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

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

CA

B

A C

B

Altitude of a Triangle

ERHS Math Geometry

Mr. Chin-Sung Lin

If BD is the altitude of ∆ ABC

then,m BDA = 90m BDC = 90

CA

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

CA

B

A C

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


ERHS Math Geometry

Mr. Chin-Sung Lin

Where is the orthocenter of a right triangle?

Orthocenter?

A C

B


ERHS Math Geometry

Mr. Chin-Sung Lin

The orthocenter is located at the vertex of the right angle

Orthocenter

A C

B


ERHS Math Geometry

Mr. Chin-Sung Lin

Where is the orthocenter of an obtuse triangle?

Orthocenter?

C

B

A


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


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

A C

B

C

A

B


ITHS Math B Term 1 (M$4)

Mr. Chin-Sung Lin

If BD is the angle bisector of ABC

then,ABD CBD

A C

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


ERHS Math Geometry

Mr. Chin-Sung Lin

A segment from a vertex to the midpoint of the opposite side

A

C

B

A C

B

C

A

B


ITHS Math B Term 1 (M$4)

Mr. Chin-Sung Lin

If BD is the median of ∆ ABC

then,

AD CD

A C

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 CDCO = OD

DO

A

C

B

~


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

A C

B

C

A

B


ERHS Math Geometry

Mr. Chin-Sung Lin

If DE is the perpendicular bisector of the side of ∆ ABC

then,

AD CD

DE ACA C

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

CED F

BD: AltitudeBE: Angle bisectorBF: MedianFG: 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

CA

B

D

BD: AltitudeBD: Angle bisectorBD: MedianBD: Perpendicular

Bisector

Scalene Triangle (Indirect Proof)

ERHS Math Geometry

Mr. Chin-Sung Lin

Given: ∆ ABC is scalene, BD bisects ABCProve: BD is not perpendicular to AC

A

B

CD

1 2

3 4

Scalene Triangle

ERHS Math Geometry

Mr. Chin-Sung Lin

Statements ReasonsA

B

CD

1 2

3 4

Scalene Triangle

ERHS Math Geometry

Mr. Chin-Sung Lin

Statements Reasons1. BD AC 1. Assume the opposite is true

A

B

CD

1 2

3 4

Scalene Triangle

ERHS Math Geometry

Mr. Chin-Sung Lin

Statements Reasons1. BD AC 1. Assume the opposite is true2. ∆ ABC is scalene, BD is angle 2. Given bisector

A

B

CD

1 2

3 4

Scalene Triangle

ERHS Math Geometry

Mr. Chin-Sung Lin

Statements Reasons1. BD AC 1. Assume the opposite is true2. ∆ ABC is scalene, BD is angle 2. Given bisector3. 1 2 3. Definition of angle bisector

A

B

CD

1 2

3 4

Scalene Triangle

ERHS Math Geometry

Mr. Chin-Sung Lin

Statements Reasons1. BD AC 1. Assume the opposite is true2. ∆ ABC is scalene, BD is angle 2. Given bisector3. 1 2 3. Definition of angle bisector4. 3 = 90o, 4 = 90o 4. Definition of perpendicular

A

B

CD

1 2

3 4

Scalene Triangle

ERHS Math Geometry

Mr. Chin-Sung Lin

Statements Reasons1. BD AC 1. Assume the opposite is true2. ∆ ABC is scalene, BD is angle 2. Given bisector3. 1 2 3. Definition of angle bisector4. 3 = 90o, 4 = 90o 4. Definition of perpendicular5. 3 4 5. Substitution postulate

A

B

CD

1 2

3 4

Scalene Triangle

ERHS Math Geometry

Mr. Chin-Sung Lin

Statements Reasons1. BD AC 1. Assume the opposite is true2. ∆ ABC is scalene, BD is angle 2. Given bisector3. 1 2 3. Definition of angle bisector4. 3 = 90o, 4 = 90o 4. Definition of perpendicular5. 3 4 5. Substitution postulate6. BD BD 6. Reflexive property

A

B

CD

1 2

3 4

Scalene Triangle

ERHS Math Geometry

Mr. Chin-Sung Lin

Statements Reasons1. BD AC 1. Assume the opposite is true2. ∆ ABC is scalene, BD is angle 2. Given bisector3. 1 2 3. Definition of angle bisector4. 3 = 90o, 4 = 90o 4. Definition of perpendicular5. 3 4 5. Substitution postulate6. BD BD 6. Reflexive property7. ∆ ABD ∆ CBD 7. ASA postulate

A

B

CD

1 2

3 4

Scalene Triangle

ERHS Math Geometry

Mr. Chin-Sung Lin

Statements Reasons1. BD AC 1. Assume the opposite is true2. ∆ ABC is scalene, BD is angle 2. Given bisector3. 1 2 3. Definition of angle bisector4. 3 = 90o, 4 = 90o 4. Definition of perpendicular5. 3 4 5. Substitution postulate6. BD BD 6. Reflexive property7. ∆ ABD ∆ CBD 7. ASA postulate8. AB = CB 8. CPCTC

A

B

CD

1 2

3 4

Scalene Triangle

ERHS Math Geometry

Mr. Chin-Sung Lin

Statements Reasons1. BD AC 1. Assume the opposite is true2. ∆ ABC is scalene, BD is angle 2. Given bisector3. 1 2 3. Definition of angle bisector4. 3 = 90o, 4 = 90o 4. Definition of perpendicular5. 3 4 5. Substitution postulate6. BD BD 6. Reflexive property7. ∆ ABD ∆ CBD 7. ASA postulate8. AB = CB 8. CPCTC9. AB ≠ CB 9. Definition of scalene triangle

A

B

CD

1 2

3 4

Scalene Triangle

ERHS Math Geometry

Mr. Chin-Sung Lin

Statements Reasons1. BD AC 1. Assume the opposite is true2. ∆ ABC is scalene, BD is angle 2. Given bisector3. 1 2 3. Definition of angle bisector4. 3 = 90o, 4 = 90o 4. Definition of perpendicular5. 3 4 5. Substitution postulate6. BD BD 6. Reflexive property7. ∆ ABD ∆ CBD 7. ASA postulate8. AB = CB 8. CPCTC9. 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

CD

1 2

3 4

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

Prove Congruent Triangles

Mr. Chin-Sung Lin

Statements Reasons

ERHS Math Geometry

A

B

C

D

E

F


Mr. Chin-Sung Lin

Statements Reasons

1. B C , and AB AC 1. Given

ERHS Math Geometry

A

B

C

D

E

F


Mr. Chin-Sung Lin

Statements Reasons

1. B C , and AB AC 1. Given2. A A 2. Reflexive property

ERHS Math Geometry

A

B

C

D

E

F


Mr. Chin-Sung Lin

Statements Reasons

1. B C , and AB AC 1. Given2. A A 2. Reflexive property• ∆ ABF ∆ ACE 3. ASA

ERHS Math Geometry

A

B

C

D

E

F


Mr. Chin-Sung Lin

Statements Reasons

1. B C , and AB AC 1. Given2. A A 2. Reflexive property• ∆ ABF ∆ ACE 3. ASA• 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

A C

B

Parts of an Isosceles Triangle

ERHS Math Geometry

Mr. Chin-Sung Lin

Leg: the two congruent sidesBase: the third sideVertex Angle: the angle formed by the two

congruent sideBase Angle: the angles whose vertices are the

endpoints of the base

A C

B

Base

LegLegBase 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 belowGiven: AB CB Prove: A C

A C

B

Base Angle Theorem

ERHS Math Geometry

Mr. Chin-Sung Lin

Statements Reasons1. 1. 2. 2. 3. 3. 4. 4. 5. 5. 6. 6.

A C

B

D

Base Angle Theorem

ERHS Math Geometry

Mr. Chin-Sung Lin

Statements Reasons1. 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 bisector2. ABD CBD 2. Definition of angle bisector3. AB CB 3. Given4. BD BD 4. Reflexive property5. ∆ ABD = ∆ CBD 5. SAS Postulate6. A C 6. CPCTC

A C

B

D

Base Angle Theorem - Example 1

ERHS Math Geometry

Mr. Chin-Sung Lin

Given: AB CB and AD CEProve: ∆ ABD = ∆ CBE

A C

B

D E


ERHS Math Geometry

Mr. Chin-Sung Lin

Statements Reasons1. 1. 2. 2. 3. 3.

A C

B

D E


ERHS Math Geometry

Mr. Chin-Sung Lin

Statements Reasons1. AB CB 1. Given AD CE 2. A C 2. Base Angle Theorem3. ∆ ABD = ∆ CBE 3. SAS Postulate

A C

B

D E


ERHS Math Geometry

Mr. Chin-Sung Lin

Given: 1 2 and 5 6Prove: 3 4

A

C

B

D

O12

56

3

4


ERHS Math Geometry

Mr. Chin-Sung Lin

Statements Reasons1. 1. 2. 2. 3. 3. 4. 4.5. 5.6. 6.

A

C

B

D

O12

56

3

4


ERHS Math Geometry

Mr. Chin-Sung Lin

Statements Reasons1. 1 2 1. Given 5 6 2. AB AB 2. Reflexive Property3. ∆ ACB = ∆ ADB 3. ASA Postulate4. AC AD 4. CPCTC5. ∆ ADC is an isosceles triangle 5. Def. of Isosceles Triangle 6. 3 4 6. Base Angle Theorem

A

C

B

D

O12

56

3

4

Base Angle Theorem - Exercise

ERHS Math Geometry

Mr. Chin-Sung Lin

Given: BD BE and AD CEProve: AB = CB

A C

B

D E

Converse of Base Angle Theorem

(Converse of Isosceles Triangle Theorem)

ERHS Math Geometry

Mr. Chin-Sung Lin


ERHS Math Geometry

Mr. Chin-Sung Lin

If two angles of a triangle are congruent, then the sides opposite these angles are congruent


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 belowGiven: A C Prove: AB CB

A C

B


ERHS Math Geometry

Mr. Chin-Sung Lin

Statements Reasons1. 1.

2. 2. 3. 3. 4. 4. 5. 5. 6. 6.

A C

B

D


ERHS Math Geometry

Mr. Chin-Sung Lin

Statements Reasons1. 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 bisector2. ABD CBD 2. Definition of angle bisector3. A C 3. Given4. BD BD 4. Reflexive property5. ∆ ABD = ∆ CBD 5. AAS Postulate6. AB CB 6. CPCTC

A C

B

D


ERHS Math Geometry

Mr. Chin-Sung Lin

Given: AO BO and 1 2 Prove: AC = BD

A

C

B

D

O

1 2


ERHS Math Geometry

Mr. Chin-Sung Lin

Statements Reasons1. 1. 2. 2. 3. 3.

A

C

B

D

O

1 2


ERHS Math Geometry

Mr. Chin-Sung Lin

Statements Reasons1. 1 2 1. Given2. CO DO 2. Converse of Base Angle

Theorem3. AO BO 3. Given4. AOC BOD 4. Vertical Angles5. ∆ AOC = ∆ BOD 5. SAS Postulate6. AC BD 6. CPCTC

A

C

B

D

O

1 2

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

A C

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


ERHS Math Geometry

Mr. Chin-Sung Lin


A

C

B

D

O ∆ ADC


ERHS Math Geometry

Mr. Chin-Sung Lin


A

C

B

D

O ∆ BCD


ERHS Math Geometry

Mr. Chin-Sung Lin


A

C

B

D

O ∆ DAB


ERHS Math Geometry

Mr. Chin-Sung Lin


A

C

B

D

O ∆ CBA


ERHS Math Geometry

Mr. Chin-Sung Lin


A

C

B

D

O ∆ DOC


ERHS Math Geometry

Mr. Chin-Sung Lin


A

C

B

D

O ∆ AOB


ERHS Math Geometry

Mr. Chin-Sung Lin


A

C

B

D

O ∆ AOD


ERHS Math Geometry

Mr. Chin-Sung Lin


A

C

B

D

O ∆ BOC


ERHS Math Geometry

Mr. Chin-Sung Lin


A

C

B

D

O Total8 Triangles


ERHS Math Geometry

Mr. Chin-Sung Lin


A

C B

D OE


ERHS Math Geometry

Mr. Chin-Sung Lin


∆ BDC

A

C B

D OE


ERHS Math Geometry

Mr. Chin-Sung Lin


∆ CEB

A

C B

D OE


ERHS Math Geometry

Mr. Chin-Sung Lin


∆ AEB

A

C B

D OE


ERHS Math Geometry

Mr. Chin-Sung Lin


∆ ADC

A

C B

D OE


ERHS Math Geometry

Mr. Chin-Sung Lin


∆ DOB

A

C B

D OE


ERHS Math Geometry

Mr. Chin-Sung Lin


∆ EOC

A

C B

D OE


ERHS Math Geometry

Mr. Chin-Sung Lin


∆ BOC

A

C B

D OE


ERHS Math Geometry

Mr. Chin-Sung Lin


∆ ABC

A

C B

D OE


ERHS Math Geometry

Mr. Chin-Sung Lin


Total8 Triangles

A

C B

D OE

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


A

C

B

D

O ∆ ADC & ∆ BCD

DC

Shared Side - 2

ERHS Math Geometry

Mr. Chin-Sung Lin


A B

C OD

E F

Shared Side - 2

ERHS Math Geometry

Mr. Chin-Sung Lin


∆ ACF & ∆ BDE

EFA B

C O

D

E F

Shared Side - 3

ERHS Math Geometry

Mr. Chin-Sung Lin


A

C B D E

Shared Side - 3

ERHS Math Geometry

Mr. Chin-Sung Lin


∆ AEB & ∆ ADC

DE

A

C B 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

OED

Shared Angle - 1

ERHS Math Geometry

Mr. Chin-Sung Lin


A

B C

OED

∆ AEB & ∆ ADC

BAC

Shared Angle - 2

ERHS Math Geometry

Mr. Chin-Sung Lin


A

C B D E

Shared Angle - 2

ERHS Math Geometry

Mr. Chin-Sung Lin


∆ AEB & ∆ ADC

DAE

A

C B 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

OED


ERHS Math Geometry

Mr. Chin-Sung Lin


A

B C

OED

∆ AEB & ∆ ADC

∆ DOB & ∆ EOC


ERHS Math Geometry

Mr. Chin-Sung Lin


A

C

B

D

O


ERHS Math Geometry

Mr. Chin-Sung Lin


∆ ADC & ∆ BCD

∆ AOD & ∆ BOC

A

C

B

D

O


ERHS Math Geometry

Mr. Chin-Sung Lin


A

C B

D OE


ERHS Math Geometry

Mr. Chin-Sung Lin


∆ BDO & ∆ CEO

∆ ECB & ∆ DBC

A

C B

D OE

∆ AEB & ∆ ADC


ERHS Math Geometry

Mr. Chin-Sung Lin


A

C B D E


ERHS Math Geometry

Mr. Chin-Sung Lin


∆ AEB & ∆ ADC

∆ ADB & ∆ AEC

A

C B D E


ERHS Math Geometry

Mr. Chin-Sung Lin


A

C B E FD


ERHS Math Geometry

Mr. Chin-Sung Lin


A

C B E FD

∆ ABD & ∆ ACF

∆ ADE & ∆ AFE

∆ ABE & ∆ ACE

∆ ABF & ∆ ACD


ERHS Math Geometry

Mr. Chin-Sung Lin


A

C

B

D

O


ERHS Math Geometry

Mr. Chin-Sung Lin


∆ ABC & ∆ BAD

∆ AOC & ∆ BOD

∆ ACD & ∆ BDC

A

C

B

D

O


ERHS Math Geometry

Mr. Chin-Sung Lin


A

D

B

C

E F


ERHS Math Geometry

Mr. Chin-Sung Lin


∆ ABD & ∆ CDB

∆ ADE & ∆ CBF

∆ ABE & ∆ CDF

A

D

B

C

E F


ERHS Math Geometry

Mr. Chin-Sung Lin


A

C

B

E F D

OG H


ERHS Math Geometry

Mr. Chin-Sung Lin


∆ AGO & ∆ BHO

∆ CGE & ∆ DHF

∆ AED & ∆ BFC

A

C

B

E F D

OG H


ERHS Math Geometry

Mr. Chin-Sung Lin


A

C B

E FD


ERHS Math Geometry

Mr. Chin-Sung Lin


∆ ABD & ∆ ACF

∆ ADE & ∆ AFE

A

C B

E FD


ERHS Math Geometry

Mr. Chin-Sung Lin


A

C B

E FD

G H


ERHS Math Geometry

Mr. Chin-Sung Lin


A

C B

E FD

G H

∆ ABG & ∆ ACH

∆ AGE & ∆ AHE

∆ ABE & ∆ ACE

∆ ADE & ∆ AFE

∆ GDE & ∆ HFE


ERHS Math Geometry

Mr. Chin-Sung Lin


A

C B

E FD

G H

I JO


ERHS Math Geometry

Mr. Chin-Sung Lin


A

C B

E FD

G H

I JO

∆ 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 CDCO = OD

DO

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 = TBProve: PT is the perpendicular bisector of AB

BO

P

A

T


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 = PBProve: P lies on the perpendicular bisector of AB

BM

P

A


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 ABProve: PA = PB

BM

P

A


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

BM

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

MA

B

C

Perpendicular Bisector Concurrence Theorems

ERHS Math Geometry

Mr. Chin-Sung Lin

Statements Reasons R

P

L

S

N Q

MA

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

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

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