Transcript Geometry - Eleanor Roosevelt High School

Congruence Based on
Triangles
Eleanor Roosevelt High School
Geometry
Mr. Chin-Sung Lin
ERHS Math Geometry
Line Segments Associated
with Triangles
Mr. Chin-Sung Lin
ERHS Math Geometry
Altitude of a Triangle
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
B
A
B
B
C A
A
C
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Altitude of a Triangle
If BD is the altitude of ∆ ABC
B
then,
m BDA = 90
m BDC = 90
A
D
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Altitude - Area of a Triangle
Altitudes can be used to compute the area of a triangle:
Area = 1/2 * Base * Altitude
B
A
Base
B
B
Altitude
Altitude
C A
A
Base C
C
Base
Mr. Chin-Sung Lin
ERHS Math Geometry
Altitude - Orthocenter
Three altitudes intersect in a single point, called the
orthocenter of the triangle
B
Orthocenter
A
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Altitude - Orthocenter
Where is the orthocenter of a right triangle?
B
Orthocenter?
A
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Altitude - Orthocenter
The orthocenter is located at the vertex of the right angle
B
Orthocenter
A
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Altitude - Orthocenter
Where is the orthocenter of an obtuse triangle?
B
A
Orthocenter?
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Altitude - Orthocenter
Orthocenter
The orthocenter is outside the triangle
B
A
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Angle Bisector of
a Triangle
Mr. Chin-Sung Lin
ERHS Math Geometry
Angle Bisector of a Triangle
A line segment that bisects an angle of the triangle and
terminates in the side opposite that angle
A
B
B
A
B
A
C
C
C
Mr. Chin-Sung Lin
ITHS Math B Term 1 (M$4)
Angle Bisector of a Triangle
If BD is the angle bisector of ABC
then,
ABD  CBD
B
A
D
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Angle Bisector - Incenter
The three angle bisectors of a triangle meet in one point
called the incenter
B
Incenter
A
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Angle Bisector - Incenter
Incenter is the center of the incircle, the circle inscribed in
the triangle
B
Incenter
A
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Median of a Triangle
Mr. Chin-Sung Lin
ERHS Math Geometry
Median of a Triangle
A segment from a vertex to the midpoint of the opposite
side
A
B
B
A
B
A
C
C
C
Mr. Chin-Sung Lin
ITHS Math B Term 1 (M$4)
Median of a Triangle
If BD is the median of ∆ ABC
then,
B
AD  CD
A
D
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Median of a Triangle - Centroid
The three medians meet in the centroid or center of mass
(center of gravity)
B
Centroid
A
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Median of a Triangle - Centroid
The centroid divides each median in a ratio of 2:1.
B
2
Centroid
A
1
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Perpendicular Bisector
of a Triangle
Mr. Chin-Sung Lin
ERHS Math Geometry
Perpendicular Bisector
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
A
AB  CD
CO ~
= OD
O
C
D
B
Mr. Chin-Sung Lin
ERHS Math Geometry
Perpendicular Bisector of a Triangle
A line, a ray, or a line segment that is perpendicular to the
side of a triangle at its midpoint
A
B
B
A
B
A
C
C
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Perpendicular Bisector of a Triangle
If DE is the perpendicular bisector of the side of ∆ ABC
then,
B
AD  CD
E
DE  AC
A
D
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Perpendicular Bisector - Circumcenter
The three perpendicular bisectors meet in one point called
the circumcenter
B
Circumcenter
A
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Perpendicular Bisector - Circumcenter
Circumcenter is the center of the circumcircle, the circle
passing through the vertices of the triangle
B
Circumcenter
A
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Scalene Triangle
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
B
G
A
D
E
F
BD:
BE:
BF:
FG:
Altitude
Angle bisector
Median
Perpendicular
C Bisector
Mr. Chin-Sung Lin
ERHS Math Geometry
Isosceles & Equilateral Triangles
In isosceles & equilateral triangles, some of the
altitude, angle bisector, median, and
perpendicular bisector coincide
B
A
D
C
BD:
BD:
BD:
BD:
Altitude
Angle bisector
Median
Perpendicular
Bisector
Mr. Chin-Sung Lin
ERHS Math Geometry
Scalene Triangle (Indirect Proof)
Given: ∆ ABC is scalene, BD bisects ABC
Prove: BD is not perpendicular to AC
B
1 2
A
3 4
D
C
Mr. Chin-Sung Lin
B
ERHS Math Geometry
1 2
Scalene Triangle
Statements
A
3 4
D
C
Reasons
Mr. Chin-Sung Lin
B
ERHS Math Geometry
1 2
Scalene Triangle
Statements
1. BD  AC
A
3 4
D
C
Reasons
1. Assume the opposite is true
Mr. Chin-Sung Lin
B
ERHS Math Geometry
1 2
Scalene Triangle
Statements
1. BD  AC
2. ∆ ABC is scalene, BD is angle
bisector
A
3 4
D
C
Reasons
1. Assume the opposite is true
2. Given
Mr. Chin-Sung Lin
B
ERHS Math Geometry
1 2
Scalene Triangle
Statements
1. BD  AC
2. ∆ ABC is scalene, BD is angle
bisector
3. 1  2
A
3 4
D
C
Reasons
1. Assume the opposite is true
2. Given
3. Definition of angle bisector
Mr. Chin-Sung Lin
B
ERHS Math Geometry
1 2
Scalene Triangle
Statements
1. BD  AC
2. ∆ ABC is scalene, BD is angle
bisector
3. 1  2
4. 3 = 90o, 4 = 90o
A
3 4
D
C
Reasons
1. Assume the opposite is true
2. Given
3. Definition of angle bisector
4. Definition of perpendicular
Mr. Chin-Sung Lin
B
ERHS Math Geometry
1 2
Scalene Triangle
Statements
1. BD  AC
2. ∆ ABC is scalene, BD is angle
bisector
3. 1  2
4. 3 = 90o, 4 = 90o
5. 3  4
A
3 4
D
C
Reasons
1. Assume the opposite is true
2. Given
3. Definition of angle bisector
4. Definition of perpendicular
5. Substitution postulate
Mr. Chin-Sung Lin
B
ERHS Math Geometry
1 2
Scalene Triangle
A
Statements
1. BD  AC
2. ∆ ABC is scalene, BD is angle
bisector
3. 1  2
4. 3 = 90o, 4 = 90o
5. 3  4
6. BD  BD
3 4
D
C
Reasons
1. Assume the opposite is true
2. Given
3.
4.
5.
6.
Definition of angle bisector
Definition of perpendicular
Substitution postulate
Reflexive property
Mr. Chin-Sung Lin
B
ERHS Math Geometry
1 2
Scalene Triangle
A
Statements
1. BD  AC
2. ∆ ABC is scalene, BD is angle
bisector
3. 1  2
4. 3 = 90o, 4 = 90o
5. 3  4
6. BD  BD
7. ∆ ABD  ∆ CBD
3 4
D
C
Reasons
1. Assume the opposite is true
2. Given
3.
4.
5.
6.
7.
Definition of angle bisector
Definition of perpendicular
Substitution postulate
Reflexive property
ASA postulate
Mr. Chin-Sung Lin
B
ERHS Math Geometry
1 2
Scalene Triangle
A
Statements
1. BD  AC
2. ∆ ABC is scalene, BD is angle
bisector
3. 1  2
4. 3 = 90o, 4 = 90o
5. 3  4
6. BD  BD
7. ∆ ABD  ∆ CBD
8. AB = CB
3 4
D
C
Reasons
1. Assume the opposite is true
2. Given
3.
4.
5.
6.
7.
8.
Definition of angle bisector
Definition of perpendicular
Substitution postulate
Reflexive property
ASA postulate
CPCTC
Mr. Chin-Sung Lin
B
ERHS Math Geometry
1 2
Scalene Triangle
A
Statements
1. BD  AC
2. ∆ ABC is scalene, BD is angle
bisector
3. 1  2
4. 3 = 90o, 4 = 90o
5. 3  4
6. BD  BD
7. ∆ ABD  ∆ CBD
8. AB = CB
9. AB ≠ CB
3 4
D
C
Reasons
1. Assume the opposite is true
2. Given
3.
4.
5.
6.
7.
8.
9.
Definition of angle bisector
Definition of perpendicular
Substitution postulate
Reflexive property
ASA postulate
CPCTC
Definition of scalene triangle
Mr. Chin-Sung Lin
B
ERHS Math Geometry
1 2
Scalene Triangle
Statements
1. BD  AC
2. ∆ ABC is scalene, BD is angle
bisector
3. 1  2
4. 3 = 90o, 4 = 90o
5. 3  4
6. BD  BD
7. ∆ ABD  ∆ CBD
8. AB = CB
9. AB ≠ CB
10. BD is not perpendicular to AC
A
3 4
D
C
Reasons
1. Assume the opposite is true
2. Given
3. Definition of angle bisector
4. Definition of perpendicular
5. Substitution postulate
6. Reflexive property
7. ASA postulate
8. CPCTC
9. Definition of scalene triangle
10. Contradition in statement 8
& 9, so, assumption is false.
The negation of the
assumption is true
Mr. Chin-Sung Lin
ERHS Math Geometry
CPCTC
Mr. Chin-Sung Lin
ERHS Math Geometry
CPCTC
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
Mr. Chin-Sung Lin
ERHS Math Geometry
Congruent Triangles
Given: B  C , and AB  AC
Prove: AF  AE
B
E
D
A
F
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Congruent Triangles
Given: B  C , and AB  AC
Prove: AF  AE
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
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
4.
AF  AE
4. CPCTC
Mr. Chin-Sung Lin
ERHS Math Geometry
Isosceles Triangles
Mr. Chin-Sung Lin
ERHS Math Geometry
Isosceles Triangles
An isosceles triangle is a triangle that has two
congruent sides
B
A
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Parts of an Isosceles Triangle
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
B
Vertex Angle
Leg
Base Angle
Leg
A
Base
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Base Angle Theorem
(Isosceles Triangle Theorem)
Mr. Chin-Sung Lin
ERHS Math Geometry
Base Angle Theorem
(Isosceles Triangle Theorem)
If two sides of a triangle are congruent, then the angles
opposite these sides are congruent
(Base angles of an isosceles triangle are congruent)
Mr. Chin-Sung Lin
ERHS Math Geometry
Base Angle Theorem
If two sides of a triangle are congruent, then the angles
opposite these sides are congruent
Draw a diagram like the one below
Given:
Prove:
AB  CB
A  C
B
A
C
Mr. Chin-Sung Lin
ERHS Math Geometry
B
Base Angle Theorem
A
Statements
D
C
Reasons
1.
1.
2.
3.
4.
5.
6.
2.
3.
4.
5.
6.
Mr. Chin-Sung Lin
ERHS Math Geometry
B
Base Angle Theorem
A
Statements
1. Draw the angle bisector of
ABC and let D be the point
where it intersects AC
2. ABD  CBD
3. AB  CB
4. BD  BD
5. ∆ ABD = ∆ CBD
6. A  C
D
C
Reasons
1. Any angle of measure less
than 180 has exactly one
bisector
2. Definition of angle bisector
3. Given
4. Reflexive property
5. SAS Postulate
6. CPCTC
Mr. Chin-Sung Lin
ERHS Math Geometry
Base Angle Theorem - Example 1
Given: AB  CB and AD  CE
Prove: ∆ ABD = ∆ CBE
B
A
D
E
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Base Angle Theorem - Example 1
Given: AB  CB and AD  CE
Prove: ∆ ABD = ∆ CBE
B
A
D
E
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Base Angle Theorem - Example 1
B
A
D
C
E
Statements
Reasons
1.
1.
2.
3.
2.
3.
Mr. Chin-Sung Lin
ERHS Math Geometry
Base Angle Theorem - Example 1
B
A
Statements
1. AB  CB
AD  CE
2. A  C
3. ∆ ABD = ∆ CBE
D
E
C
Reasons
1. Given
2. Base Angle Theorem
3. SAS Postulate
Mr. Chin-Sung Lin
ERHS Math Geometry
Base Angle Theorem - Example 2
Given: 1  2 and 5  6
Prove: 3  4
C
A
1
2
3
O
4
5
6
B
D
Mr. Chin-Sung Lin
ERHS Math Geometry
Base Angle Theorem - Example 2
Given: 1  2 and 5  6
Prove: 3  4
C
A
1
2
3
O
4
5
6
B
D
Mr. Chin-Sung Lin
ERHS Math Geometry
Base Angle Theorem - Example 2
C
A
1
2
3
O
4
Statements
5
6
B
Reasons
D
1.
1.
2.
3.
4.
5.
6.
2.
3.
4.
5.
6.
Mr. Chin-Sung Lin
ERHS Math Geometry
Base Angle Theorem - Example 2
C
A
1
2
3
O
4
Statements
D
1. 1  2
5  6
2. AB  AB
3. ∆ ACB = ∆ ADB
4. AC  AD
5. ∆ ADC is an isosceles triangle
6. 3  4
5
6
B
Reasons
1. Given
2. Reflexive Property
3. ASA Postulate
4. CPCTC
5. Def. of Isosceles Triangle
6. Base Angle Theorem
Mr. Chin-Sung Lin
ERHS Math Geometry
Base Angle Theorem - Exercise
Given: BD  BE and AD  CE
Prove: AB = CB
B
A
D
E
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Converse of
Base Angle Theorem
(Converse of Isosceles
Triangle Theorem)
Mr. Chin-Sung Lin
ERHS Math Geometry
Converse of Base Angle Theorem
If two angles of a triangle are congruent, then the sides
opposite these angles are congruent
Mr. Chin-Sung Lin
ERHS Math Geometry
Converse of Base Angle Theorem
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
B
Prove:
AB  CB
A
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Converse of Base Angle Theorem
B
Statements
A
C
D
1.
1.
2.
3.
4.
5.
6.
2.
3.
4.
5.
6.
Reasons
Mr. Chin-Sung Lin
ERHS Math Geometry
Converse of Base Angle Theorem
B
Statements
A
D
1. Draw the angle bisector of
ABC and let D be the point
where it intersects AC
2. ABD  CBD
3. A  C
4. BD  BD
5. ∆ ABD = ∆ CBD
6. AB  CB
C
Reasons
1. Any angle of measure less
than 180 has exactly one
bisector
2. Definition of angle bisector
3. Given
4. Reflexive property
5. AAS Postulate
6. CPCTC
Mr. Chin-Sung Lin
ERHS Math Geometry
Base Angle Theorem - Example 3
Given: AO  BO and 1  2
Prove: AC = BD
A
B
O
C
1
2
D
Mr. Chin-Sung Lin
ERHS Math Geometry
Base Angle Theorem - Example 3
Given: AO  BO and 1  2
Prove: AC = BD
A
B
O
C
1
2
D
Mr. Chin-Sung Lin
ERHS Math Geometry
Base Angle Theorem - Example 3
A
B
O
C
1
2
D
Statements
Reasons
1.
1.
2.
3.
2.
3.
Mr. Chin-Sung Lin
ERHS Math Geometry
Base Angle Theorem - Example 3
A
B
O
C
Statements
1. 1  2
2. CO  DO
3. AO  BO
4. AOC  BOD
5. ∆ AOC = ∆ BOD
6. AC  BD
1
2
D
Reasons
1. Given
2. Converse of Base Angle
Theorem
3. Given
4. Vertical Angles
5. SAS Postulate
6. CPCTC
Mr. Chin-Sung Lin
ERHS Math Geometry
Corollaries of Base Angle Theorem
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
Mr. Chin-Sung Lin
ERHS Math Geometry
Equilateral and
Equiangular Triangles
Mr. Chin-Sung Lin
ERHS Math Geometry
Equilateral Triangles
A equilateral triangle is a triangle that has three
congruent sides
B
A
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Equilateral & Equiangular Triangles
If a triangle is an equilateral triangle, then it is an
equiangular triangle
Mr. Chin-Sung Lin
ERHS Math Geometry
Identify Overlapping
Triangles
Mr. Chin-Sung Lin
ERHS Math Geometry
Identify Triangles - 1
How many triangles can you identify in the following
diagram?
A
B
O
D
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Identify Triangles - 1
How many triangles can you identify in the following
diagram?
A
B
O
D
∆ ADC
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Identify Triangles - 1
How many triangles can you identify in the following
diagram?
A
B
O
D
∆ BCD
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Identify Triangles - 1
How many triangles can you identify in the following
diagram?
A
B
O
D
∆ DAB
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Identify Triangles - 1
How many triangles can you identify in the following
diagram?
A
B
O
D
∆ CBA
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Identify Triangles - 1
How many triangles can you identify in the following
diagram?
A
B
O
D
∆ DOC
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Identify Triangles - 1
How many triangles can you identify in the following
diagram?
A
B
O
D
∆ AOB
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Identify Triangles - 1
How many triangles can you identify in the following
diagram?
A
B
O
D
∆ AOD
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Identify Triangles - 1
How many triangles can you identify in the following
diagram?
A
B
O
D
∆ BOC
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Identify Triangles - 1
How many triangles can you identify in the following
diagram?
A
B
Total
8 Triangles
O
D
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Identify Triangles - 2
How many triangles can you identify in the following
diagram?
A
D
B
O
E
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Identify Triangles - 2
How many triangles can you identify in the following
diagram?
A
D
B
O
E
∆ BDC
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Identify Triangles - 2
How many triangles can you identify in the following
diagram?
A
D
B
O
E
∆ CEB
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Identify Triangles - 2
How many triangles can you identify in the following
diagram?
A
D
B
O
E
∆ AEB
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Identify Triangles - 2
How many triangles can you identify in the following
diagram?
A
D
B
O
E
∆ ADC
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Identify Triangles - 2
How many triangles can you identify in the following
diagram?
A
D
B
O
E
∆ DOB
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Identify Triangles - 2
How many triangles can you identify in the following
diagram?
A
D
B
O
E
∆ EOC
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Identify Triangles - 2
How many triangles can you identify in the following
diagram?
A
D
B
O
E
∆ BOC
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Identify Triangles - 2
How many triangles can you identify in the following
diagram?
A
D
B
O
E
∆ ABC
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Identify Triangles - 2
How many triangles can you identify in the following
diagram?
A
D
B
O
Total
8 Triangles
E
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Shared Sides & Angles
Mr. Chin-Sung Lin
ERHS Math Geometry
Shared Side - 1
Which two congruent-triangle candidates have a
shared side? Which line segment has been shared?
A
B
O
D
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Shared Side - 1
Which two congruent-triangle candidates have a
shared side? Which line segment has been shared?
A
B
∆ ADC & ∆ BCD
O
D
C
DC
Mr. Chin-Sung Lin
ERHS Math Geometry
Shared Side - 2
Which two congruent-triangle candidates have a
shared side? Which line segment has been shared?
C
A
E
O
D
F
B
Mr. Chin-Sung Lin
ERHS Math Geometry
Shared Side - 2
Which two congruent-triangle candidates have a
shared side? Which line segment has been shared?
C
A
E
O
D
F
∆ ACF & ∆ BDE
B
EF
Mr. Chin-Sung Lin
ERHS Math Geometry
Shared Side - 3
Which two congruent-triangle candidates have a
shared side? Which line segment has been shared?
A
B
D
E
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Shared Side - 3
Which two congruent-triangle candidates have a
shared side? Which line segment has been shared?
A
∆ AEB & ∆ ADC
B
D
E
C
DE
Mr. Chin-Sung Lin
ERHS Math Geometry
Shared Angle - 1
Which two congruent-triangle candidates have a
shared angle? Which angle has been shared?
A
D
B
O
E
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Shared Angle - 1
Which two congruent-triangle candidates have a
shared angle? Which angle has been shared?
A
∆ AEB & ∆ ADC
D
B
O
E
BAC
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Shared Angle - 2
Which two congruent-triangle candidates have a
shared angle? Which angle has been shared?
A
B
D
E
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Shared Angle - 2
Which two congruent-triangle candidates have a
shared angle? Which angle has been shared?
A
∆ AEB & ∆ ADC
B
D
E
C
DAE
Mr. Chin-Sung Lin
ERHS Math Geometry
Congruent Overlapping
Triangles
Mr. Chin-Sung Lin
ERHS Math Geometry
Congruent Triangles - 1
Name the possible congruent-triangle pairs?
A
D
B
O
E
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Congruent Triangles - 1
Name the possible congruent-triangle pairs?
A
∆ AEB & ∆ ADC
D
B
O
E
∆ DOB & ∆ EOC
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Congruent Triangles - 2
Name the possible congruent-triangle pairs?
A
B
O
D
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Congruent Triangles - 2
Name the possible congruent-triangle pairs?
A
B
O
∆ ADC & ∆ BCD
∆ AOD & ∆ BOC
D
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Congruent Triangles - 3
Name the possible congruent-triangle pairs?
A
D
B
O
E
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Congruent Triangles - 3
Name the possible congruent-triangle pairs?
A
D
B
O
∆ AEB & ∆ ADC
E
∆ BDO & ∆ CEO
C
∆ ECB & ∆ DBC
Mr. Chin-Sung Lin
ERHS Math Geometry
Congruent Triangles - 4
Name the possible congruent-triangle pairs?
A
B
D
E
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Congruent Triangles - 4
Name the possible congruent-triangle pairs?
A
B
D
∆ AEB & ∆ ADC
E
C
∆ ADB & ∆ AEC
Mr. Chin-Sung Lin
ERHS Math Geometry
Congruent Triangles - 5
Name the possible congruent-triangle pairs?
A
B
D
E
F
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Congruent Triangles - 5
Name the possible congruent-triangle pairs?
∆ ABD & ∆ ACF
A
∆ ADE & ∆ AFE
∆ ABE & ∆ ACE
∆ ABF & ∆ ACD
B
D
E
F
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Congruent Triangles - 6
Name the possible congruent-triangle pairs?
A
B
O
C
D
Mr. Chin-Sung Lin
ERHS Math Geometry
Congruent Triangles - 6
Name the possible congruent-triangle pairs?
A
B
∆ ABC & ∆ BAD
∆ AOC & ∆ BOD
O
C
D
∆ ACD & ∆ BDC
Mr. Chin-Sung Lin
ERHS Math Geometry
Congruent Triangles - 7
Name the possible congruent-triangle pairs?
A
B
E
D
F
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Congruent Triangles - 7
Name the possible congruent-triangle pairs?
A
B
E
D
∆ ABD & ∆ CDB
∆ ADE & ∆ CBF
F
C
∆ ABE & ∆ CDF
Mr. Chin-Sung Lin
ERHS Math Geometry
Congruent Triangles - 8
Name the possible congruent-triangle pairs?
A
G
C
E
B
O
H
F
D
Mr. Chin-Sung Lin
ERHS Math Geometry
Congruent Triangles - 8
Name the possible congruent-triangle pairs?
A
G
C
E
B
O
∆ AGO & ∆ BHO
H
∆ CGE & ∆ DHF
F
D
∆ AED & ∆ BFC
Mr. Chin-Sung Lin
ERHS Math Geometry
Congruent Triangles - 9
Name the possible congruent-triangle pairs?
A
B
C
D
E
F
Mr. Chin-Sung Lin
ERHS Math Geometry
Congruent Triangles - 9
Name the possible congruent-triangle pairs?
A
B
C
∆ ABD & ∆ ACF
∆ ADE & ∆ AFE
D
E
F
Mr. Chin-Sung Lin
ERHS Math Geometry
Congruent Triangles - 10
Name the possible congruent-triangle pairs?
A
B
C
G
D
H
E
F
Mr. Chin-Sung Lin
ERHS Math Geometry
Congruent Triangles - 10
Name the possible congruent-triangle pairs?
∆ ABG & ∆ ACH
A
∆ AGE & ∆ AHE
B
C
G
D
E
∆ ABE & ∆ ACE
H
∆ ADE & ∆ AFE
F
∆ GDE & ∆ HFE
Mr. Chin-Sung Lin
ERHS Math Geometry
Congruent Triangles - 11
Name the possible congruent-triangle pairs?
A
B
I
D
H
G
O
E
C
J
F
Mr. Chin-Sung Lin
ERHS Math Geometry
Congruent Triangles - 11
Name the possible congruent-triangle pairs?
∆ AGO & ∆ AHO
A
B
I
D
H
G
O
E
C
∆ BGI & ∆ CHJ
∆ IDE & ∆ JFE
J
F
∆ BOE & ∆ COE
∆ AIE & ∆ AJE
∆ ADE & ∆ AFE
Mr. Chin-Sung Lin
ERHS Math Geometry
Theorems about
Perpendicular Bisector
Mr. Chin-Sung Lin
ERHS Math Geometry
Perpendicular Bisector
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
A
AB  CD
CO ~
= OD
O
C
D
B
Mr. Chin-Sung Lin
ERHS Math Geometry
Theorems of Perpendicular Bisector
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
P
O
A
B
T
Mr. Chin-Sung Lin
ERHS Math Geometry
Theorems of Perpendicular Bisector
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
P
M
A
B
Mr. Chin-Sung Lin
ERHS Math Geometry
Theorems of Perpendicular Bisector
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
P
M
A
B
Mr. Chin-Sung Lin
ERHS Math Geometry
Theorems of Perpendicular Bisector
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
P
M
A
B
Mr. Chin-Sung Lin
ERHS Math Geometry
Perpendicular Bisector Concurrence
Theorems
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
N
B
L
Q
P
A
S
M
C
R
Mr. Chin-Sung Lin
B
ERHS Math Geometry
Perpendicular Bisector
Concurrence Theorems
Statements
L
Q
N
P
A
Reasons
C
S
M
R
Mr. Chin-Sung Lin
ERHS Math Geometry
Construction
Mr. Chin-Sung Lin
ERHS Math Geometry
Construction of Perpendicular Bisector
M
A
B
Mr. Chin-Sung Lin
ERHS Math Geometry
Q&A
Mr. Chin-Sung Lin
ERHS Math Geometry
The End
Mr. Chin-Sung Lin