Download Triangle Angles Triangle Sum Conjecture The sum of the measures

Ch 4 Summary L1 Key
Name ____________________________
Chapter 4: Discovering and Proving Triangle Properties
Triangle Angles
Triangle Sum Conjecture
The sum of the measures of the
angles in every triangle is 180°.
Third Angle Conjecture
If two angles of one triangle are
congruent to two angles of another
triangle, then the third angles of the
triangles are congruent.
A
I
C
T
Z
P
m∠A + m∠C + m∠T = 180
m∠A = m∠I and m∠T = m∠P ,
then m∠ C = m∠ Z .
If
C
Triangle Exterior Angle Conjecture
The measure of an exterior angle of
a triangle is equal to the sum of the
measures of the remote interior
angles.
D
B
A
Triangles Inequalities
Triangle Inequality Conjecture
The sum of the lengths of any two sides of a
triangle is greater than the length of the third
side.
Side-Angle Inequality Conjecture (only applies
to triangles!)
In a triangle, if one side is the longest side, then
the angle opposite the longest side is the largest
angle.
m∠CBD = m∠A + m∠C
5
2
10
Since 2 + 5 < 10 , you cannot
make a triangle.
Lg
Medium
Short
Md
Longes t
Sm
Likewise, if one side is the shortest side, then the
angle opposite the shortest side is the smallest
angle.
S. Stirling
Page 1 of 6
Ch 4 Summary L1 Key
Name ____________________________
B
Isosceles Triangle Properties
(also for Equilateral-equiangular)
Vocab: leg, base, base angles, vertex angle.
S
V
Isosceles Triangle Conjecture
If a triangle is isosceles, then its base angles
are congruent.
Converse of the Isosceles Triangle Conj.
If a triangle has two congruent angles, then it
is an isosceles triangle.
Equilateral/Equiangular Triangle
Conjecture
Every equilateral triangle is equiangular.
Conversely, every equiangular triangle is
equilateral.
Vertex Angle Bisector Conjecture
In an isosceles triangle, the bisector of the
vertex angle is also the altitude and the
median to the base and the perpendicular
bisector of the base.
ΔVBS is isosceles,
then ∠ B ≅ ∠ S .
If
∠B ≅ ∠S ,
then ΔVBS is isosceles.
If
ΔEQU is equilateral,
then ΔEQU is
E
If
equiangular.
U
Q
ΔVBS is isosceles
VX bisects ∠BVS ,
then VX is an altitude,
If
median and
perpendicular bisector of
BS
B
X
S
20
20
V
.
A
Medians to the Congruent Sides Thm
In an isosceles triangle, the medians to the
congruent sides are congruent.
Angle Bisectors to the Congruent Sides
Thm
In an isosceles triangle, the angle bisectors to
the congruent sides are congruent.
If BN and MC are
altitudes of isosceles
ΔABC ,
then BN = MC .
C
B
If BS and CP are
angle bisectors of
A
ΔABC ,
SB = CP .
P
isosceles
then
N
M
S
C
B
A
Altitudes to the Congruent Sides Thm
In an isosceles triangle, the altitudes to the
congruent sides are congruent.
S. Stirling
If BT and LC are
altitudes of isosceles
ΔABC ,
then BT = LC .
L
T
C
B
Page 2 of 6
Ch 4 Summary L1 Key
Name ____________________________
Definition of Congruence (aka CPCTC)
Two polygons are congruent if they have the
Same shape → corresponding angles are congruent
Same size → corresponding sides are congruent.
Triangle Congruence Shortcuts
SSS Congruence Conjecture
If the three sides of one triangle are congruent to the
three sides of another triangle, then the triangles are
congruent.
SAS Congruence Conjecture
If two sides and the included angle of one triangle are
congruent to two sides and the included angle of
another triangle, then the triangles are congruent.
ASA Congruence Conjecture
If two angles and the included side of one triangle are
congruent to two angles and the included side of
another triangle, then the triangles are congruent.
SAA or AAS Congruence Conjecture
If two angles and a non-included side of one triangle
are congruent to the corresponding angles and side of
another triangle, then the triangles are congruent.
A
C
AAA Congruence Conjecture
If three angles of one triangle are congruent to the
corresponding angles of another triangle, then the
triangles are NOT necessarily congruent.
S. Stirling
T
Z
P
A
I
C
T
Z
P
A
I
C
T
Z
P
A
I
C
Hypotenuse Leg Theorem
If the hypotenuse and one leg of a right triangle are
congruent to the hypotenuse and one leg of another
right triangle, then the two triangles are congruent.
What DOES NOT guarantee Triangles Congruence
SSA or ASS Congruence?
If two sides and the non-included angle of one
triangle are congruent to two sides and the nonincluded angle of another triangle, then the triangles
are NOT necessarily congruent.
I
T
Z
P
A
I
C
T
Z
P
Y
B
A
X
C
X
B
Y
A
C
X
Z
Page 3 of 6
Z
Ch 4 Summary L1 Key
Name ____________________________
Ways to Get Equal Angles [to Get Congruent Triangles]
Isosceles Triangle Conjecture
Corresponding Angles Conjecture,
or CA Conjecture
If a triangle is isosceles,
then its base angles are congruent.
V
ΔVBS
If
isosceles
or VB ≅ VS ,
then ∠ B ≅ ∠ S
.
If two parallel lines are cut by a transversal,
then corresponding angles are congruent.
S
HJJG HJJG
If AB & DC ,
then ∠ EFA ≅ ∠ FGD .
B
E
B
A
F
Converse
If you have an angle bisector, then the ray cuts
the angle into two equal angles.
A
JJJG
If BD bisects ∠ ABC
then ∠ ABD ≅ ∠ DBC .
Alternate Interior Angles Conjecture,
or AIA Conjecture
C
If two parallel lines are cut by a transversal, then
alternate interior angles are congruent..
If ∠ABD ≅ ∠DBC ,
HJJG HJJG
If LK & NJ ,
then ∠ KHM ≅ ∠HMN
JJJG
then BD bisects ∠ ABC .
Def. perpendicular lines
L
.
Converse
B
∠LHM ≅ ∠HMJ
HJJG HJJG
then LK & NJ .
If
M
D
A
CD ⊥ AB , then
m∠CMA = m∠CMB = 90° .
M
N
,
Vertical Angle Conjecture
If two angles are vertical,
then they are congruent.
Converse
m∠AMD = 90° ,
then CD ⊥ AB .
If
T
V
E
R
C
∠VET and ∠CER are vertical,
then ∠ VET ≅ ∠ CER .
If
Def. Altitude
R
No Converse
I
A
RA is an altitude of ΔTRI
then m∠RAT = m∠RAI = 90° .
If
Converse
m∠RAT = 90° ,
then RA is an altitude of ΔTRI
“Same Angle”
Use when two triangles share the exact same angle.
∠TIA ≅ ∠RIN
I
A
R
G
If
S. Stirling
K
H
J
C
If
A segment in a triangle that goes
from a vertex perpendicular to
the line that contain the
opposite side.
T
G
D
D
B
Converse
Two lines that intersect to form
equal 90° angles.
C
∠EFB ≅ ∠FGC ,
HJJG HJJG
then AB & DC .
If
Definition of Angle Bisector
T
N
.
Page 4 of 6
Ch 4 Summary L1 Key
Name ____________________________
Ways to Get Equal Segments [to Get Congruent Triangles]
Converse of the Isosceles Triangle Conj.
Def. Isosceles Triangle
If a triangle has two congruent angles,
then it is an isosceles triangle.
If a triangle is isosceles,
then its legs are congruent.
If ∠ B ≅ ∠ S
,
then VB ≅ VS
or ΔVBS is isosceles.
V
ΔVBS isosceles
then VB ≅ VS .
If
S
V
S
B
B
Perpendicular Bisector Conjecture
If a point is on the perpendicular bisector of a segment,
then it is equidistant from the endpoints.
Def. Midpoint
A point that divides a segment into two equal
segments.
If M is the midpoint of SG ,
If C is on perp. bisect of AB , C
then AC = CB .
M
S
then SM ≅ MG .
Converse
Converse
G
If SM = MG
then M is the midpoint of SG .
CM bisects AB , then
AM ≅ MB .
If
AM ≅ MB ,
then CM bisects AB .
then D is on perp. bisect of AB .
A median is a segment that connects a vertex of a
triangle to the midpoint of the opposite side.
A line (or part of a line) that passes through the
midpoint of the segment.
C
If
D
Def. Median
Def. segment bisector
Converse
A
If AD = DB
M
B
If AM is median in ΔABC ,
then BM ≅ MC
B
.
M
B
M
A
Converse
C
A
If BM = MC
then AM is median in ΔABC .
“Same Segment”
Use when two triangles share the exact same segment.
ET ≅ TE
B
W
O
E
S. Stirling
T
Page 5 of 6
Ch 4 Summary L1 Key
Ways to Get equal Angles and
Sides [to Get Congruent Triangles]
Name ____________________________
Def. perpendicular bisector
A line (or part of a line) that
passes through the midpoint of
a segment and is perpendicular
to the segment.
C
B
M
A
D
CD is the perp. bisector of AB ,
then ∠ CMA = ∠ CMB = 90°
and AM ≅ MB .
If
Converse
AM ≅ MB and ∠CMA = 90° ,
then CD is the perp. bisector of AB .
If
Basic Procedure for Proofs
“parts” refers to sides and/or angles.
1. Get equal parts by using given info. and known definitions and conjectures.
2. State the triangles are congruent by SSS, SAS, ASA or AAS.
3. Use CPCTC to get more equal parts.
4. Connect that info. to what you were trying to prove .
Hints [if you get stuck]:
Mark the diagram with what you have stated as congruent in your proof.
( If given M is the midpoint of
midpoint before marking the diagram!)
AB , convert it to AM = MB by def. of
Look at the diagram to find equal parts.
Brainstorm and then apply previous conjectures and definitions.
Work (or think) backwards!
Draw overlapping triangles separately.
Re-draw figures without all of the “extra segments” in there.
Draw an auxillary line.
Break a problem into smaller parts.
S. Stirling
Page 6 of 6