Congruence Based on Triangles Eleanor Roosevelt High School Geometry Mr. Chin-Sung Lin.

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Presentation transcript:

Congruence Based on Triangles Eleanor Roosevelt High School Geometry Mr. Chin-Sung Lin

Line Segments Associated with Triangles ERHS Math Geometry Mr. Chin-Sung Lin

Altitude of a Triangle ERHS Math Geometry Mr. Chin-Sung Lin An altitude of a triangle is a line segment drawn from any vertex of the triangle, perpendicular to and ending in the line that contains the opposite side A C B C A B AC B

Altitude of a Triangle ERHS Math Geometry Mr. Chin-Sung Lin If BD is the altitude of ∆ ABC then, m  BDA = 90 m  BDC = 90 C A B D

Altitude - Area of a Triangle ERHS Math Geometry Mr. Chin-Sung Lin Altitudes can be used to compute the area of a triangle: A C B C A B AC B Base Altitude Base Altitude Base Area = 1/2 * Base * Altitude

Altitude - Orthocenter ERHS Math Geometry Mr. Chin-Sung Lin Three altitudes intersect in a single point, called the orthocenter of the triangle C Orthocenter A B

Altitude - Orthocenter ERHS Math Geometry Mr. Chin-Sung Lin Where is the orthocenter of a right triangle? Orthocenter? AC B

Altitude - Orthocenter ERHS Math Geometry Mr. Chin-Sung Lin The orthocenter is located at the vertex of the right angle Orthocenter A C B

Altitude - Orthocenter ERHS Math Geometry Mr. Chin-Sung Lin Where is the orthocenter of an obtuse triangle? Orthocenter? C B A

Altitude - Orthocenter ERHS Math Geometry Mr. Chin-Sung Lin Orthocenter C B A The orthocenter is outside the triangle

Angle Bisector of a Triangle ERHS Math Geometry Mr. Chin-Sung Lin

Angle Bisector of a Triangle ERHS Math Geometry Mr. Chin-Sung Lin A line segment that bisects an angle of the triangle and terminates in the side opposite that angle A C B AC B C A B

Angle Bisector of a Triangle ITHS Math B Term 1 (M$4) Mr. Chin-Sung Lin If BD is the angle bisector of  ABC then,  ABD   CBD AC B D

Angle Bisector - Incenter ERHS Math Geometry Mr. Chin-Sung Lin The three angle bisectors of a triangle meet in one point called the incenter A B Incenter C

Angle Bisector - Incenter ERHS Math Geometry Mr. Chin-Sung Lin Incenter is the center of the incircle, the circle inscribed in the triangle A B Incenter C

Median of a Triangle ERHS Math Geometry Mr. Chin-Sung Lin

Median of a Triangle ERHS Math Geometry Mr. Chin-Sung Lin A segment from a vertex to the midpoint of the opposite side A C B AC B C A B

Median of a Triangle ITHS Math B Term 1 (M$4) Mr. Chin-Sung Lin If BD is the median of ∆ ABC then, AD  CD AC B D

Median of a Triangle - Centroid ERHS Math Geometry Mr. Chin-Sung Lin The three medians meet in the centroid or center of mass (center of gravity) A B Centroid C

Median of a Triangle - Centroid ERHS Math Geometry Mr. Chin-Sung Lin The centroid divides each median in a ratio of 2:1. A B Centroid C 2 1

Perpendicular Bisector of a Triangle ERHS Math Geometry Mr. Chin-Sung Lin

Perpendicular Bisector ERHS Math Geometry Mr. Chin-Sung Lin The perpendicular bisector of a line segment is a line, a ray, or a line segment that is perpendicular to the line segment at its midpoint AB  CD CO = OD D O A C B ~

Perpendicular Bisector of a Triangle ERHS Math Geometry Mr. Chin-Sung Lin A line, a ray, or a line segment that is perpendicular to the side of a triangle at its midpoint A C B AC B C A B

Perpendicular Bisector of a Triangle ERHS Math Geometry Mr. Chin-Sung Lin If DE is the perpendicular bisector of the side of ∆ ABC then, AD  CD DE  AC AC B D E

Perpendicular Bisector - Circumcenter ERHS Math Geometry Mr. Chin-Sung Lin The three perpendicular bisectors meet in one point called the circumcenter A B Circumcenter C

Perpendicular Bisector - Circumcenter ERHS Math Geometry Mr. Chin-Sung Lin Circumcenter is the center of the circumcircle, the circle passing through the vertices of the triangle A B Circumcenter C

Scalene Triangle ERHS Math Geometry Mr. Chin-Sung Lin In a scalene triangle, the altitude, angle bisector, median drawn from any common vertex, and the perpendicular bisector of the opposite side are four distinct line segments A B C E DF BD: Altitude BE: Angle bisector BF: Median FG:Perpendicular Bisector G

Isosceles & Equilateral Triangles ERHS Math Geometry Mr. Chin-Sung Lin In isosceles & equilateral triangles, some of the altitude, angle bisector, median, and perpendicular bisector coincide C A B D BD: Altitude BD: Angle bisector BD: Median BD:Perpendicular Bisector

Scalene Triangle (Indirect Proof) ERHS Math Geometry Mr. Chin-Sung Lin Given: ∆ ABC is scalene, BD bisects ABC Prove: BD is not perpendicular to AC A B C D 12 34

Scalene Triangle ERHS Math Geometry Mr. Chin-Sung Lin StatementsReasons A B C D 12 34

Scalene Triangle ERHS Math Geometry Mr. Chin-Sung Lin StatementsReasons 1. BD  AC 1. Assume the opposite is true A B C D 12 34

Scalene Triangle ERHS Math Geometry Mr. Chin-Sung Lin StatementsReasons 1. BD  AC 1. Assume the opposite is true 2. ∆ ABC is scalene, BD is angle 2. Given bisector A B C D 12 34

Scalene Triangle ERHS Math Geometry Mr. Chin-Sung Lin StatementsReasons 1. BD  AC 1. Assume the opposite is true 2. ∆ ABC is scalene, BD is angle 2. Given bisector 3. 1  2 3. Definition of angle bisector A B C D 12 34

Scalene Triangle ERHS Math Geometry Mr. Chin-Sung Lin StatementsReasons 1. BD  AC 1. Assume the opposite is true 2. ∆ ABC is scalene, BD is angle 2. Given bisector 3. 1  2 3. Definition of angle bisector 4. 3 = 90 o, 4 = 90 o 4. Definition of perpendicular A B C D 12 34

Scalene Triangle ERHS Math Geometry Mr. Chin-Sung Lin StatementsReasons 1. BD  AC 1. Assume the opposite is true 2. ∆ ABC is scalene, BD is angle 2. Given bisector 3. 1  2 3. Definition of angle bisector 4. 3 = 90 o, 4 = 90 o 4. Definition of perpendicular 5. 3  4 5. Substitution postulate A B C D 12 34

Scalene Triangle ERHS Math Geometry Mr. Chin-Sung Lin StatementsReasons 1. BD  AC 1. Assume the opposite is true 2. ∆ ABC is scalene, BD is angle 2. Given bisector 3. 1  2 3. Definition of angle bisector 4. 3 = 90 o, 4 = 90 o 4. Definition of perpendicular 5. 3  4 5. Substitution postulate 6. BD  BD 6. Reflexive property A B C D 12 34

Scalene Triangle ERHS Math Geometry Mr. Chin-Sung Lin StatementsReasons 1. BD  AC 1. Assume the opposite is true 2. ∆ ABC is scalene, BD is angle 2. Given bisector 3. 1  2 3. Definition of angle bisector 4. 3 = 90 o, 4 = 90 o 4. Definition of perpendicular 5. 3  4 5. Substitution postulate 6. BD  BD 6. Reflexive property 7. ∆ ABD  ∆ CBD 7. ASA postulate A B C D 12 34

Scalene Triangle ERHS Math Geometry Mr. Chin-Sung Lin StatementsReasons 1. BD  AC 1. Assume the opposite is true 2. ∆ ABC is scalene, BD is angle 2. Given bisector 3. 1  2 3. Definition of angle bisector 4. 3 = 90 o, 4 = 90 o 4. Definition of perpendicular 5. 3  4 5. Substitution postulate 6. BD  BD 6. Reflexive property 7. ∆ ABD  ∆ CBD 7. ASA postulate 8. AB = CB 8. CPCTC A B C D 12 34

Scalene Triangle ERHS Math Geometry Mr. Chin-Sung Lin StatementsReasons 1. BD  AC 1. Assume the opposite is true 2. ∆ ABC is scalene, BD is angle 2. Given bisector 3. 1  2 3. Definition of angle bisector 4. 3 = 90 o, 4 = 90 o 4. Definition of perpendicular 5. 3  4 5. Substitution postulate 6. BD  BD 6. Reflexive property 7. ∆ ABD  ∆ CBD 7. ASA postulate 8. AB = CB 8. CPCTC 9. AB ≠ CB 9. Definition of scalene triangle A B C D 12 34

Scalene Triangle ERHS Math Geometry Mr. Chin-Sung Lin StatementsReasons 1. BD  AC 1. Assume the opposite is true 2. ∆ ABC is scalene, BD is angle 2. Given bisector 3. 1  2 3. Definition of angle bisector 4. 3 = 90 o, 4 = 90 o 4. Definition of perpendicular 5. 3  4 5. Substitution postulate 6. BD  BD 6. Reflexive property 7. ∆ ABD  ∆ CBD 7. ASA postulate 8. AB = CB 8. CPCTC 9. AB ≠ CB 9. Definition of scalene triangle 10. BD is not perpendicular to AC10. Contradition in statement 8 & 9, so, assumption is false. The negation of the assumption is true A B C D 12 34

CPCTC ERHS Math Geometry Mr. Chin-Sung Lin

CPCTC ERHS Math Geometry Mr. Chin-Sung Lin Corresponding Parts of Congruent Triangles are Congruent After proving that two triangles are congruent, we can conclude that their corresponding parts (angles & sides) are congruent

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

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

Prove Congruent Triangles Mr. Chin-Sung Lin StatementsReasons ERHS Math Geometry A B C D E F

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

Prove Congruent Triangles Mr. Chin-Sung Lin StatementsReasons 1.  B   C, and AB  AC 1. Given 2. A  A2. Reflexive property ERHS Math Geometry A B C D E F

Prove Congruent Triangles Mr. Chin-Sung Lin StatementsReasons 1.  B   C, and AB  AC 1. Given 2. A  A2. Reflexive property 3.∆ ABF  ∆ ACE 3. ASA ERHS Math Geometry A B C D E F

Prove Congruent Triangles Mr. Chin-Sung Lin StatementsReasons 1.  B   C, and AB  AC 1. Given 2. A  A2. Reflexive property 3.∆ ABF  ∆ ACE 3. ASA 4.AF  AE 4. CPCTC ERHS Math Geometry A B C D E F

Isosceles Triangles ERHS Math Geometry Mr. Chin-Sung Lin

Isosceles Triangles ERHS Math Geometry Mr. Chin-Sung Lin An isosceles triangle is a triangle that has two congruent sides AC B

Parts of an Isosceles Triangle ERHS Math Geometry Mr. Chin-Sung Lin Leg: the two congruent sides Base: the third side Vertex Angle: the angle formed by the two congruent side Base Angle: the angles whose vertices are the endpoints of the base AC B Base Leg Base Angle Vertex Angle

Base Angle Theorem (Isosceles Triangle Theorem) ERHS Math Geometry Mr. Chin-Sung Lin

Base Angle Theorem (Isosceles Triangle Theorem) ERHS Math Geometry Mr. Chin-Sung Lin If two sides of a triangle are congruent, then the angles opposite these sides are congruent (Base angles of an isosceles triangle are congruent)

Base Angle Theorem ERHS Math Geometry Mr. Chin-Sung Lin If two sides of a triangle are congruent, then the angles opposite these sides are congruent Draw a diagram like the one below Given: A B  CB Prove:A  C AC B

Base Angle Theorem ERHS Math Geometry Mr. Chin-Sung Lin StatementsReasons AC B D

Base Angle Theorem ERHS Math Geometry Mr. Chin-Sung Lin StatementsReasons 1. Draw the angle bisector of 1. Any angle of measure less ABC and let D be the point than 180 has exactly one where it intersects AC bisector 2. ABD  CBD 2. Definition of angle bisector 3. A B  CB 3. Given 4. BD  BD 4. Reflexive property 5. ∆ ABD = ∆ CBD 5. SAS Postulate 6. A  C6. CPCTC AC B D

Base Angle Theorem - Example 1 ERHS Math Geometry Mr. Chin-Sung Lin Given: A B  CB and AD  CE Prove: ∆ ABD = ∆ CBE A C B DE

Base Angle Theorem - Example 1 ERHS Math Geometry Mr. Chin-Sung Lin Given: A B  CB and AD  CE Prove: ∆ ABD = ∆ CBE A C B DE

Base Angle Theorem - Example 1 ERHS Math Geometry Mr. Chin-Sung Lin StatementsReasons A C B DE

Base Angle Theorem - Example 1 ERHS Math Geometry Mr. Chin-Sung Lin StatementsReasons 1. A B  CB 1. Given AD  CE 2. A  C 2. Base Angle Theorem 3. ∆ ABD = ∆ CBE 3. SAS Postulate A C B DE

Base Angle Theorem - Example 2 ERHS Math Geometry Mr. Chin-Sung Lin Given: 1  2 and 5  6 Prove: 3  4 A C B D O

Base Angle Theorem - Example 2 ERHS Math Geometry Mr. Chin-Sung Lin Given: 1  2 and 5  6 Prove: 3  4 A C B D O

Base Angle Theorem - Example 2 ERHS Math Geometry Mr. Chin-Sung Lin StatementsReasons A C B D O

Base Angle Theorem - Example 2 ERHS Math Geometry Mr. Chin-Sung Lin StatementsReasons 1. 1  2 1. Given 5  6 2. A B  AB 2. Reflexive Property 3. ∆ ACB = ∆ ADB 3. ASA Postulate 4. A C  AD 4. CPCTC 5. ∆ ADC is an isosceles triangle 5. Def. of Isosceles Triangle 6. 3  4 6. Base Angle Theorem A C B D O

Base Angle Theorem - Exercise ERHS Math Geometry Mr. Chin-Sung Lin Given: BD  BE and AD  CE Prove: AB = CB A C B DE

Converse of Base Angle Theorem (Converse of Isosceles Triangle Theorem) ERHS Math Geometry Mr. Chin-Sung Lin

Converse of Base Angle Theorem ERHS Math Geometry Mr. Chin-Sung Lin If two angles of a triangle are congruent, then the sides opposite these angles are congruent

Converse of Base Angle Theorem ERHS Math Geometry Mr. Chin-Sung Lin If two angles of a triangle are congruent, then the sides opposite these angles are congruent Draw a diagram like the one below Given:A  C Prove: A B  CB AC B

Converse of Base Angle Theorem ERHS Math Geometry Mr. Chin-Sung Lin StatementsReasons AC B D

Converse of Base Angle Theorem ERHS Math Geometry Mr. Chin-Sung Lin StatementsReasons 1. Draw the angle bisector of 1. Any angle of measure less ABC and let D be the point than 180 has exactly one where it intersects AC bisector 2. ABD  CBD 2. Definition of angle bisector 3. A  C 3. Given 4. BD  BD 4. Reflexive property 5. ∆ ABD = ∆ CBD 5. AAS Postulate 6. A B  CB 6. CPCTC AC B D

Base Angle Theorem - Example 3 ERHS Math Geometry Mr. Chin-Sung Lin Given: A O  BO and 1  2 Prove: AC = BD A C B D O 12

Base Angle Theorem - Example 3 ERHS Math Geometry Mr. Chin-Sung Lin Given: A O  BO and 1  2 Prove: AC = BD A C B D O 12

Base Angle Theorem - Example 3 ERHS Math Geometry Mr. Chin-Sung Lin StatementsReasons A C B D O 12

Base Angle Theorem - Example 3 ERHS Math Geometry Mr. Chin-Sung Lin StatementsReasons 1. 1  2 1. Given 2. C O  DO 2. Converse of Base Angle Theorem 3. A O  BO 3. Given 4. AOC  BOD 4. Vertical Angles 5. ∆ AOC = ∆ BOD 5. SAS Postulate 6. AC  BD6. CPCTC A C B D O 12

Corollaries of Base Angle Theorem ERHS Math Geometry Mr. Chin-Sung Lin The median from the vertex angle of an isosceles triangle bisects the vertex angle The median from the vertex angle of an isosceles triangle is perpendicular to the base

Equilateral and Equiangular Triangles ERHS Math Geometry Mr. Chin-Sung Lin

Equilateral Triangles ERHS Math Geometry Mr. Chin-Sung Lin A equilateral triangle is a triangle that has three congruent sides AC B

Equilateral & Equiangular Triangles ERHS Math Geometry Mr. Chin-Sung Lin If a triangle is an equilateral triangle, then it is an equiangular triangle

Identify Overlapping Triangles ERHS Math Geometry Mr. Chin-Sung Lin

Identify Triangles - 1 ERHS Math Geometry Mr. Chin-Sung Lin How many triangles can you identify in the following diagram? A C B D O

Identify Triangles - 1 ERHS Math Geometry Mr. Chin-Sung Lin How many triangles can you identify in the following diagram? A C B D O ∆ ADC

Identify Triangles - 1 ERHS Math Geometry Mr. Chin-Sung Lin How many triangles can you identify in the following diagram? A C B D O ∆ BCD

Identify Triangles - 1 ERHS Math Geometry Mr. Chin-Sung Lin How many triangles can you identify in the following diagram? A C B D O ∆ DAB

Identify Triangles - 1 ERHS Math Geometry Mr. Chin-Sung Lin How many triangles can you identify in the following diagram? A C B D O ∆ CBA

Identify Triangles - 1 ERHS Math Geometry Mr. Chin-Sung Lin How many triangles can you identify in the following diagram? A C B D O ∆ DOC

Identify Triangles - 1 ERHS Math Geometry Mr. Chin-Sung Lin How many triangles can you identify in the following diagram? A C B D O ∆ AOB

Identify Triangles - 1 ERHS Math Geometry Mr. Chin-Sung Lin How many triangles can you identify in the following diagram? A C B D O ∆ AOD

Identify Triangles - 1 ERHS Math Geometry Mr. Chin-Sung Lin How many triangles can you identify in the following diagram? A C B D O ∆ BOC

Identify Triangles - 1 ERHS Math Geometry Mr. Chin-Sung Lin How many triangles can you identify in the following diagram? A C B D O Total 8 Triangles

Identify Triangles - 2 ERHS Math Geometry Mr. Chin-Sung Lin How many triangles can you identify in the following diagram? A C B D O E

Identify Triangles - 2 ERHS Math Geometry Mr. Chin-Sung Lin How many triangles can you identify in the following diagram? ∆ BDC A C B D O E

Identify Triangles - 2 ERHS Math Geometry Mr. Chin-Sung Lin How many triangles can you identify in the following diagram? ∆ CEB A C B D O E

Identify Triangles - 2 ERHS Math Geometry Mr. Chin-Sung Lin How many triangles can you identify in the following diagram? ∆ AEB A C B D O E

Identify Triangles - 2 ERHS Math Geometry Mr. Chin-Sung Lin How many triangles can you identify in the following diagram? ∆ ADC A C B D O E

Identify Triangles - 2 ERHS Math Geometry Mr. Chin-Sung Lin How many triangles can you identify in the following diagram? ∆ DOB A C B D O E

Identify Triangles - 2 ERHS Math Geometry Mr. Chin-Sung Lin How many triangles can you identify in the following diagram? ∆ EOC A C B D O E

Identify Triangles - 2 ERHS Math Geometry Mr. Chin-Sung Lin How many triangles can you identify in the following diagram? ∆ BOC A C B D O E

Identify Triangles - 2 ERHS Math Geometry Mr. Chin-Sung Lin How many triangles can you identify in the following diagram? ∆ ABC A C B D O E

Identify Triangles - 2 ERHS Math Geometry Mr. Chin-Sung Lin How many triangles can you identify in the following diagram? Total 8 Triangles A C B D O E

Shared Sides & Angles ERHS Math Geometry Mr. Chin-Sung Lin

Shared Side - 1 ERHS Math Geometry Mr. Chin-Sung Lin Which two congruent-triangle candidates have a shared side? Which line segment has been shared? A C B D O

Shared Side - 1 ERHS Math Geometry Mr. Chin-Sung Lin Which two congruent-triangle candidates have a shared side? Which line segment has been shared? A C B D O ∆ ADC & ∆ BCD DC

Shared Side - 2 ERHS Math Geometry Mr. Chin-Sung Lin Which two congruent-triangle candidates have a shared side? Which line segment has been shared? AB C O D EF

Shared Side - 2 ERHS Math Geometry Mr. Chin-Sung Lin Which two congruent-triangle candidates have a shared side? Which line segment has been shared? ∆ ACF & ∆ BDE EF AB C O D EF

Shared Side - 3 ERHS Math Geometry Mr. Chin-Sung Lin Which two congruent-triangle candidates have a shared side? Which line segment has been shared? A CB D E

Shared Side - 3 ERHS Math Geometry Mr. Chin-Sung Lin Which two congruent-triangle candidates have a shared side? Which line segment has been shared? ∆ AEB & ∆ ADC DE A CB D E

Shared Angle - 1 ERHS Math Geometry Mr. Chin-Sung Lin Which two congruent-triangle candidates have a shared angle? Which angle has been shared? A B C O E D

Shared Angle - 1 ERHS Math Geometry Mr. Chin-Sung Lin Which two congruent-triangle candidates have a shared angle? Which angle has been shared? A B C O E D ∆ AEB & ∆ ADC  BAC

Shared Angle - 2 ERHS Math Geometry Mr. Chin-Sung Lin Which two congruent-triangle candidates have a shared angle? Which angle has been shared? A CB D E

Shared Angle - 2 ERHS Math Geometry Mr. Chin-Sung Lin Which two congruent-triangle candidates have a shared angle? Which angle has been shared? ∆ AEB & ∆ ADC  DAE A CB D E

Congruent Overlapping Triangles ERHS Math Geometry Mr. Chin-Sung Lin

Congruent Triangles - 1 ERHS Math Geometry Mr. Chin-Sung Lin Name the possible congruent-triangle pairs? A B C O E D

Congruent Triangles - 1 ERHS Math Geometry Mr. Chin-Sung Lin Name the possible congruent-triangle pairs? A B C O E D ∆ AEB & ∆ ADC ∆ DOB & ∆ EOC

Congruent Triangles - 2 ERHS Math Geometry Mr. Chin-Sung Lin Name the possible congruent-triangle pairs? A C B D O

Congruent Triangles - 2 ERHS Math Geometry Mr. Chin-Sung Lin Name the possible congruent-triangle pairs? ∆ ADC & ∆ BCD ∆ AOD & ∆ BOC A C B D O

Congruent Triangles - 3 ERHS Math Geometry Mr. Chin-Sung Lin Name the possible congruent-triangle pairs? A C B D O E

Congruent Triangles - 3 ERHS Math Geometry Mr. Chin-Sung Lin Name the possible congruent-triangle pairs? ∆ BDO & ∆ CEO ∆ ECB & ∆ DBC A C B D O E ∆ AEB & ∆ ADC

Congruent Triangles - 4 ERHS Math Geometry Mr. Chin-Sung Lin Name the possible congruent-triangle pairs? A CB D E

Congruent Triangles - 4 ERHS Math Geometry Mr. Chin-Sung Lin Name the possible congruent-triangle pairs? ∆ AEB & ∆ ADC ∆ ADB & ∆ AEC A CB D E

Congruent Triangles - 5 ERHS Math Geometry Mr. Chin-Sung Lin Name the possible congruent-triangle pairs? A CB EFD

Congruent Triangles - 5 ERHS Math Geometry Mr. Chin-Sung Lin Name the possible congruent-triangle pairs? A CB EFD ∆ ABD & ∆ ACF ∆ ADE & ∆ AFE ∆ ABE & ∆ ACE ∆ ABF & ∆ ACD

Congruent Triangles - 6 ERHS Math Geometry Mr. Chin-Sung Lin Name the possible congruent-triangle pairs? A C B D O

Congruent Triangles - 6 ERHS Math Geometry Mr. Chin-Sung Lin Name the possible congruent-triangle pairs? ∆ ABC & ∆ BAD ∆ AOC & ∆ BOD ∆ ACD & ∆ BDC A C B D O

Congruent Triangles - 7 ERHS Math Geometry Mr. Chin-Sung Lin Name the possible congruent-triangle pairs? A D B C E F

Congruent Triangles - 7 ERHS Math Geometry Mr. Chin-Sung Lin Name the possible congruent-triangle pairs? ∆ ABD & ∆ CDB ∆ ADE & ∆ CBF ∆ ABE & ∆ CDF A D B C E F

Congruent Triangles - 8 ERHS Math Geometry Mr. Chin-Sung Lin Name the possible congruent-triangle pairs? A C B EF D O G H

Congruent Triangles - 8 ERHS Math Geometry Mr. Chin-Sung Lin Name the possible congruent-triangle pairs? ∆ AGO & ∆ BHO ∆ CGE & ∆ DHF ∆ AED & ∆ BFC A C B EF D O G H

Congruent Triangles - 9 ERHS Math Geometry Mr. Chin-Sung Lin Name the possible congruent-triangle pairs? A C B EFD

Congruent Triangles - 9 ERHS Math Geometry Mr. Chin-Sung Lin Name the possible congruent-triangle pairs? ∆ ABD & ∆ ACF ∆ ADE & ∆ AFE A C B EFD

Congruent Triangles - 10 ERHS Math Geometry Mr. Chin-Sung Lin Name the possible congruent-triangle pairs? A CB EFD GH

Congruent Triangles - 10 ERHS Math Geometry Mr. Chin-Sung Lin Name the possible congruent-triangle pairs? A CB EFD GH ∆ ABG & ∆ ACH ∆ AGE & ∆ AHE ∆ ABE & ∆ ACE ∆ ADE & ∆ AFE ∆ GDE & ∆ HFE

Congruent Triangles - 11 ERHS Math Geometry Mr. Chin-Sung Lin Name the possible congruent-triangle pairs? A C B EF D G H IJ O

Congruent Triangles - 11 ERHS Math Geometry Mr. Chin-Sung Lin Name the possible congruent-triangle pairs? A C B EF D G H IJ O ∆ AGO & ∆ AHO ∆ BGI & ∆ CHJ ∆ IDE & ∆ JFE ∆ AIE & ∆ AJE ∆ ADE & ∆ AFE∆ BOE & ∆ COE

Theorems about Perpendicular Bisector ERHS Math Geometry Mr. Chin-Sung Lin

Perpendicular Bisector ERHS Math Geometry Mr. Chin-Sung Lin The perpendicular bisector of a line segment is a line, a ray, or a line segment that is perpendicular to the line segment at its midpoint AB  CD CO = OD D O A C B ~

Theorems of Perpendicular Bisector ERHS Math Geometry Mr. Chin-Sung Lin If two points are each equidistant from the endpoints of a line segment, then the points determine the perpendicular bisector of the line segment Given: AB and points P and T such that PA = PB and TA = TB Prove: PT is the perpendicular bisector of AB B O P A T

Theorems of Perpendicular Bisector ERHS Math Geometry Mr. Chin-Sung Lin If a point is equidistant from the endpoints of a line segment, then it is on the perpendicular bisector of the line segment Given: Point P such that PA = PB Prove: P lies on the perpendicular bisector of AB B M P A

Theorems of Perpendicular Bisector ERHS Math Geometry Mr. Chin-Sung Lin If a point is on the perpendicular bisector of a line segmenton, then it is equidistant from the endpoints of the line segment Given: Point P on the perpendicular bisector of AB Prove: PA = PB B M P A

Theorems of Perpendicular Bisector ERHS Math Geometry Mr. Chin-Sung Lin A point is on the perpendicular bisector of a line segmenton if and only if it is equidistant from the endpoints of the line segment B M P A

Perpendicular Bisector Concurrence Theorems ERHS Math Geometry Mr. Chin-Sung Lin The perpendicular bisectors of the sides of a triangle are concurrent (intersect in one point) Given: MQ, the perpendicular bisector of AB NR, the perpendicular bisector of AC LS, the perpendicular bisector of BC Prove: MQ, NR, and LS intersect in P R P L S N Q M A B C

Perpendicular Bisector Concurrence Theorems ERHS Math Geometry Mr. Chin-Sung Lin StatementsReasons R P L S N Q M A B C

Construction ERHS Math Geometry Mr. Chin-Sung Lin

Construction of Perpendicular Bisector ERHS Math Geometry Mr. Chin-Sung Lin B M A

Q & A ERHS Math Geometry Mr. Chin-Sung Lin

The End ERHS Math Geometry Mr. Chin-Sung Lin