Chapter 4: Lessons 1,2,3, & 6 BY MAI MOHAMMAD. Lesson 1: Coordinates & Distance Quadrants: I, II,...
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Transcript of Chapter 4: Lessons 1,2,3, & 6 BY MAI MOHAMMAD. Lesson 1: Coordinates & Distance Quadrants: I, II,...
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