NOTES geom chap 04 congruent triangles · (CPCTC) Notes Chapter 04: ... 4.6 4.7 Notes on your desk...

NotesNotes Chapter 04: Congruent TrianglesUnit 1: Corresponding Parts in a CongruenceSection 1: Congruent Figures

Whenever two figures have the same size and shape, they are called congruent.congruent.congruent.congruent.

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4.5 Triangles ABC and DEF are congruent. You can match up vertices like

AAAA BBBB

CCCC

DDDD EEEE

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Triangles ABC and DEF are congruent. You can match up vertices likeA!D B!E C!F

This means that:Corresponding AnglesCorresponding AnglesCorresponding AnglesCorresponding Angles Corresponding SidesCorresponding SidesCorresponding SidesCorresponding Sides∠A !_____ segment AB !_____∠ B! _____ segment BC !_____∠ C! _____ segment AC !_____

Corresponding parts of a congruent triangles are congruent. (CPCTC)Corresponding parts of a congruent triangles are congruent. (CPCTC)Corresponding parts of a congruent triangles are congruent. (CPCTC)Corresponding parts of a congruent triangles are congruent. (CPCTC)

NotesNotes Chapter 04: Congruent TrianglesUnit 1: Corresponding Parts in a CongruenceSection 1: Congruent Figures

Example 1Example 1Example 1Example 1 Two triangles are congruent.on your deskon your deskon your deskon your desk

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DDDD CCCC

OOOO

a. ∆ABON_________b. ∠BN_________c. ∠AN _________d. segment AON _________e. segment ABN _________

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Example 2Example 2Example 2Example 2 Two triangles are congruent.

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MMMMBBBB

EEEE PPPP

NNNNCCCCDDDD OOOO

a. B corresponds to ______b. m∠DN_________c. ∠DN _________d. segment AEN _________e. AE= _________4cm

43!

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on your deskon your deskon your deskon your desk Postulate 12 (SSS Postulate)Postulate 12 (SSS Postulate)Postulate 12 (SSS Postulate)Postulate 12 (SSS Postulate)If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent.

Chapter 04: Congruent TrianglesUnit 1: Corresponding Parts in a CongruenceSection 2: Some Ways to Prove Triangles Congruent

Postulate 13 (SAS Postulate)Postulate 13 (SAS Postulate)Postulate 13 (SAS Postulate)Postulate 13 (SAS Postulate)If two sides and an included angle of one triangle are congruent to two sides and an included angle of another triangle, then the triangles are

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sides and an included angle of another triangle, then the triangles are congruent.

Postulate 14 (ASA Postulate)Postulate 14 (ASA Postulate)Postulate 14 (ASA Postulate)Postulate 14 (ASA Postulate)If two angles and an included side of one triangle are congruent to two angles and an included side of another triangle, then the triangles are congruent.

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Chapter 04: Congruent TrianglesUnit 1: Corresponding Parts in a CongruenceSection 2: Some Ways to Prove Triangles Congruent

Example 1Example 1Example 1Example 1 Which of the three postulates do you use?

a.

b.

d.

e.

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b.

c.

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Chapter 04: Congruent TrianglesUnit 1: Corresponding Parts in a CongruenceSection 3: Using Congruent Triangles

Example 1Example 1Example 1Example 1Given: segment AB and segment CD

bisect each other at MProve: segment AD || segment BC

Statements Reasons

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Example 2Example 2Example 2Example 2Given: segment PO ⊥ plane X;

segment PO bisects ∠APBProve: segment AD || segment BC

Statements Reasons

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PPPP

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Example 3Example 3Example 3Example 3Given: m∠1=m∠2; m∠3=m∠4Prove: M is the midpoint of segment JK

Statements Reasons

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JJJJ KKKKMMMM

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Chapter 04: Congruent TrianglesUnit 2: Some Theorems Based on Congruent TrianglesSection 4: The Isosceles Triangle

Theorem 4Theorem 4Theorem 4Theorem 4----1 (The Isosceles Triangle Thm)1 (The Isosceles Triangle Thm)1 (The Isosceles Triangle Thm)1 (The Isosceles Triangle Thm)If two sides of a triangles are congruent, then the angles opposite those sides are congruent. AAAA

BBBB CCCCDDDD

ProofProofProofProofGiven: segment AB N segment ACProve: ∠BN∠C

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Theorem 4Theorem 4Theorem 4Theorem 4----1 (The Isosceles Triangle Thm)1 (The Isosceles Triangle Thm)1 (The Isosceles Triangle Thm)1 (The Isosceles Triangle Thm)If two sides of a triangles are congruent, then the angles opposite those sides are congruent.

Corollary 1Corollary 1Corollary 1Corollary 1An equilateral triangle is also equiangular.

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vertexvertexvertexvertex

vertex vertex vertex vertex anglesanglesanglesangles

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Corollary 2Corollary 2Corollary 2Corollary 2An equilateral triangle has three 60!angles.

Corollary 3Corollary 3Corollary 3Corollary 3The bisector of the vertex angle of an isosceles triangle is perpendicular to the base at its midpoint.

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legleglegleg legleglegleg

basebasebasebase

base base base base anglesanglesanglesangles

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Theorem 4Theorem 4Theorem 4Theorem 4----2 (The Converse of Isosceles Triangle Thm)2 (The Converse of Isosceles Triangle Thm)2 (The Converse of Isosceles Triangle Thm)2 (The Converse of Isosceles Triangle Thm)If two angles of a triangles are congruent, then the sides opposite those angles are congruent.

Corollary 4Corollary 4Corollary 4Corollary 4An equiangular triangle is also equilateral.

How would How would How would How would you prove you prove you prove you prove this?this?this?this?

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Guided PracticeGuided PracticeGuided PracticeGuided PracticeFind the value of x.(1) (2) (3)

30! x!2x+2

x+5

2x-4

56! 62!

42 41

x

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1111

(4) Place the statements in an appropriate order for a proof.

Given: ∠1N∠2Prove: segment OK N segment OJ

(a) ∠3N∠4(b) ∠2N∠4; ∠3N∠1(c) segment OK N segment OJ(d) ∠1N∠2

O

J

K

3

1

4

2

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Guided PracticeGuided PracticeGuided PracticeGuided Practice(5) Do the proof of the followingGiven: segment XZ N segment XY;

ray YO bisects ∠XYZ;ray ZO bisects ∠XZY

Prove: segment YO N segment ZO

Statements Reasons

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Y

X

Z

O12

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Chapter 04: Congruent TrianglesUnit 2: Some Theorems Based on Congruent TrianglesSection 5: Other Methods of Proving Triangles Congruent

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Theorem 4Theorem 4Theorem 4Theorem 4----3 (AAS Thm)3 (AAS Thm)3 (AAS Thm)3 (AAS Thm)If two sides and a non-included angle of one triangle are congruent to the corresponding parts of another triangle, then the triangles are congruent.

Statements Reasons

Given: ∆ABC and ∆DEF; ∠BN∠E;∠CN∠F; segment AC N segment DF

Prove: ∆ABC N ∆DEF

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Statements Reasons

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Chapter 04: Congruent TrianglesUnit 2: Some Theorems Based on Congruent TrianglesSection 5: Other Methods of Proving Triangles Congruent

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Theorem 4Theorem 4Theorem 4Theorem 4----4 (HL Thm)4 (HL Thm)4 (HL Thm)4 (HL Thm)If the hypotenuse and a leg of one right triangle are congruent to the corresponding parts of another right triangle, then the triangles are congruent.

Statements Reasons

Given: ∆ABC and ∆DEF; ∠C and ∠F are right ∠s

Prove: ∆ABC N ∆DEF

AB DE; BC EF≅ ≅

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Statements Reasons

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Chapter 04: Congruent TrianglesUnit 2: Some Theorems Based on Congruent TrianglesSection 6: Using More than One Pair of Congruent Triangle

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Statements Reasons

ExampleExampleExampleExampleGiven: ∠1N∠2; ∠3N∠4Prove: segment TU N segment TW 1

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U

T V

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Q

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Statements Reasons

ExampleExampleExampleExampleGiven: segment RT N segment RV

segment NT N segment NVProve: segment TS N segment VS

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T

R S

V

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ExampleExampleExampleExampleGiven: segment RT N segment RV

segment NT N segment NVProve: segment TS N segment VS

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N

T

R S

VProve by Paragraph ProofProve by Paragraph ProofProve by Paragraph ProofProve by Paragraph Proof

Given that segment RT N segment RV, segment NT N segment NV, and

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Given that segment RT N segment RV, segment NT N segment NV, and segment RN N segment RN by reflexive property, ∆RNTN∆RNV by SSS.We note that segment RS N segment RS by reflexive property and ∠1N∠2 by CPCTC, so ∆RTSN∆RVS by SAS.Therefore, segment TS N segment VS by CPCTC.

NotesNotes


Chapter 04: Congruent TrianglesUnit 2: Some Theorems Based on Congruent TrianglesSection 7: Medians, Altitudes, and Perpendicular Bisectors

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A C

B

A C

B

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B

A median median median median of a triangle is a segment from a vertex of a triangle to the midpoint of the opposite side.

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B

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An altitude altitude altitude altitude of a triangle is a perpendicular segment from a vertex of a triangle to the line containing the opposite side.

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Draw three altitude on the following right triangle and obtuse triangle.

A C

B

A C

B

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A perpendicular bisector perpendicular bisector perpendicular bisector perpendicular bisector of a segment is a line (or ray or segment) that is perpendicular to the segment at its midpoint.

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Theorem 4Theorem 4Theorem 4Theorem 4----5 5 5 5 If a point lies on the perpendicular bisector of a segment, then the point is equidistant from the endpoints of the segment.

A

B CX m

2020

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Theorem 4Theorem 4Theorem 4Theorem 4----6 6 6 6 If a point is equidistant from the endpoints of a segment, then the point lies on the perpendicular bisector of the segment.

A

B CX1 2

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Theorem 4Theorem 4Theorem 4Theorem 4----5 5 5 5 If a point lies on the perpendicular bisector of a segment, then the point is equidistant from the endpoints of the segment.

A

B CX mStatements Reasons

Given: line m is the ⊥ bisector of segment BC;A is on m

Prove: AB N AC

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Theorem 4Theorem 4Theorem 4Theorem 4----6 6 6 6 If a point is equidistant from the endpoints of a segment, then the point lies on the perpendicular bisector of the segment.

A

B CX1 2

Statements Reasons

Given: AB=ACProve: A is on the ⊥ bisector of segment BC

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The distance from a point to a line distance from a point to a line distance from a point to a line distance from a point to a line (or plane) is defined to be the length of the perpendicular segment from the point to the line (or plane).

RRRR

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The length of segment RS, denoted RS, is the distance between the point P and the line t.

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Theorem 4Theorem 4Theorem 4Theorem 4----7 7 7 7 If a point lies on the perpendicular bisector of an angle, then the point is equidistant from the sides of the angle.

B Y

X

P

C

Z

A

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Theorem 4Theorem 4Theorem 4Theorem 4----8 8 8 8 If a point is equidistant from the sides of an angle, then the point lies on the bisector of the angle.

B Y

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P

C

A

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Theorem 4Theorem 4Theorem 4Theorem 4----7 7 7 7 If a point lies on the perpendicular bisector of an angle, then the point is equidistant from the sides of the angle.

B Y

X

P

C

Z

A

Given: ray BZ bisects ∠ABC; P lies on ray BZ; segment PX ⊥ ray BA; segment PY ⊥ ray BC

Prove: PX=PYStatements Reasons

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Theorem 4Theorem 4Theorem 4Theorem 4----8 8 8 8 If a point is equidistant from the sides of an angle, then the point lies on the bisector of the angle.

B Y

X

P

C

A

Statements Reasons

Given: segment PX ⊥ ray BA; segment PY ⊥ ray BC; PX=PY

Prove: ray BP bisects ∠ABC

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Chalkboard ExamplesChalkboard ExamplesChalkboard ExamplesChalkboard ExamplesFill in the blank with always, sometimes, or never.1. An altitude is ____________ perpendicular to the opposite side.2. A median is ____________ perpendicular to the opposite side.3. An altitude is ____________ an angle bisector.4. An angle bisector is ____________ perpendicular to the opposite side.5. A perpendicular bisector of a segment is ____________ equidistant form

the endpoints of the segment.

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the endpoints of the segment.

6. Suppose ray OG bisects angle TOY. What can you deduce if you also know that:a. Point J lies on ray OG?

b. A point K is such that the distance from K to ray OT is 13 cm and the distance from K to ray OY is 13 cm?

NotesNotes



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ExamplesExamplesExamplesExamplesGiven: ray DP bisects ∠ADE;

ray EP bisects ∠DEC;Prove: ray BP bisects ∠ABC

B E C

DA

PYou do!!You do!!You do!!You do!!

Statements Reasons

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NOTES geom chap 04 congruent triangles · (CPCTC) Notes Chapter 04: ... 4.6 4.7 Notes on your desk...

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