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Congruence of Line Segments, Angles, and Triangles Eleanor Roosevelt High School Geometry Mr. Chin-Sung Lin ERHS Math Geometry Postulates of Lines, Line Segments, and Angles Mr. Chin-Sung Lin 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 B D Mr. Chin-Sung Lin 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 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 C M B A D Mr. Chin-Sung Lin 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 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 P A B Mr. Chin-Sung Lin 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 P A B Mr. Chin-Sung Lin 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 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 B A C Mr. Chin-Sung Lin 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 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 D B C Mr. Chin-Sung Lin ERHS Math Geometry Conditional Statements and Proof Mr. Chin-Sung Lin 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 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 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 D Prove: BD AC A B C Mr. Chin-Sung Lin 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 ERHS Math Geometry Rewrite the Conditionals for Proof If a triangle is equilateral, then the measures of the sides are equal A Given: ΔABC is equilateral Prove: AB = BC = CA C B Mr. Chin-Sung Lin ERHS Math Geometry Using Postulates and Definitions in Proofs Mr. Chin-Sung Lin ERHS Math Geometry Two Column Proof Example Given: DR is the bisector of ADC, 3 C ≅ 1 and 4 ≅ 2. B Proof: Statements 3 4 Prove: 3 ≅ 4 R 1 D 2 A Reasons Mr. Chin-Sung Lin ERHS Math Geometry Two Column Proof Example Given: DR is the bisector of ADC, 3 C ≅ 1 and 4 ≅ 2. B Proof: Statements 1.DR is the bisector of ABC. 3 ≅ 1 and 4 ≅ 2 3 4 Prove: 3 ≅ 4 R 1 D 2 A Reasons 1. Given. Mr. Chin-Sung Lin ERHS Math Geometry Two Column Proof Example Given: DR is the bisector of ADC, 3 C ≅ 1 and 4 ≅ 2. 3 4 Prove: 3 ≅ 4 B Proof: Statements R 1 D 2 A Reasons 1.DR is the bisector of ABC. 3 ≅ 1 and 4 ≅ 2 1. Given. 2.1 ≅ 2 2. Definition of angle bisector. Mr. Chin-Sung Lin ERHS Math Geometry Two Column Proof Example Given: DR is the bisector of ADC, 3 C ≅ 1 and 4 ≅ 2. 3 4 Prove: 3 ≅ 4 B Proof: Statements R 1 D 2 A Reasons 1.DR is the bisector of ABC. 3 ≅ 1 and 4 ≅ 2 1. Given. 2. 1 ≅ 2 2. Definition of angle bisector. 3. 3 ≅ 4 3. Substitution postulate. Mr. Chin-Sung Lin ERHS Math Geometry Two Column Proof Exercise Given: M is the midpoint of AB. M A B Prove: AM = ½ AB and MB = ½ AB Proof: Statements Reasons Mr. Chin-Sung Lin ERHS Math Geometry Two Column Proof Exercise Given: M is the midpoint of AB. M A B Prove: AM = ½ AB and MB = ½ AB Proof: Statements 1.M is the midpoint of AB. Reasons 1. Given. Mr. Chin-Sung Lin ERHS Math Geometry Two Column Proof Exercise Given: M is the midpoint of AB. M A B Prove: AM = ½ AB and MB = ½ AB Proof: Statements 1.M is the midpoint of AB. 2.AM ≅ MB Reasons 1. Given. 2. Definition of midpoint. Mr. Chin-Sung Lin ERHS Math Geometry Two Column Proof Exercise Given: M is the midpoint of AB. M A B Prove: AM = ½ AB and MB = ½ AB Proof: Statements 1.M is the midpoint of AB. 2.AM ≅ MB 3.AM = MB Reasons 1. Given. 2. Definition of midpoint. 3. Definition of congruent segments. Mr. Chin-Sung Lin ERHS Math Geometry Two Column Proof Exercise Given: M is the midpoint of AB. M A B Prove: AM = ½ AB and MB = ½ AB Proof: Statements 1.M is the midpoint of AB. 2.AM ≅ MB Reasons 1. Given. 2. Definition of midpoint. 3.AM = MB 3. Definition of congruent segments. 4.AM + MB = AB 4. Partition postulate. Mr. Chin-Sung Lin ERHS Math Geometry Two Column Proof Exercise Given: M is the midpoint of AB. M A B Prove: AM = ½ AB and MB = ½ AB Proof: Statements 1.M is the midpoint of AB. 2.AM ≅ MB Reasons 1. Given. 2. Definition of midpoint. 3.AM = MB 3. Definition of congruent segments. 4.AM + MB = AB 4. Partition postulate. 5.2AM = AB and 2 MB = AB 5. Substitution postulate. Mr. Chin-Sung Lin ERHS Math Geometry Two Column Proof Exercise Given: M is the midpoint of AB. M A B Prove: AM = ½ AB and MB = ½ AB Proof: Statements 1.M is the midpoint of AB. 2.AM ≅ MB Reasons 1. Given. 2. Definition of midpoint. 3.AM = MB 3. Definition of congruent segments. 4.AM + MB = AB 4. Partition postulate. 5.2AM = AB and 2 MB = AB 5. Substitution postulate. 6.AM = ½ AB and MB = ½ AB 6. Division postulate. Mr. Chin-Sung Lin ERHS Math Geometry Angles & Angle Pairs Mr. Chin-Sung Lin ERHS Math Geometry Congruent Angles Congruent angles are angles that have the same measure ~ ABC DOE = m DOE = m ABC D O A E B C Mr. Chin-Sung Lin ERHS Math Geometry Right Angles Perpendicular lines are two lines that intersect to form right angles AB CD A O C D 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 A C O 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 A C O B D 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 A O A B O C B S R T 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 B A O A O S R T C 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 B A O C Mr. Chin-Sung Lin ERHS Math Geometry Theorems of Congruent Angle Pairs Mr. Chin-Sung Lin 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 Statements 2 Reasons 1. 1 and 2 are linear pair 1. Given 2. m1 + m2 = 180 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 ~ 2 Prove: 1 = 1 2 Mr. Chin-Sung Lin ERHS Math Geometry Theorem - Right Angles 1 Statements 2 Reasons 1. 1 and 2 are right angles 1. Given 2. m1 = 90; m2 = 90 2. Definition of right angle 3. m1 = m2 3. Substitution postulate 4. 1 ~ = 2 4. Definition of congruent angles Mr. Chin-Sung Lin ERHS Math Geometry Theorem - Complementary Angles If two angles are complementary to the same angle, then these angles are congruent Mr. Chin-Sung Lin 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 1 2 3 Mr. Chin-Sung Lin ERHS Math Geometry Theorem - Complementary Angles 1 2 Statements 1. 1 and 2 are complementary 3 and 2 are complementary 2. m1 + m2 = 90 m3 + m2 = 90 3. m1 + m2 = m3 + m2 4. m2 = m2 5. m1 = m3 ~ 3 6. 1 = 3 Reasons 1. Given 2. Definition of complementary angles 3. Substitution postulate 4. Reflexive property 5. Subtraction postulate 6. Definition of congruent angles Mr. Chin-Sung Lin ERHS Math Geometry Theorem - Supplementary Angles If two angles are supplementary to the same angle, then these angles are congruent Mr. Chin-Sung Lin 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 ~ 3 Prove: 1 = 2 1 3 Mr. Chin-Sung Lin ERHS Math Geometry Theorem - Supplementary Angles 2 1 1. 2. 3. 4. 5. 6. Statements 1 and 2 are supplementary 3 and 2 are supplementary m1 + m2 = 180 m3 + m2 = 180 m1 + m2 = m3 + m2 m2 = m2 m1 = m3 ~ 3 1 = 3 Reasons 1. Given 2. Definition of supplementary angles 3. Substitution postulate 4. Reflexive property 5. Subtraction postulate 6. Definition of congruent angles Mr. Chin-Sung Lin ERHS Math Geometry Theorem - Vertical Angles If two angles are vertical angles, then these angles are congruent Mr. Chin-Sung Lin 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 2 3 1 4 Mr. Chin-Sung Lin ERHS Math Geometry 2 Theorem - Vertical Angles 3 1 4 Statements 1. 1 and 3 are vertical angles 2. 1 and 2 are linear pair 3 and 2 are linear pair 3. m1 + m2 = 180 m3 + m2 = 180 4. m1 + m2 = m3 + m2 5. m2 = m2 6. m1 = m3 ~ 3 7. 1 = Reasons 1. Given 2. Definition of vertical angle 3. Definition of linear pair 4. 5. 6. 7. Substitution postulate Reflexive property Subtraction postulate Definition of congruent angles Mr. Chin-Sung Lin ERHS Math Geometry Theorems of Angle Pairs Review Linear pair Right Angles Complementary Angles Supplementary Angles Vertical Angles Mr. Chin-Sung Lin ERHS Math Geometry Exercise: Theorems of Congruent Angle Pairs Mr. Chin-Sung Lin ERHS Math Geometry Theorem - Complementary Angles If two angles are congruent, then their complements are congruent Mr. Chin-Sung Lin 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 ERHS Math Geometry Congruent Polygons & Congruent Triangles Mr. Chin-Sung Lin ERHS Math Geometry Congruent Polygons Polygons are congruent if and only if there is a one-toone correspondence between their vertices such that all corresponding sides and corresponding angles are congruent Mr. Chin-Sung Lin ERHS Math Geometry Congruent Polygons Corresponding parts of congruent polygons are congruent ABCD ≅ WXYZ AB ≅ WX A = W B = X C = Y D = Z BC ≅ XY CD ≅ YZ DA ≅ ZW Mr. Chin-Sung Lin 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 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 A Y C Z X Mr. Chin-Sung Lin ERHS Math Geometry Congruent Triangles Corresponding parts of congruent triangles are congruent Corresponding parts of congruent triangles are equal in measure B A Y C Z X Mr. Chin-Sung Lin ERHS Math Geometry Congruent Triangles Corresponding Sides Corresponding Angles A ≅ X B ≅ Y C ≅ Z mA = mX mB = mY mC = mZ AB ≅ XY BC ≅ YZ CA ≅ ZX AB = XY BC = YZ CA = ZX B ∆ ABC ≅ ∆ XYZ Y A C X Z Mr. Chin-Sung Lin ERHS Math Geometry Congruent Triangles Congruence can be represented by more than one way, as long as the corresponding vertices in the same order B A ∆ ABC ≅ ∆ XYZ ∆ BAC ≅ ∆ YXZ ∆ CAB ≅ ∆ ZXY C Z Y X 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 ∆ 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 ERHS Math Geometry Equivalence Relation of Congruence Mr. Chin-Sung Lin ERHS Math Geometry Reflexive Property Any geometric figure is congruent to itself B A ∆ ABC ≅ ∆ ABC C Mr. Chin-Sung Lin ERHS Math Geometry Symmetric Property A congruence may be expressed in either order B A If ∆ ABC ≅ ∆ XYZ then ∆ XYZ ≅ ∆ ABC C Z Y X Mr. Chin-Sung Lin ERHS Math Geometry Transitive Property Two geometric figures congruent to the same geometric figure are congruent to each other B A If ∆ ABC ≅ ∆ RST and ∆ RST ≅ ∆ XYZ then ∆ ABC ≅ ∆ XYZ C S R Y Z X T Mr. Chin-Sung Lin ERHS Math Geometry Postulates that Prove Triangles Congruent Mr. Chin-Sung Lin 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 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 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 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 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 ERHS Math Geometry Side-Side-Angle Case (SSA) Mr. Chin-Sung Lin 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 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 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 ERHS Math Geometry Identify the Postulate Mr. Chin-Sung Lin ERHS Math Geometry Postulate for Proving Congruence Given A ≅ X, B ≅ Y and AB ≅ XY Prove ∆ ABC ≅ ∆ XYZ B A Y X C Z Mr. Chin-Sung Lin ERHS Math Geometry Postulate for Proving Congruence Given A ≅ X, B ≅ Y and AB ≅ XY Prove ∆ ABC ≅ ∆ XYZ ASA B A Y X C Z Mr. Chin-Sung Lin ERHS Math Geometry Postulate for Proving Congruence Given O is the midpoint of AX and BY Prove ∆ ABO ≅ ∆ XYO A B Y O X Mr. Chin-Sung Lin ERHS Math Geometry Postulate for Proving Congruence Given O is the midpoint of AX and BY Prove ∆ ABO ≅ ∆ XYO SAS A B Y O X Mr. Chin-Sung Lin 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 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 ERHS Math Geometry Postulate for Proving Congruence Given ∆ ABC is an isosceles triangle and BD is the median Prove ∆ ABD ≅ ∆ CBD B A D C Mr. Chin-Sung Lin ERHS Math Geometry Postulate for Proving Congruence Given ∆ ABC is an isosceles triangle and BD is the median Prove ∆ ABD ≅ ∆ CBD SSS B A D C Mr. Chin-Sung Lin ERHS Math Geometry Postulate for Proving Congruence Given DE ≅ AE, BE ≅ CE, and 1 ≅ 2 Prove ∆ DBC ≅ ∆ ACB A D E B 1 2 C Mr. Chin-Sung Lin ERHS Math Geometry Postulate for Proving Congruence Given DE ≅ AE, BE ≅ CE, and 1 ≅ 2 Prove ∆ DBC ≅ ∆ ACB A D SAS E B 1 2 C Mr. Chin-Sung Lin ERHS Math Geometry Identify Congruent Triangles & the Postulate Mr. Chin-Sung Lin ERHS Math Geometry Congruent Triangles Given: AB XY, BC YZ, and B Y Prove: A B Z Y C X Mr. Chin-Sung Lin ERHS Math Geometry Congruent Triangles Given: AB XY, BC YZ, and B Y Prove: ∆ ABC ∆ XYZ A B Z Y C X Mr. Chin-Sung Lin ERHS Math Geometry Congruent Triangles Given: AB XY, BC YZ, and B Y Prove: ∆ ABC ∆ XYZ A Z Y SAS B C X Mr. Chin-Sung Lin ERHS Math Geometry Congruent Triangles Given: AB AC, and BD CD Prove: A B D C Mr. Chin-Sung Lin ERHS Math Geometry Congruent Triangles Given: AB AC, and BD CD Prove: ∆ ABD ∆ ACD A B D C Mr. Chin-Sung Lin ERHS Math Geometry Congruent Triangles Given: AB AC, and BD CD Prove: ∆ ABD ∆ ACD A SSS B D C Mr. Chin-Sung Lin ERHS Math Geometry Congruent Triangles Given: AO XO, and BO YO Prove: A B Y O X Mr. Chin-Sung Lin ERHS Math Geometry Congruent Triangles Given: AO XO, and BO YO Prove: ∆ AOB ∆ XOY A B Y O X Mr. Chin-Sung Lin ERHS Math Geometry Congruent Triangles Given: AO XO, and BO YO Prove: ∆ AOB ∆ XOY A SAS B Y O X Mr. Chin-Sung Lin ERHS Math Geometry Congruent Triangles Given: D B , and DAC BAC Prove: D A C B Mr. Chin-Sung Lin ERHS Math Geometry Congruent Triangles Given: D B , and DAC BAC Prove: ∆ ABC ∆ ADC D A C B Mr. Chin-Sung Lin ERHS Math Geometry Congruent Triangles Given: D B , and DAC BAC Prove: ∆ ABC ∆ ADC D AAS A C B Mr. Chin-Sung Lin ERHS Math Geometry Congruent Triangles Given: B C , and AB AC Prove: B E D A F C Mr. Chin-Sung Lin ERHS Math Geometry Congruent Triangles Given: B C , and AB AC Prove: ∆ ABF ∆ ACE B E D A F C Mr. Chin-Sung Lin ERHS Math Geometry Congruent Triangles Given: B C , and AB AC Prove: ∆ ABF ∆ ACE B E ASA D A F C Mr. Chin-Sung Lin ERHS Math Geometry Two-Column Proof of Congruent Triangles Mr. Chin-Sung Lin ERHS Math Geometry Prove Congruent Triangles Given: AB AC, and BD CD Prove: ∆ ABD ∆ ACD A B D C Mr. Chin-Sung Lin ERHS Math Geometry Prove Congruent Triangles Given: AB AC, and BD CD Prove: ∆ ABD ∆ ACD A B D C Mr. Chin-Sung Lin ERHS Math Geometry A Prove Congruent Triangles B Statements D C Reasons Mr. Chin-Sung Lin ERHS Math Geometry A Prove Congruent Triangles B Statements 1. AB AC, and BD CD D C Reasons 1. Given Mr. Chin-Sung Lin ERHS Math Geometry A Prove Congruent Triangles B Statements D C Reasons 1. AB AC, and BD CD 1. Given 2. AD AD 2. Reflexive property Mr. Chin-Sung Lin ERHS Math Geometry A Prove Congruent Triangles B Statements D C Reasons 1. AB AC, and BD CD 1. Given 2. AD AD 2. Reflexive property 3. ∆ ABD ∆ ACD 3. SSS Mr. Chin-Sung Lin ERHS Math Geometry Congruent Triangles Given: B C , and AB AC Prove: ∆ ABF ∆ ACE B E D A F C Mr. Chin-Sung Lin ERHS Math Geometry Congruent Triangles Given: B C , and AB AC Prove: ∆ ABF ∆ ACE B E D A F C Mr. Chin-Sung Lin ERHS Math Geometry B Prove Congruent Triangles E D A F Statements C Reasons Mr. Chin-Sung Lin ERHS Math Geometry B Prove Congruent Triangles E D A F Statements 1. B C , and AB AC C Reasons 1. Given Mr. Chin-Sung Lin ERHS Math Geometry B Prove Congruent Triangles E D A F Statements C Reasons 1. B C , and AB AC 1. Given 2. A A 2. Reflexive property Mr. Chin-Sung Lin ERHS Math Geometry B Prove Congruent Triangles E D A F Statements C Reasons 1. B C , and AB AC 1. Given 2. A A 2. Reflexive property 3. ∆ ABF ∆ ACE 3. 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 D C 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 D C Mr. Chin-Sung Lin ERHS Math Geometry A Prove Congruent Triangles B Statements D C Reasons Mr. Chin-Sung Lin ERHS Math Geometry A Prove Congruent Triangles B Statements 1. ∆ ABC, AD is the bisector of BC, and AD D C Reasons 1. Given BC Mr. Chin-Sung Lin ERHS Math Geometry A Prove Congruent Triangles B Statements D C Reasons 1. ∆ ABC, AD is the bisector of BC, 1. Given 2. and AD AD AD 2. Reflexive property BC Mr. Chin-Sung Lin ERHS Math Geometry A Prove Congruent Triangles B Statements D C Reasons 1. ∆ ABC, AD is the bisector of BC, 1. Given 2. and AD AD AD 2. Reflexive property 3. D is the midpoint of BC BC 3. Definition of segment bisector Mr. Chin-Sung Lin ERHS Math Geometry A Prove Congruent Triangles B Statements D C Reasons 1. ∆ ABC, AD is the bisector of BC, 1. Given 2. and AD AD AD 2. Reflexive property 3. 4. D is the midpoint of BC BD DC BC 3. Definition of segment bisector 4. Definition of midpoint Mr. Chin-Sung Lin ERHS Math Geometry A Prove Congruent Triangles B Statements D C Reasons 1. ∆ ABC, AD is the bisector of BC, 1. Given 2. and AD AD AD 2. Reflexive property 3. 4. D is the midpoint of BC BD DC 3. Definition of segment bisector 4. Definition of midpoint 5. ADB and ADC are right 5. Definition of perpendicular angles BC lines Mr. Chin-Sung Lin ERHS Math Geometry A Prove Congruent Triangles B Statements D C Reasons 1. ∆ ABC, AD is the bisector of BC, 1. Given 2. and AD AD AD 2. Reflexive property 3. 4. D is the midpoint of BC BD DC 3. Definition of segment bisector 4. Definition of midpoint 5. ADB and ADC are right 5. Definition of perpendicular 6. angles ADB ADC lines 6. Right angles are congruent BC Mr. Chin-Sung Lin ERHS Math Geometry A Prove Congruent Triangles B Statements D C Reasons 1. ∆ ABC, AD is the bisector of BC, 1. Given 2. and AD AD AD 2. Reflexive property 3. 4. D is the midpoint of BC BD DC 3. Definition of segment bisector 4. Definition of midpoint 5. ADB and ADC are right 5. Definition of perpendicular 6. 7. angles ADB ADC ∆ ABD ∆ ACD lines 6. Right angles are congruent 7. SAS Mr. Chin-Sung Lin BC ERHS Math Geometry Congruent Triangles Given: O is the midpoint of AX, and B Y Prove: ∆ AOB ∆ XOY A B Y O X Mr. Chin-Sung Lin ERHS Math Geometry Congruent Triangles Given: O is the midpoint of AX, and B Y Prove: ∆ AOB ∆ XOY A B Y O X Mr. Chin-Sung Lin ERHS Math Geometry A Prove Congruent Triangles O B Statements Y X Reasons Mr. Chin-Sung Lin ERHS Math Geometry A Prove Congruent Triangles O B Statements 1. O is the midpoint of AX, and B Y Y X Reasons 1. Given Mr. Chin-Sung Lin ERHS Math Geometry A Prove Congruent Triangles O B Statements 1. 2. O is the midpoint of AX, and B Y AO XO Y X Reasons 1. Given 2. Definition of midpoint Mr. Chin-Sung Lin ERHS Math Geometry A Prove Congruent Triangles O B Statements Y X Reasons 1. Given 2. O is the midpoint of AX, and B Y AO XO 3. AOB XOY 3. Vertical angle theorem 1. 2. Definition of midpoint Mr. Chin-Sung Lin ERHS Math Geometry A Prove Congruent Triangles O B Statements Y X Reasons 1. Given 2. O is the midpoint of AX, and B Y AO XO 3. 4. AOB XOY ∆ AOB ∆ XOY 3. Vertical angle theorem 4. AAS 1. 2. Definition of midpoint Mr. Chin-Sung Lin ERHS Math Geometry Q&A Mr. Chin-Sung Lin ERHS Math Geometry The End Mr. Chin-Sung Lin
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