Geometry
Section 4.1 - Congruent Figures
Congruent figures – two figures with the same size and shape
A
B
C
D
Congruent segments
Congruent angles
---------------------------------------------------------------------------------------------------------------------Congruent Triangles
Sides -> size of triangle
Angles -> shape of triangle
B
C
Corresponding Vertices
A <---> D
C <---> F
B <----> E
Corresponding Sides
AC      DF
CB      FE
CB      FE
D
A
( ABC fits exactly over DEF )
E
F
Corresponding Angles
A      D
C      F
B      E
Congruent Triangles (definition) – Two triangles are congruent iff their vertices can be matched so that
corresponding sides and angles are congruent.
ABC  DEF ( ABC fits exactly over DEF after flipping it)
A
F
E
BCA  EFD - also valid
CAB  FDE - also valid
B
C
D
Must line up corresponding vertices! We don’t write CAB  DEF !
----------------------------------------------------------------------------------------------------------------------
GH  JK
HI  KL
IG  LJ
G  J
H  K
I  L
D
A
If given GHI  JKL , what can you conclude?
C
B
F
E
The reason you can make these conclusions (e.g GH  JK or G  J ), is:
Corresponding Parts of Congruent Triangles are Congruent
Or
Corr parts of  s are 
Or
CPCTC
Note: This is the same as the definition of congruent triangles so instead of saying “Definition of Congruent Triangles” in a
proof you can use the above.
Congruent polygons (definition) – Two polygons are congruent iff their vertices can be matched so that
corresponding parts are congruent.
Geometry
4.2 & 4.3 - Proving Triangles are Congruent
If two triangles are congruent, the 6 parts of one triangle are congruent to the 6 corresponding parts of
the other triangle. However we don’t necessarily need to have all 6 parts given to us in order to
guarantee that two triangles are congruent. We will discover some triangle congruency postulates today.
_____________ Postulate
If _________________________________________of one triangle are congruent to __________________________________________________
of another triangle, then the triangles are congruent.
________________Postulate
If _________________________________________of one triangle are congruent to __________________________________________________
of another triangle, then the triangles are congruent.
________________Postulate
If _________________________________________of one triangle are congruent to __________________________________________________
of another triangle, then the triangles are congruent.
Example 1
Statements
Reasons
1)
1)
2)
2) Reflexive Prop.
3)
3) _____________________ Postulate
4)
4) Corresponding Parts of Congruent Triangles are Congruent
Example 2
Given: AB and CD bisect each other
at M
Prove: AD P BC
Statements
Reasons
1)
2) M is the midpoint of _________and_______
2) Def. of a bisector of a segment
3)
3) Def. of midpoint
4)
4) Vertical angles are congruent
5)
5) ________________ Postulate
6)
6) CPCTC
7)
7) If ___________________________ are ≅, then the lines are parallel.
Example 3:
Given: mÐ1 = mÐ2; mÐ3 = mÐ4
Prove: M is the midpoint of JK
Statements
Reasons
A line and plane are perpendicular iff (a) they intersect and (b) the line is perpendicular to all lines in the plane
that pass through the point of intersection.
suur
If XY ^ plane P, then
(a)
(b)
(c)
suur sur
XY
suur ^ YA
sur
XY
^
YB
suur sur
XY ^ YC
-------------------------------------------------------------------------------------------------------------------Example 4:
Given: PO ^ plane X
AO @ BO
Prove: PA @ PB
Statements
Reasons
Geometry
Section 4.4 - The Isosceles Triangle Theorems
Isosceles Triangle (definition) – a triangle with at least two equal sides
Isosceles Triangle Theorem – If two sides of a triangle are congruent, then the
sides are congruent.
angles opposite the
If XY @ XZ then ÐY @ ÐZ
Converse of the Isosceles Triangle Theorem – If two angles of a triangle
congruent, then the sides opposite the angles are congruent.
If ÐY @ ÐZ then XY @ XZ
How would you write these two theorems as a biconditional?
Corollary: An equilateral triangle is also equiangular
Corollary: An equiangular triangle is also equilateral
(is this different than what we discovered about polygons in general?)
o
Corollary: An equilateral triangle has three 60 angles
Corollary: The bisector of the vertex angle of an isosceles triangle is perpendicular
to the base at its midpoint. (Draw this.)
are
Geometry
Section 4.5 - Other Methods of Proving Triangles Congruent
Two additional theorems to prove triangles congruent:
AAS Theorem - If two angles and a non-included side of one triangle are congruent to the corresponding parts of
another triangle, then the triangles are congruent.
DABC @ DDEF by AAS Thm.
HL Theorem – If the hypotenuse and one leg of one right triangle are congruent to the corresponding parts of
another right triangle, then the triangles are congruent.
DGHJ @ DKLM by HL Thm.
HL Theorem is a special case of SSA. Why do you think it yields congruent triangles in this special case?
Overlapping Triangles – some problems involve triangles that overlap. In these cases you need to:
SEPARATE THE TRIANGLES AND REDRAW THE DIAGRAM!!!
Example 1:
Given: BD @ BE ;
ÐA @ ÐC
Prove: DABE @ DCBD
Example 2:
Given: XY ^ AB ; XA @ XB
Prove: Ð1 @ Ð2
Example 3:
(Challenging problem – need to prove one pair of triangles are congruent first. Then use CPCTC to help
prove the second pair congruent).
Given: ÐABC @ ÐACB; AE ^ EC ; AD ^ DB
Prove: DABD @ DACE
Geometry
Section 4.6 - Using More Than One Pair of Congruent Triangles
Objective: Prove two triangles congruent by first proving two other triangles are congruent and then using
CPCTC.
1. Given: Ð1 @ Ð2 ; Ð3 @ Ð4
Prove: TU @ TW
Statements
Reasons
2. Given: Ð1 @ Ð2 ; Ð5 @ Ð6
Prove: AC ^ BD
Statements
Reasons
3. Given: PQ ^ QR ; PQ @ PS
Prove: O is the midpoint of QS
Statements
Reasons
4. Given: RP bisects ÐSPT and ÐSRT
Prove: RP bisects ÐSQT
Statements
Reasons
Section 4.7 - Medians, Altitudes, and Perpendicular Bisectors
A median of a triangle is a segment from a vertex to the midpoint of the opposite side.
An altitude of a triangle is the perpendicular segment from a vertex to the line that contains the opposite
side.
A perpendicular bisector of a segment is a line that is perpendicular to the segment at its midpoint.
Theorems




If a point lies on the perpendicular bisector of a segment, then the point is equidistant from the
endpoints of the segment.
If a point is equidistant from the endpoints of a segment, then the point lies on the perpendicular
bisector of the segment.
If a point lies on the bisector of an angle, then the point is equidistant from the sides of the angle.
If a point is equidistant from the sides of an angle, then the point lies on the bisector of the angle.
Examples
In the diagram, BD is the perpendicular bisector of AC.
a.
What segment lengths are equal?
b.
What is the value of x?
c.
Find AB.
Refer to the diagram and name each of the following.
a.
a median of
b.
an altitude of
c.
a bisector of an angle of
What kind of triangle has three angle bisectors that are also altitudes and medians?
4.7 Medians, Altitudes, and Perpendicular Bisectors
Objectives: (1) Define and apply altitude, median and perpendicular bisector.
(2) State and apply theorem about point on perpendicular bisector of a
segment.
(3) State and apply theorem about point on angle bisector of a segment.
Know the difference between an altitude and a median of a triangle.
Median: segment from a vertex to midpoint of opposite side. Every triangle has 3 different medians.
CM is a median
AM is a median
BM is a median
Altitude: perpendicular segment in a triangle from a vertex to the line that contains the opposite side.
(1) In acute triangles, the three altitudes are inside the triangle.
(2) In right triangles, two of the altitudes are part of the triangle, one altitude is inside the triangle.
(3) In obtuse triangles, two of the altitudes are outside the triangle, one altitude is inside the triangle.
Perpendicular Bisector of a Segment (definition): line, ray or segment that is perpendicular to a segment at its midpoint.
Line k is a perpendicular bisector of AB , with midpoint M.
Theorem: A point lies on the perpendicular bisector of a segment iff the point is equidistant from the endpoints of the
segment.
Point Y lies on the perpendicular bisector
of XZ and is equidistant from the endpoints
of XZ .
Distance from a point to a line (definition): length of the perpendicular segment from the point to the line.
Example: The distance from point Y to XZ (in the figure above) is YM (the length of YM ).
Theorem: A point lies on the bisector of an angle iff the point is equidistant from the sides of the angle.
Point S lies on the bisector of ÐQPR
and is equidistant from the sides of
the angle. ÐQPR