Post on 20-Jan-2018
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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
Altitude - Orthocenter
ERHS Math Geometry
Mr. Chin-Sung Lin
Where is the orthocenter of a right triangle?
Orthocenter?
A C
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
A C
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
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
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
A C
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
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
~
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
A C
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 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
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
Prove Congruent Triangles
Mr. Chin-Sung Lin
Statements Reasons
1. B C , and AB AC 1. Given
ERHS Math Geometry
A
B
C
D
E
F
Prove Congruent Triangles
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
Prove Congruent Triangles
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
Prove Congruent Triangles
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
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
Base Angle Theorem - Example 1
ERHS Math Geometry
Mr. Chin-Sung Lin
Statements Reasons1. 1. 2. 2. 3. 3.
A C
B
D E
Base Angle Theorem - Example 1
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
Base Angle Theorem - Example 2
ERHS Math Geometry
Mr. Chin-Sung Lin
Given: 1 2 and 5 6Prove: 3 4
A
C
B
D
O12
56
3
4
Base Angle Theorem - Example 2
ERHS Math Geometry
Mr. Chin-Sung Lin
Given: 1 2 and 5 6Prove: 3 4
A
C
B
D
O12
56
3
4
Base Angle Theorem - Example 2
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
Base Angle Theorem - Example 2
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
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 belowGiven: A C Prove: AB CB
A C
B
Converse of 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
Converse of 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. A C 3. Given4. BD BD 4. Reflexive property5. ∆ ABD = ∆ CBD 5. AAS Postulate6. AB CB 6. CPCTC
A C
B
D
Base Angle Theorem - Example 3
ERHS Math Geometry
Mr. Chin-Sung Lin
Given: AO BO and 1 2 Prove: AC = BD
A
C
B
D
O
1 2
Base Angle Theorem - Example 3
ERHS Math Geometry
Mr. Chin-Sung Lin
Given: AO BO and 1 2 Prove: AC = BD
A
C
B
D
O
1 2
Base Angle Theorem - Example 3
ERHS Math Geometry
Mr. Chin-Sung Lin
Statements Reasons1. 1. 2. 2. 3. 3.
A
C
B
D
O
1 2
Base Angle Theorem - Example 3
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
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 Total8 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 OE
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 OE
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 OE
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 OE
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 OE
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 OE
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 OE
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 OE
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 OE
Identify Triangles - 2
ERHS Math Geometry
Mr. Chin-Sung Lin
How many triangles can you identify in the following diagram?
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
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?
A B
C OD
E F
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
EFA B
C O
D
E F
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
C B 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
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
Which two congruent-triangle candidates have a shared angle? Which angle has been shared?
A
B C
OED
∆ 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
C B 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
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
Congruent Triangles - 1
ERHS Math Geometry
Mr. Chin-Sung Lin
Name the possible congruent-triangle pairs?
A
B C
OED
∆ 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 OE
Congruent Triangles - 3
ERHS Math Geometry
Mr. Chin-Sung Lin
Name the possible congruent-triangle pairs?
∆ BDO & ∆ CEO
∆ ECB & ∆ DBC
A
C B
D OE
∆ AEB & ∆ ADC
Congruent Triangles - 4
ERHS Math Geometry
Mr. Chin-Sung Lin
Name the possible congruent-triangle pairs?
A
C B D E
Congruent Triangles - 4
ERHS Math Geometry
Mr. Chin-Sung Lin
Name the possible congruent-triangle pairs?
∆ AEB & ∆ ADC
∆ ADB & ∆ AEC
A
C B D E
Congruent Triangles - 5
ERHS Math Geometry
Mr. Chin-Sung Lin
Name the possible congruent-triangle pairs?
A
C B E FD
Congruent Triangles - 5
ERHS Math Geometry
Mr. Chin-Sung Lin
Name the possible congruent-triangle pairs?
A
C B E FD
∆ 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
E F D
OG 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
E F D
OG H
Congruent Triangles - 9
ERHS Math Geometry
Mr. Chin-Sung Lin
Name the possible congruent-triangle pairs?
A
C B
E FD
Congruent Triangles - 9
ERHS Math Geometry
Mr. Chin-Sung Lin
Name the possible congruent-triangle pairs?
∆ ABD & ∆ ACF
∆ ADE & ∆ AFE
A
C B
E FD
Congruent Triangles - 10
ERHS Math Geometry
Mr. Chin-Sung Lin
Name the possible congruent-triangle pairs?
A
C B
E FD
G H
Congruent Triangles - 10
ERHS Math Geometry
Mr. Chin-Sung Lin
Name the possible congruent-triangle pairs?
A
C B
E FD
G H
∆ 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
E FD
G H
I JO
Congruent Triangles - 11
ERHS Math Geometry
Mr. Chin-Sung Lin
Name the possible congruent-triangle pairs?
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
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 = PBProve: P lies on the perpendicular bisector of AB
BM
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 ABProve: PA = PB
BM
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
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