Congruence of Line Segments, Angles, and Triangles

Congruence of Line Segments, Angles, and Triangles
Chapter 1 - Essential of Geometry Congruence of Line Segments, Angles, and Triangles Eleanor Roosevelt High School Geometry Mr. Chin-Sung Lin

Postulates of Lines, Line Segments, and Angles
Chapter 1 - Essential of Geometry ERHS Math Geometry Postulates of Lines, Line Segments, and Angles Mr. Chin-Sung Lin

L10_Inductive Reasoning ERHS Math Geometry Postulates of Lines, Line Segments, and Angles A line segment can be extended to any length in either direction We can choose some point of AB that is not a point of AB to form a line segment of any length A D B Mr. Chin-Sung Lin

L10_Inductive Reasoning ERHS Math Geometry Postulates of Lines, Line Segments, and Angles Through two given points, one and only one line can be drawn, i.e., two points determine a line Through given points A and B, one and only one line can be drawn A B Mr. Chin-Sung Lin

L10_Inductive Reasoning ERHS Math Geometry Postulates of Lines, Line Segments, and Angles Two lines cannot intersect in more than one point AMB and CMD intersect at M and cannot intersect at any other point A M B C D Mr. Chin-Sung Lin

L10_Inductive Reasoning ERHS Math Geometry Postulates of Lines, Line Segments, and Angles One and only one circle can be drawn with any given point as center and the length of any given segment as a radius Only one circle can be drawn that has point O as its center and a radius equal in length to segment r O r Mr. Chin-Sung Lin

L10_Inductive Reasoning ERHS Math Geometry Postulates of Lines, Line Segments, and Angles At a given point on a given line, one and only one perpendicular can be drawn to the line At point P on APB, exactly one line, PD, can be drawn perpendicular to APB and no other line through P is perpendicular to APB D A P B Mr. Chin-Sung Lin

L10_Inductive Reasoning ERHS Math Geometry Postulates of Lines, Line Segments, and Angles From a given point not on a given line, one and only one perpendicular can be drawn to the line From point D not on AB, exactly one line DP, can be drawn perpendicular to AB and no other linefrom D is perpendicular to AB. D A P B Mr. Chin-Sung Lin

L10_Inductive Reasoning ERHS Math Geometry Postulates of Lines, Line Segments, and Angles Distance Postulate - For any two distinct points, there is only one positive real number that is the length of the line segment joining the two points For any distinct points A and B, there is only one positive real number, represented by AB, that is the length of AB A B Mr. Chin-Sung Lin

L10_Inductive Reasoning ERHS Math Geometry Postulates of Lines, Line Segments, and Angles The shortest distance between two points is the length of the line segment joining these two points The measure of the shortest path from A to B is the distance AB A C B Mr. Chin-Sung Lin

L10_Inductive Reasoning ERHS Math Geometry Postulates of Lines, Line Segments, and Angles A line segment has one and only one midpoint AB has a midpoint, point P, and no other point is a midpoint of AB A P B Mr. Chin-Sung Lin

L10_Inductive Reasoning ERHS Math Geometry Postulates of Lines, Line Segments, and Angles An angle has one and only one bisector Angle ABC has one bisector, BD, and no other ray bisects ABC A B D C Mr. Chin-Sung Lin

Conditional Statements and Proof
Chapter 1 - Essential of Geometry ERHS Math Geometry Conditional Statements and Proof Mr. Chin-Sung Lin

Conditionals and Proof
Chapter 1 - Essential of Geometry ERHS Math Geometry Conditionals and Proof When the information needed for a proof is presented in a conditional statement, we use the information in the hypothesis to form a given statement, and the information in the conclusion to form a prove statement Mr. Chin-Sung Lin

Rewrite the Conditionals for Proof
Chapter 1 - Essential of Geometry ERHS Math Geometry Rewrite the Conditionals for Proof If a ray bisects a straight angle, it is perpendicular to the line determined by the straight angle Mr. Chin-Sung Lin

Chapter 1 - Essential of Geometry ERHS Math Geometry Rewrite the Conditionals for Proof If a ray bisects a straight angle, it is perpendicular to the line determined by the straight angle Given: ABC is an straight angle and BD bisects ABC Prove: BD  AC D B C A Mr. Chin-Sung Lin

Chapter 1 - Essential of Geometry ERHS Math Geometry Rewrite the Conditionals for Proof If a triangle is equilateral, then the measures of the sides are equal Mr. Chin-Sung Lin

Chapter 1 - Essential of Geometry ERHS Math Geometry Rewrite the Conditionals for Proof If a triangle is equilateral, then the measures of the sides are equal Given: ΔABC is equilateral Prove: AB = BC = CA A C B Mr. Chin-Sung Lin

Using Postulates and Definitions in Proofs
Chapter 1 - Essential of Geometry ERHS Math Geometry Using Postulates and Definitions in Proofs Mr. Chin-Sung Lin

Two Column Proof Example
Chapter 1 - Essential of Geometry ERHS Math Geometry Two Column Proof Example C A B D R 1 2 3 4 Given: DR is the bisector of ADC, 3 ≅ 1 and 4 ≅ 2. Prove: 3 ≅ 4 Proof: Statements Reasons Mr. Chin-Sung Lin

Chapter 1 - Essential of Geometry ERHS Math Geometry Two Column Proof Example C A B D R 1 2 3 4 Given: DR is the bisector of ADC, 3 ≅ 1 and 4 ≅ 2. Prove: 3 ≅ 4 Proof: Statements Reasons DR is the bisector of ABC Given. 3 ≅ 1 and 4 ≅ 2 Mr. Chin-Sung Lin

Chapter 1 - Essential of Geometry ERHS Math Geometry Two Column Proof Example C A B D R 1 2 3 4 Given: DR is the bisector of ADC, 3 ≅ 1 and 4 ≅ 2. Prove: 3 ≅ 4 Proof: Statements Reasons DR is the bisector of ABC Given. 3 ≅ 1 and 4 ≅ 2 1 ≅  Definition of angle bisector. Mr. Chin-Sung Lin

Chapter 1 - Essential of Geometry ERHS Math Geometry Two Column Proof Example C A B D R 1 2 3 4 Given: DR is the bisector of ADC, 3 ≅ 1 and 4 ≅ 2. Prove: 3 ≅ 4 Proof: Statements Reasons DR is the bisector of ABC Given. 3 ≅ 1 and 4 ≅ 2 2. 1 ≅  Definition of angle bisector. 3. 3 ≅  Substitution postulate. Mr. Chin-Sung Lin

Two Column Proof Exercise
Chapter 1 - Essential of Geometry ERHS Math Geometry Two Column Proof Exercise Given: M is the midpoint of AB. Prove: AM = ½ AB and MB = ½ AB Proof: Statements Reasons A M B Mr. Chin-Sung Lin

Chapter 1 - Essential of Geometry ERHS Math Geometry Two Column Proof Exercise Given: M is the midpoint of AB. Prove: AM = ½ AB and MB = ½ AB Proof: Statements Reasons M is the midpoint of AB Given. A M B Mr. Chin-Sung Lin

Chapter 1 - Essential of Geometry ERHS Math Geometry Two Column Proof Exercise Given: M is the midpoint of AB. Prove: AM = ½ AB and MB = ½ AB Proof: Statements Reasons M is the midpoint of AB Given. AM ≅ MB Definition of midpoint. A M B Mr. Chin-Sung Lin

Chapter 1 - Essential of Geometry ERHS Math Geometry Two Column Proof Exercise Given: M is the midpoint of AB. Prove: AM = ½ AB and MB = ½ AB Proof: Statements Reasons M is the midpoint of AB Given. AM ≅ MB Definition of midpoint. AM = MB Definition of congruent segments. A M B Mr. Chin-Sung Lin

Chapter 1 - Essential of Geometry ERHS Math Geometry Two Column Proof Exercise Given: M is the midpoint of AB. Prove: AM = ½ AB and MB = ½ AB Proof: Statements Reasons M is the midpoint of AB Given. AM ≅ MB Definition of midpoint. AM = MB Definition of congruent segments. AM + MB = AB Partition postulate. A M B Mr. Chin-Sung Lin

Chapter 1 - Essential of Geometry ERHS Math Geometry Two Column Proof Exercise Given: M is the midpoint of AB. Prove: AM = ½ AB and MB = ½ AB Proof: Statements Reasons M is the midpoint of AB Given. AM ≅ MB Definition of midpoint. AM = MB Definition of congruent segments. AM + MB = AB Partition postulate. 2AM = AB and 2 MB = AB Substitution postulate. A M B Mr. Chin-Sung Lin

Chapter 1 - Essential of Geometry ERHS Math Geometry Two Column Proof Exercise Given: M is the midpoint of AB. Prove: AM = ½ AB and MB = ½ AB Proof: Statements Reasons M is the midpoint of AB Given. AM ≅ MB Definition of midpoint. AM = MB Definition of congruent segments. AM + MB = AB Partition postulate. 2AM = AB and 2 MB = AB Substitution postulate. AM = ½ AB and MB = ½ AB 6. Division postulate. A M B Mr. Chin-Sung Lin

L6_Congruent Angle Pairs
ERHS Math Geometry Angles & Angle Pairs Mr. Chin-Sung Lin

ERHS Math Geometry Congruent Angles Congruent angles are angles that have the same measure DOE = ABC m DOE = m ABC ~ E O D C B A Mr. Chin-Sung Lin

ERHS Math Geometry Right Angles Perpendicular lines are two lines that intersect to form right angles AB  CD D O A C B Mr. Chin-Sung Lin

ERHS Math Geometry Adjacent Angles Adjacent angles are two angles in the same plane that have a common vertex and a common side but do not have any interior points in common AOC and COD O A C D Mr. Chin-Sung Lin

ERHS Math Geometry Vertical Angles Vertical angles are two angles in which the sides of one angle are opposite rays to the sides of the second angle AOC and BOD AOB and COD D O A C B Mr. Chin-Sung Lin

ERHS Math Geometry Complementary Angles Complementary angles are two angles the sum of whose degree measure is 90 AOB and BOC AOB and RST B T O A S R B C O A Mr. Chin-Sung Lin

ERHS Math Geometry Supplementary Angles Supplementary angles are two angles the sum of whose degree measure is 180 AOB and BOC AOB and RST B O S R C A B O A T Mr. Chin-Sung Lin

ERHS Math Geometry Linear Pair A linear pair of angles are two adjacent angles whose sum is a straight angle AOB and BOC C A B O Mr. Chin-Sung Lin

Theorems of Congruent Angle Pairs
Chapter 1 - Essential of Geometry ERHS Math Geometry Theorems of Congruent Angle Pairs Mr. Chin-Sung Lin

Theorems of Angle Pairs
L6_Congruent Angle Pairs ERHS Math Geometry Theorems of Angle Pairs Linear pair Right Angles Complementary Angles Supplementary Angles Vertical Angles Mr. Chin-Sung Lin

ERHS Math Geometry Theorem - Linear Pair If two angles form a linear pair, then these angles are supplementary Mr. Chin-Sung Lin

ERHS Math Geometry Theorem - Linear Pair If two angles form a linear pair, then these angles are supplementary Draw a diagram like the one below Given: 1 and 2 are linear pair Prove: 1 and 2 are supplementary 1 2 Mr. Chin-Sung Lin

ERHS Math Geometry Theorem - Linear Pair 1 2 Statements Reasons 1. 1 and 2 are linear pair 1. Given 2. m1 + m2 = Definition of linear pair 3. 1 and 2 are supplementary 3. Definition of supplementary angles Mr. Chin-Sung Lin

ERHS Math Geometry Theorem - Right Angles If two angles are right angles, then these angles are congruent Mr. Chin-Sung Lin

ERHS Math Geometry Theorem - Right Angles If two angles are right angles, then these angles are congruent Draw a diagram like the one below Given: 1 and 2 are right angles Prove: 1 = 2 ~ 2 1 Mr. Chin-Sung Lin

ERHS Math Geometry Theorem - Right Angles 2 1 Statements Reasons 1. 1 and 2 are right angles 1. Given 2. m1 = 90; m2 = Definition of right angle 3. m1 = m Substitution postulate 4. 1 =  Definition of congruent angles ~ Mr. Chin-Sung Lin

Theorem - Complementary Angles
L6_Congruent Angle Pairs ERHS Math Geometry Theorem - Complementary Angles If two angles are complementary to the same angle, then these angles are congruent Mr. Chin-Sung Lin

L6_Congruent Angle Pairs ERHS Math Geometry Theorem - Complementary Angles If two angles are complementary to the same angle, then these angles are congruent Draw a diagram like the one below Given: 1 and 2 are complementary 3 and 2 are complementary Prove: 1 = 3 ~ 2 3 1 Mr. Chin-Sung Lin

L6_Congruent Angle Pairs ERHS Math Geometry Theorem - Complementary Angles 2 3 1 Statements Reasons 1. 1 and 2 are complementary 1. Given 3 and 2 are complementary 2. m1 + m2 = Definition of complementary m3 + m2 = angles 3. m1 + m2 = m3 + m2 3. Substitution postulate 4. m2 = m Reflexive property 5. m1 = m Subtraction postulate 6. 1 =  Definition of congruent angles ~ Mr. Chin-Sung Lin

Theorem - Supplementary Angles
L6_Congruent Angle Pairs ERHS Math Geometry Theorem - Supplementary Angles If two angles are supplementary to the same angle, then these angles are congruent Mr. Chin-Sung Lin

L6_Congruent Angle Pairs ERHS Math Geometry Theorem - Supplementary Angles If two angles are supplementary to the same angle, then these angles are congruent Draw a diagram like the one below Given: 1 and 2 are supplementary 3 and 2 are supplementary Prove: 1 = 3 ~ 1 3 2 Mr. Chin-Sung Lin

L6_Congruent Angle Pairs ERHS Math Geometry Theorem - Supplementary Angles 1 3 2 Statements Reasons 1. 1 and 2 are supplementary 1. Given 3 and 2 are supplementary 2. m1 + m2 = Definition of supplementary m3 + m2 = angles 3. m1 + m2 = m3 + m2 3. Substitution postulate 4. m2 = m Reflexive property 5. m1 = m Subtraction postulate 6. 1 =  Definition of congruent angles ~ Mr. Chin-Sung Lin

Theorem - Vertical Angles
L6_Congruent Angle Pairs ERHS Math Geometry Theorem - Vertical Angles If two angles are vertical angles, then these angles are congruent Mr. Chin-Sung Lin

L6_Congruent Angle Pairs ERHS Math Geometry Theorem - Vertical Angles If two angles are vertical angles, then these angles are congruent Draw a diagram like the one below Given: 1 and 3 are vertical angles Prove: 1 = 3 ~ 1 3 2 4 Mr. Chin-Sung Lin

L6_Congruent Angle Pairs ERHS Math Geometry 1 3 2 4 Theorem - Vertical Angles Statements Reasons 1. 1 and 3 are vertical angles 1. Given 2. 1 and 2 are linear pair 2. Definition of vertical angle 3 and 2 are linear pair 3. m1 + m2 = Definition of linear pair m3 + m2 = 180 4. m1 + m2 = m3 + m2 4. Substitution postulate 5. m2 = m Reflexive property 6. m1 = m Subtraction postulate 7. 1 =  Definition of congruent angles ~ Mr. Chin-Sung Lin

Theorems of Angle Pairs Review
L6_Congruent Angle Pairs ERHS Math Geometry Theorems of Angle Pairs Review Linear pair Right Angles Complementary Angles Supplementary Angles Vertical Angles Mr. Chin-Sung Lin

Exercise: Theorems of Congruent Angle Pairs
Chapter 1 - Essential of Geometry ERHS Math Geometry Exercise: Theorems of Congruent Angle Pairs Mr. Chin-Sung Lin

L6_Congruent Angle Pairs ERHS Math Geometry Theorem - Complementary Angles If two angles are congruent, then their complements are congruent Mr. Chin-Sung Lin

L6_Congruent Angle Pairs ERHS Math Geometry Theorem - Supplementary Angles If two angles are congruent, then their supplements are congruent Mr. Chin-Sung Lin

ERHS Math Geometry Theorem - Right Angles If two lines intersect to form congruent angles, then they are perpendicular Mr. Chin-Sung Lin

Congruent Polygons & Congruent Triangles
ERHS Math Geometry Congruent Polygons & Congruent Triangles Mr. Chin-Sung Lin

Congruent Triangles ERHS Math Geometry Congruent Polygons Polygons are congruent if and only if there is a one-to-one correspondence between their vertices such that all corresponding sides and corresponding angles are congruent Mr. Chin-Sung Lin

Congruent Polygons ERHS Math Geometry
Congruent Triangles ERHS Math Geometry Congruent Polygons Corresponding parts of congruent polygons are congruent ABCD ≅ WXYZ A = W B = X C = Y D = Z AB ≅ WX BC ≅ XY CD ≅ YZ DA ≅ ZW Mr. Chin-Sung Lin

Congruent Triangles ERHS Math Geometry Congruent Polygons The polygons will have the same shape and size, but one may be a rotated, or be the mirror image of the other Mr. Chin-Sung Lin

Congruent Triangles ERHS Math Geometry Congruent Triangles Two triangles are congruent if the vertices of one triangle can be matched with the vertices of the other triangle such that corresponding angles are congruent and the corresponding sides are congruent B Y A C Z X Mr. Chin-Sung Lin

Congruent Triangles ERHS Math Geometry Congruent Triangles Corresponding parts of congruent triangles are congruent Corresponding parts of congruent triangles are equal in measure B Y A C Z X Mr. Chin-Sung Lin

Congruent Triangles Corresponding Angles Corresponding Sides
ERHS Math Geometry Congruent Triangles Corresponding Angles A ≅ X mA = mX B ≅ Y mB = mY C ≅ Z mC = mZ Corresponding Sides AB ≅ XY AB = XY BC ≅ YZ BC = YZ CA ≅ ZX CA = ZX ∆ ABC ≅ ∆ XYZ B Y A C Z X Mr. Chin-Sung Lin

Congruent Triangles ERHS Math Geometry Congruent Triangles Congruence can be represented by more than one way, as long as the corresponding vertices in the same order ∆ ABC ≅ ∆ XYZ ∆ BAC ≅ ∆ YXZ ∆ CAB ≅ ∆ ZXY B Y A C Z X Mr. Chin-Sung Lin

Congruent Triangles – Exercise
ERHS Math Geometry Congruent Triangles – Exercise Congruence can be represented by more than one way, as long as the corresponding vertices in the same order If ∆ OPQ  ∆ DEF ∆ POQ  ∆ FED  ∆ OQP  ∆ EFD  ∆ QOP  Mr. Chin-Sung Lin

ERHS Math Geometry Congruent Triangles – Exercise Congruence can be represented by more than one way, as long as the corresponding vertices in the same order If ∆ OPQ  ∆ DEF ∆ POQ  ∆ EDF ∆ FED  ∆ OQP  ∆ EFD  ∆ QOP  Mr. Chin-Sung Lin

ERHS Math Geometry Congruent Triangles – Exercise Congruence can be represented by more than one way, as long as the corresponding vertices in the same order If ∆ OPQ  ∆ DEF ∆ POQ  ∆ EDF ∆ FED  ∆ QPO ∆ OQP  ∆ EFD  ∆ QOP  Mr. Chin-Sung Lin

ERHS Math Geometry Congruent Triangles – Exercise Congruence can be represented by more than one way, as long as the corresponding vertices in the same order If ∆ OPQ  ∆ DEF ∆ POQ  ∆ EDF ∆ FED  ∆ QPO ∆ OQP  ∆ DFE ∆ EFD  ∆ QOP  Mr. Chin-Sung Lin

ERHS Math Geometry Congruent Triangles – Exercise Congruence can be represented by more than one way, as long as the corresponding vertices in the same order If ∆ OPQ  ∆ DEF ∆ POQ  ∆ EDF ∆ FED  ∆ QPO ∆ OQP  ∆ DFE ∆ EFD  ∆ PQO ∆ QOP  Mr. Chin-Sung Lin

ERHS Math Geometry Congruent Triangles – Exercise Congruence can be represented by more than one way, as long as the corresponding vertices in the same order If ∆ OPQ  ∆ DEF ∆ POQ  ∆ EDF ∆ FED  ∆ QPO ∆ OQP  ∆ DFE ∆ EFD  ∆ PQO ∆ QOP  ∆ FDE Mr. Chin-Sung Lin

Equivalence Relation of Congruence
Congruent Triangles ERHS Math Geometry Equivalence Relation of Congruence Mr. Chin-Sung Lin

Reflexive Property Any geometric figure is congruent to itself
Congruent Triangles ERHS Math Geometry Reflexive Property Any geometric figure is congruent to itself ∆ ABC ≅ ∆ ABC B A C Mr. Chin-Sung Lin

Symmetric Property A congruence may be expressed in either order
Congruent Triangles ERHS Math Geometry Symmetric Property A congruence may be expressed in either order If ∆ ABC ≅ ∆ XYZ then ∆ XYZ ≅ ∆ ABC B Y A C Z X Mr. Chin-Sung Lin

Congruent Triangles ERHS Math Geometry Transitive Property Two geometric figures congruent to the same geometric figure are congruent to each other If ∆ ABC ≅ ∆ RST and ∆ RST ≅ ∆ XYZ then ∆ ABC ≅ ∆ XYZ B Y S A C Z X R T Mr. Chin-Sung Lin

Postulates that Prove Triangles Congruent
Congruent Triangles ERHS Math Geometry Postulates that Prove Triangles Congruent Mr. Chin-Sung Lin

Congruent Triangles ERHS Math Geometry Postulates that Prove Triangles Congruent Side-Side-Side Congruence (SSS) Side-Angle-Side Congruence (SAS) Angle-Side-Angle Congruence (ASA) Angle-Angle-Side Congruence (AAS) Mr. Chin-Sung Lin

Side-Side-Side Congruence (SSS)
Congruent Triangles ERHS Math Geometry Side-Side-Side Congruence (SSS) If the three sides of one triangle are congruent, respectively, to the three sides of a second triangle, then two triangles are congruent Mr. Chin-Sung Lin

Side-Angle-Side Congruence (SAS)
Congruent Triangles ERHS Math Geometry Side-Angle-Side Congruence (SAS) If two sides and the included angle of one triangle are congruent, respectively, to the ones of another triangle, then two triangles are congruent Mr. Chin-Sung Lin

Angle-Side-Angle Congruence (ASA)
Congruent Triangles ERHS Math Geometry Angle-Side-Angle Congruence (ASA) If two angles and the included side of one triangle are congruent, respectively, to the ones of another triangle, then two triangles are congruent Mr. Chin-Sung Lin

Angle-Angle-Side Congruence (AAS)
Congruent Triangles ERHS Math Geometry Angle-Angle-Side Congruence (AAS) If two of corresponding angles and a not-included side are congruent, respectively, to the ones of another triangle, then the triangles are congruent Mr. Chin-Sung Lin

Side-Side-Angle Case (SSA)
Congruent Triangles ERHS Math Geometry Side-Side-Angle Case (SSA) Mr. Chin-Sung Lin

Side-Side-Angle Case (SSA)
Congruent Triangles ERHS Math Geometry Side-Side-Angle Case (SSA) The condition does not guarantee congruence, because it is possible to have two incongruent triangles. This is known as the ambiguous case Mr. Chin-Sung Lin

Angle-Angle-Angle Case (AAA)
Congruent Triangles ERHS Math Geometry Angle-Angle-Angle Case (AAA) The AAA says nothing about the size of the two triangles and hence shows only similarity and not congruence Mr. Chin-Sung Lin

Congruent Triangles ERHS Math Geometry Postulates that Prove Triangles Congruent Side-Side-Side Congruence (SSS) Side-Angle-Side Congruence (SAS) Angle-Side-Angle Congruence (ASA) Angle-Angle-Side Congruence (AAS) Side-Side-Angle Congruence (SSA) Angle-Angle-Angle Congruence (AAA) Mr. Chin-Sung Lin

Identify the Postulate
Congruent Triangles ERHS Math Geometry Identify the Postulate Mr. Chin-Sung Lin

Postulate for Proving Congruence
Congruent Triangles ERHS Math Geometry Postulate for Proving Congruence Given A ≅ X, B ≅ Y and AB ≅ XY Prove ∆ ABC ≅ ∆ XYZ A C B X Z Y Mr. Chin-Sung Lin

Congruent Triangles ERHS Math Geometry Postulate for Proving Congruence Given A ≅ X, B ≅ Y and AB ≅ XY Prove ∆ ABC ≅ ∆ XYZ ASA A C B X Z Y Mr. Chin-Sung Lin

Congruent Triangles ERHS Math Geometry Postulate for Proving Congruence Given O is the midpoint of AX and BY Prove ∆ ABO ≅ ∆ XYO A Y O B X Mr. Chin-Sung Lin

Congruent Triangles ERHS Math Geometry Postulate for Proving Congruence Given O is the midpoint of AX and BY Prove ∆ ABO ≅ ∆ XYO SAS A Y O B X Mr. Chin-Sung Lin

Congruent Triangles ERHS Math Geometry Postulate for Proving Congruence Given CA is an angle bisector of DCB, and B ≅ D Prove ∆ ACD = ∆ ACB D A C B Mr. Chin-Sung Lin

Congruent Triangles ERHS Math Geometry Postulate for Proving Congruence Given CA is an angle bisector of DCB, and B ≅ D Prove ∆ ACD = ∆ ACB AAS D A C B Mr. Chin-Sung Lin

Congruent Triangles ERHS Math Geometry Postulate for Proving Congruence Given ∆ ABC is an isosceles triangle and BD is the median Prove ∆ ABD ≅ ∆ CBD A C B D Mr. Chin-Sung Lin

Congruent Triangles ERHS Math Geometry Postulate for Proving Congruence Given ∆ ABC is an isosceles triangle and BD is the median Prove ∆ ABD ≅ ∆ CBD SSS A C B D Mr. Chin-Sung Lin

Congruent Triangles ERHS Math Geometry Postulate for Proving Congruence Given DE ≅ AE, BE ≅ CE, and 1 ≅ 2 Prove ∆ DBC ≅ ∆ ACB A D E 1 2 B C Mr. Chin-Sung Lin

Congruent Triangles ERHS Math Geometry Postulate for Proving Congruence Given DE ≅ AE, BE ≅ CE, and 1 ≅ 2 Prove ∆ DBC ≅ ∆ ACB SAS A D E 1 2 B C Mr. Chin-Sung Lin

Identify Congruent Triangles & the Postulate
ERHS Math Geometry Identify Congruent Triangles & the Postulate Mr. Chin-Sung Lin

Congruent Triangles Given: AB  XY, BC  YZ, and B  Y Prove:
ERHS Math Geometry Congruent Triangles Given: AB  XY, BC  YZ, and B  Y Prove: Z A Y C B X Mr. Chin-Sung Lin

Congruent Triangles Given: AB  XY, BC  YZ, and B  Y
ERHS Math Geometry Congruent Triangles Given: AB  XY, BC  YZ, and B  Y Prove: ∆ ABC  ∆ XYZ A B C X Y Z Mr. Chin-Sung Lin

SAS Congruent Triangles Given: AB  XY, BC  YZ, and B  Y
ERHS Math Geometry Congruent Triangles Given: AB  XY, BC  YZ, and B  Y Prove: ∆ ABC  ∆ XYZ A B C X Y Z SAS Mr. Chin-Sung Lin

Congruent Triangles Given: AB  AC, and BD  CD Prove:
ERHS Math Geometry Congruent Triangles Given: AB  AC, and BD  CD Prove: A B C D Mr. Chin-Sung Lin

Congruent Triangles Given: AB  AC, and BD  CD Prove: ∆ ABD  ∆ ACD
ERHS Math Geometry Congruent Triangles Given: AB  AC, and BD  CD Prove: ∆ ABD  ∆ ACD A B C D Mr. Chin-Sung Lin

SSS Congruent Triangles Given: AB  AC, and BD  CD
ERHS Math Geometry Congruent Triangles Given: AB  AC, and BD  CD Prove: ∆ ABD  ∆ ACD A B C D SSS Mr. Chin-Sung Lin

Congruent Triangles Given: AO  XO, and BO  YO Prove:
ERHS Math Geometry Congruent Triangles Given: AO  XO, and BO  YO Prove: A B X O Y Mr. Chin-Sung Lin

Congruent Triangles Given: AO  XO, and BO  YO Prove: ∆ AOB  ∆ XOY
ERHS Math Geometry Congruent Triangles Given: AO  XO, and BO  YO Prove: ∆ AOB  ∆ XOY A B X O Y Mr. Chin-Sung Lin

SAS Congruent Triangles Given: AO  XO, and BO  YO
ERHS Math Geometry Congruent Triangles Given: AO  XO, and BO  YO Prove: ∆ AOB  ∆ XOY A B X O Y SAS Mr. Chin-Sung Lin

Congruent Triangles Given: D  B , and DAC  BAC Prove:
ERHS Math Geometry Congruent Triangles Given: D  B , and DAC  BAC Prove: D A C B Mr. Chin-Sung Lin

Congruent Triangles Given: D  B , and DAC  BAC
ERHS Math Geometry Congruent Triangles Given: D  B , and DAC  BAC Prove: ∆ ABC  ∆ ADC A B C D Mr. Chin-Sung Lin

AAS Congruent Triangles Given: D  B , and DAC  BAC
ERHS Math Geometry Congruent Triangles Given: D  B , and DAC  BAC Prove: ∆ ABC  ∆ ADC A B C D AAS Mr. Chin-Sung Lin

Congruent Triangles Given: B  C , and AB  AC Prove:
ERHS Math Geometry Congruent Triangles Given: B  C , and AB  AC Prove: A B C D E F Mr. Chin-Sung Lin

Congruent Triangles Given: B  C , and AB  AC Prove: ∆ ABF  ∆ ACE
ERHS Math Geometry Congruent Triangles Given: B  C , and AB  AC Prove: ∆ ABF  ∆ ACE A B C D E F Mr. Chin-Sung Lin

ASA Congruent Triangles Given: B  C , and AB  AC
ERHS Math Geometry Congruent Triangles Given: B  C , and AB  AC Prove: ∆ ABF  ∆ ACE A B C D E F ASA Mr. Chin-Sung Lin

Two-Column Proof of Congruent Triangles
Chapter 1 - Essential of Geometry ERHS Math Geometry Two-Column Proof of Congruent Triangles Mr. Chin-Sung Lin

Prove Congruent Triangles
ERHS Math Geometry Prove Congruent Triangles Given: AB  AC, and BD  CD Prove: ∆ ABD  ∆ ACD A B C D Mr. Chin-Sung Lin

L6_Congruent Angle Pairs ERHS Math Geometry A B C D Prove Congruent Triangles Statements Reasons Mr. Chin-Sung Lin

L6_Congruent Angle Pairs ERHS Math Geometry A B C D Prove Congruent Triangles Statements Reasons 1. AB  AC, and BD  CD 1. Given Mr. Chin-Sung Lin

L6_Congruent Angle Pairs ERHS Math Geometry A B C D Prove Congruent Triangles Statements Reasons 1. AB  AC, and BD  CD 1. Given 2. AD  AD Reflexive property Mr. Chin-Sung Lin

L6_Congruent Angle Pairs ERHS Math Geometry A B C D Prove Congruent Triangles Statements Reasons 1. AB  AC, and BD  CD 1. Given 2. AD  AD Reflexive property 3. ∆ ABD  ∆ ACD SSS Mr. Chin-Sung Lin

Congruent Triangles Given: B  C , and AB  AC Prove: ∆ ABF  ∆ ACE
ERHS Math Geometry Congruent Triangles Given: B  C , and AB  AC Prove: ∆ ABF  ∆ ACE A B C D E F Mr. Chin-Sung Lin

L6_Congruent Angle Pairs ERHS Math Geometry A B C D E F Prove Congruent Triangles Statements Reasons Mr. Chin-Sung Lin

L6_Congruent Angle Pairs ERHS Math Geometry A B C D E F Prove Congruent Triangles Statements Reasons B  C , and AB  AC 1. Given Mr. Chin-Sung Lin

L6_Congruent Angle Pairs ERHS Math Geometry A B C D E F Prove Congruent Triangles Statements Reasons B  C , and AB  AC 1. Given 2. A  A Reflexive property Mr. Chin-Sung Lin

L6_Congruent Angle Pairs ERHS Math Geometry A B C D E F Prove Congruent Triangles Statements Reasons B  C , and AB  AC 1. Given 2. A  A Reflexive property 3. ∆ ABF  ∆ ACE ASA Mr. Chin-Sung Lin

ERHS Math Geometry Prove Congruent Triangles Given: ∆ ABC, AD is the bisector of BC, and AD  BC Prove: ∆ ABD  ∆ ACD A B C D Mr. Chin-Sung Lin

L6_Congruent Angle Pairs ERHS Math Geometry A B C D Prove Congruent Triangles Statements Reasons Mr. Chin-Sung Lin

L6_Congruent Angle Pairs ERHS Math Geometry A B C D Prove Congruent Triangles Statements Reasons ∆ ABC, AD is the bisector of BC, 1. Given and AD  BC Mr. Chin-Sung Lin

L6_Congruent Angle Pairs ERHS Math Geometry A B C D Prove Congruent Triangles Statements Reasons ∆ ABC, AD is the bisector of BC, 1. Given and AD  BC 2. AD  AD Reflexive property Mr. Chin-Sung Lin

L6_Congruent Angle Pairs ERHS Math Geometry A B C D Prove Congruent Triangles Statements Reasons ∆ ABC, AD is the bisector of BC, 1. Given and AD  BC 2. AD  AD Reflexive property D is the midpoint of BC 3. Definition of segment bisector Mr. Chin-Sung Lin

L6_Congruent Angle Pairs ERHS Math Geometry A B C D Prove Congruent Triangles Statements Reasons ∆ ABC, AD is the bisector of BC, 1. Given and AD  BC 2. AD  AD Reflexive property D is the midpoint of BC 3. Definition of segment bisector BD  DC Definition of midpoint Mr. Chin-Sung Lin

L6_Congruent Angle Pairs ERHS Math Geometry A B C D Prove Congruent Triangles Statements Reasons ∆ ABC, AD is the bisector of BC, 1. Given and AD  BC 2. AD  AD Reflexive property D is the midpoint of BC 3. Definition of segment bisector BD  DC Definition of midpoint ADB and ADC are right 5. Definition of perpendicular angles lines Mr. Chin-Sung Lin

L6_Congruent Angle Pairs ERHS Math Geometry A B C D Prove Congruent Triangles Statements Reasons ∆ ABC, AD is the bisector of BC, 1. Given and AD  BC 2. AD  AD Reflexive property D is the midpoint of BC 3. Definition of segment bisector BD  DC Definition of midpoint ADB and ADC are right 5. Definition of perpendicular angles lines ADB  ADC 6. Right angles are congruent Mr. Chin-Sung Lin

L6_Congruent Angle Pairs ERHS Math Geometry A B C D Prove Congruent Triangles Statements Reasons ∆ ABC, AD is the bisector of BC, 1. Given and AD  BC 2. AD  AD Reflexive property D is the midpoint of BC 3. Definition of segment bisector BD  DC Definition of midpoint ADB and ADC are right 5. Definition of perpendicular angles lines ADB  ADC 6. Right angles are congruent ∆ ABD  ∆ ACD SAS Mr. Chin-Sung Lin

Congruent Triangles Given: O is the midpoint of AX, and B  Y
ERHS Math Geometry Congruent Triangles Given: O is the midpoint of AX, and B  Y Prove: ∆ AOB  ∆ XOY A B X O Y Mr. Chin-Sung Lin

L6_Congruent Angle Pairs ERHS Math Geometry A B X O Y Prove Congruent Triangles Statements Reasons Mr. Chin-Sung Lin

L6_Congruent Angle Pairs ERHS Math Geometry A B X O Y Prove Congruent Triangles Statements Reasons O is the midpoint of AX, and 1. Given B  Y Mr. Chin-Sung Lin

L6_Congruent Angle Pairs ERHS Math Geometry A B X O Y Prove Congruent Triangles Statements Reasons O is the midpoint of AX, and 1. Given B  Y 2. AO  XO Definition of midpoint Mr. Chin-Sung Lin

L6_Congruent Angle Pairs ERHS Math Geometry A B X O Y Prove Congruent Triangles Statements Reasons O is the midpoint of AX, and 1. Given B  Y 2. AO  XO Definition of midpoint AOB  XOY Vertical angle theorem Mr. Chin-Sung Lin

L6_Congruent Angle Pairs ERHS Math Geometry A B X O Y Prove Congruent Triangles Statements Reasons O is the midpoint of AX, and 1. Given B  Y 2. AO  XO Definition of midpoint AOB  XOY Vertical angle theorem ∆ AOB  ∆ XOY AAS Mr. Chin-Sung Lin

Chapter 1 - Essential of Geometry
ERHS Math Geometry Q & A Mr. Chin-Sung Lin

Chapter 1 - Essential of Geometry
ERHS Math Geometry The End Mr. Chin-Sung Lin

Congruence of Line Segments, Angles, and Triangles

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Congruence of Line Segments, Angles, and Triangles

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Presentation on theme: "Congruence of Line Segments, Angles, and Triangles"— Presentation transcript:

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