5A 12.5

BC

18

11 14

24A

B

C

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”

Triangle Inequality

11 14

24A

B

C

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!

Assumptions

You should Assume You should NOT Assume

*Straight lines and angles

*Collinearity of points

*Betweenness of points

*Relative positions of points

*Right angles

*Congruent segments

*Congruent angles

*Relative sizes of segments and angles

Lesson 1.3 Worksheet

Homework

Lesson 1.4Beginning Proofs

Objective:

Write simple two-column proofs

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.

Introducing…

1.We present a theorem(s).

2.We prove the theorem(s).

3.We use the theorems to help prove sample problems.

4. You use the theorems to prove homework problems.

Theorem Procedure…

Note:

The sooner you learn the theorems, the

easier your homework will be!

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

Theorem 1

A

BGiven: <A is a right <.

<B is a right <.

Prove: A B

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

5. A B 5. If 2 <‘s have the same measure then they are congruent.

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

Theorem 2

Given: <NAU is a straight <.

<PHS is a straight <.

Prove: NAU PHS

Statement Reason

U

A

N

SH

P

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!

Practice Makes Perfect…

Example #1

Given: <RST = 50°

<TSV = 40°

<X is a right angle

Prove: RSV X

Statement Reason

X

TR

S V

Example #2

Given: <ABD = 30°

<ABC = 90°

<EFY = 50° 20’

<XFY = 9° 40’

Prove: DBC XFE

Statement Reason

A

B C

E

XY

F

D

Lesson 1.4 Worksheet

Homework

Lesson 1.5Division of Segments and Angles

Objective:

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

trisectors.

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.

Definitions

Note:

Only segments have midpoints!

X

Y

X is not a midpoint

Y is not a midpoint

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

A

Y

B

X

M

Conclusions:

Example

F

G

ED

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

Point D bisects

bisects

FD DE

FE

DG FE

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.

Definitions

Note:

One again, only segments have trisection points!

If , what conclusions can we draw?

Examples

A

S

C

R

AR RS SC

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

H

D E F G

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.

Definitions

Examples

DA

B C

If , then is the bisector of ABCABC DBC BD ��������������

40°40°

D

A

B E

35°35°

C

35°

If ,

then and trisect ABE

ABC CBD DBE

BC BD

����������������������������

Example #1

MO P

x + 8 2x - 6

44

Does M bisect segment OP?

Example #2 A

BGiven: B is a midpoint of

Prove:

Statement Reason

AC

AB BCC

D

Example #3

FE G

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.

H

Classwork

1.1-1.3 Review Worksheet

Lesson 1.5 Worksheet

Homework