Chapter 4: Lessons 1,2,3, & 6

BY MAI MOHAMMAD

Lesson 1: Coordinates & Distance

Quadrants: I, II, III, IVAxes: x-axis, y-axisOrigin: O (0,0)Coordinates: A (6,3),

B (-8,7)C (-3,-5), D (3,-2)

A one-dimensional coordinate system is used to choose an origin

A two-dimensional coordinate system to locate points in the plane

The Pythagorean Theorem gives us

the distance formula:

- The length of AB and BC are

given (using the grid)

- AB² + BC² = AC²

- AC is the distance

The Distance Formula:The distance formula is used to

find the distance from one

point to another using their

coordinates

Lesson 2: Polygons and Congruence

Definition of a polygon:

A connected set of at least three line segments in the

same plane such that each segment intersects exactly

two others, one at each endpoint

Not polygons:

Polygons:

Definition of congruent triangles:

Two triangles are congruent iff there is a

correspondence between their vertices such that all of

their corresponding sides and angles are equal

Corollary to the definition of congruent

triangles:

Two triangles congruent to a third triangle are

congruent to each other

Lesson 3: ASA and SAS Congruence

The ASA Postulate:If two angles and the included side of one triangle are equal to two angles and the included side of another triangle, the triangles are congruent(a side included by 2 angles)

The SAS Postulate:If two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, the triangles are congruent(an angle included by 2 sides)

Lesson 6: SSS Congruence

The SSS Theorem:If the three sides of one triangle are equal to the three sides of another triangle, the triangles are congruent

Lab: Proving Triangles Congruent

At least three pieces of the criteria are necessary to prove congruence (two angles and a segment, two segments and an angle, three segments, etc.)

Proves why ASA, AAS, SSS work and other combinations, like AAA, do not

Summary:

To find the distance between two points, the Pythagorean Theorem or the distance formula can be used

Polygons are made up of at least three line segments of the same plane that intersect exactly two other segments, one at each endpoint (triangle, square, pentagon, etc.)

ASA, SAS, SSS, and AAS prove triangle congruence

BY CLARE STRICKLAND

CHAPTER 4LESSONS 4, 5, 7 &

PROOFS

Lesson 4: Congruence Proofs

Two triangles are congruent iff there is a correspondence between their vertices such that all of their corresponding sides and angles are equal:

Corresponding Parts of Congruent Triangles are Equal (CPCTE)Generally proved using SAS, ASA, or SSS Can go in many different orders

EXAMPLE PROOF

Lesson 5: Isosceles and Equilateral Triangles

A triangle is: Scalene iff it has no equal sides Isosceles iff it has at least 2 equal sides Equilateral iff all of its sides are equal

Lesson 5: Isosceles and Equilateral Triangles

A triangle is Obtuse iff it has an obtuse angle Right iff it has a right angle Acute iff all of its angles are acute Equiangular iff all of its angles are equal

Lesson 5: Isosceles and Equilateral Triangles

Theorems: If two sides of a triangle are equal, the angles

opposite them are equal.

If two angles of a triangle are equal, the sides opposite them are equal.

Lesson 5: Isosceles and Equilateral Triangles

Corollaries: An equilateral triangle is equiangular

An equiangular triangle is equilateral

Lesson 7: Constructions

How to copy a line segment: Set the radius of the compass to the length of AB.

Draw line l and mark point P. With P as center, draw an arc of radius AB that intersects line l and draw point Q.

Lesson 7: Constructions

How to copy an angle: Draw PQ as one ray of the

angle. With point A as its center, draw an arc to create points B and C. Using that same radius on the compass, draw an arc on line PQ. Set the radius on your compass to length BC. Use that compass setting to draw an arc with point R at its center. Mark the intersection of the arcs as point S. Draw line segment PS

Lesson 7: Constructions

How to copy a triangle: Construct line segment XY

equal to AB. Set the compass length of CB, and with point Y as its center construct an arc of that length. Set the compass length of CA, and with point X as its center construct an arc of that length. Mark the point of intersection of the two arcs as point Z. Use a straightedge to construct XZ and YZ

Proofs

Tips for Proofs: Set up the two columns (Statements & Reasons) and

number each step Mark up your figure with your given Identify what you’re looking for When you name an angle, use three letters Be careful of when you’re using arrows

VERSUS

Use different colors to help visualize

Reasons for Two Column Proofs

Segments

Definition, Postulate, or Theorem

Definition of midpoint Midpt = parts

Definition of betweenness of points Def. of BOP

Definition of segment bisector Segment bisector = parts

Ruler Postulate Ruler Post.

Betweeness of Points Theorm BOP Thm.

A line segment had exactly one midpoint Segment 1 midpt.

Reasons for Two Column Proofs

Angles

Definition, Postulate, or Theorem

Definition of Betweeness of Rays Def. of BOR

Definition of Perpendicular Lines right angle

Definition of straight angle Straight 180º

Definition of right angle Right 90º

Definition of angle bisector bisector = parts

Definition of a linear pair Lin. Pr. Opp rays &

Definition of supplementary angles Suppl. Sum = 180º

Definition of complementary angles Compl. Sum = 90º

Protractor Postulate Protractor Post.

Reasons for Two Column Proofs

Angles

Definition, Postulate, or Theorem

An angle has exactly one ray that bisects it 1 bisector

Betweeness of Rays Theorem BOR Theorem

If 2 angles are complementary to the same angle, they are equal

compl same =

If 2 angles are supplementary to the same angle, they are equal

suppl same =

If two angles form a linear pair, they are supplementary

Lin pr suppl

Vertical angles are equal Vertical =

If lines are perpendicular, they form 4 right angles

4 right

All right angles are equal Right =

If two angles in a linear pair are equal, their sides are perpendicular

Lin pr = sides

Given: BD is a bisector of AC, BD is perpendicular to ACProve: ABC is isoscelesStatements:

1. BD is a bisector of AC, BD is perpendicular to AC2. AD=AC3. ADB & CDB are right

angles4. ADB= CDB5. BD = BD6. ADB = CDB7. AB=CB8. ABC is isosceles

Reasons:1. Given

2. Bisector 2 = parts3. Perp right angles4. All right angles =5. Reflexive Property6. SAS (steps 2, 4, 5)7. CPCTE8. Def. of isosceles