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4.1 Triangles and Angles
Triangle: a figure formed by three segments joining three noncollinear
points. It can be classified by its sides and by its angles.
Classification by Sides
Equilateral
Isosceles
3 congruent sides
at least 2 congruent sides no congruent sides
Classification by Angles
Acute
Equiangular
3 acute angles
Scalene
3 congruent angles
Right
Obtuse
1 right angle
1 obtuse angle
Vertex: each of three points joining the sides of a triangle.
A, B, and C are vertices of triangle ABC.
A
B
C
Adjacent sides: two sides sharing a common vertex. AB and BC share vertex B.
Right and Isosceles Triangles
Right Triangle
Isosceles Triangle
hypotenuse
leg
leg
leg
leg
base
Triangle Sum Theorem:
The sum of the measures of the interior angles of a triangle is 180*.
m<A + m<B + m<C = 180*
A
B
C
Exterior Angle Theorem:
The measure of an exterior angle of a triangle is equal to the sum of
the measures of the two nonadjacent interior angles.
m<1 = m<A + m<B
A
B
C
Corollary to the Triangle Sum Theorem:
The acute angles of a right triangle are complementary.
m<A + m<B = 90*
A
C
B
4.2 Congruence and Triangles
Congruent: two geometric figures that have exactly the same size and
shape. When two figures are congruent, there is a correspondence
between their angles and sides such that corresponding sides are
congruent and corresponding angles are congruent.
ABC = PQR
A
B
Corresponding Angles
<A = <P
<B = <Q
<C = <R
P
C
R
Corresponding Sides
AB = PQ
BC = QR
CA = RP
Q
Third Angle Theorem:
If two angles of one triangle are congruent to two angles of another
triangle, then the third angles are also congruent.
If <A = <P and <B = <Q, then <C = <R
Reflexive Property of Congruent Triangles:
Every triangle is congruent to itself.
Symmetric Property of Congruent Triangles:
If ABC = DEF, then DEF = ABC.
Transitive Property of Congruent Triangles:
If ABC = DEF and DEF = JKL, then ABC = JKL
4.3 Proving Triangles are Congruent: SSS and SAS
SSS Congruence Postulate
If three sides of one triangle are congruent to three sides of a second
triangle, then the two triangles are congruent.
If Side AB = DE
C
F
Side BC = EF
Side CA = FD
Then ABC = DEF
A
B
E
D
SAS Congruence Postulate
If two sides and the included angle of one triangle are congruent to two
sides and the included angle of a second triangle, then the two triangles
are congruent.
If Side AB = DE
C
F
Angle <B = <E
Side BC = EF
Then ABC = DEF
A
B
E
D
Choosing Which Congruence Postulate to Use
Name the included angle between the given pair of sides.
AB and BC
CE and DC
AC and BC
4.4 Proving Triangles are Congruent: ASA and AAS
ASA Congruence Postulate
If two angles and the included side on one triangle are congruent to two
angles and the included side of a second triangle, then the two triangles
are congruent.
If Angle <C = <F
C
F
Side BC = EF
Angle <B = <E
Then ABC = DEF
A
B
E
D
AAS Congruence Postulate
If two angles and the nonincluded side of one triangle are congruent to
two angles and the nonincluded side of a second triangle, then the two
triangles are congruent.
If Angle <C = <F
C
F
Angle <A = <D
Side AB = DE
Then ABC = DEF
4.5 Using Congruent Triangles
EQ: What does the acronym CPCTC represent?
Knowing that all pairs of corresponding parts of congruent triangles are
congruent can help one prove congruent parts of triangles.
Given: PS = RS, PQ = RQ
Prove: <PQS = <RQS
Statements
Reasons
1. PS = RS
1. Given
2. PQ = RQ
2. Given
3. QS = QS
3. Reflexive
4. PQS = RQS
4. SSS
5. <PQS = <RQS
5. CPCTC
Q
P
R
S
Once one proves two triangles are congruent, then any pair of congruent
parts are congruent by CPCTC: Corresponding Parts of Congruent
Triangles are Congruent. These may be corresponding sides or angles.
4.6 Isosceles, Equilateral, and Right Triangles
Base Angle Theorem: if two sides of a triangle are
congruent, then the angles opposite them are congruent.
If AB = AC, then <B = <C
Base Angle Converse: if two angles of a triangle are
Congruent, then the sides opposite them are congruent.
If <B = <C, then AB = AC
Corollary: if a triangle is equilateral, then it is equiangular.
Corollary: if a triangle is equiangular, then it is equilateral.
HL: Hypotenuse-Leg: if the hypotenuse and a leg
of one right triangle are congruent to the hypotenuse
and a leg of a second right triangle, then the two
triangles are congruent.
In the two right triangles
if leg BC = EF
and hypotenuse AC = DF
then ABC = DEF by HL
Triangle may be proved congruent by any of five ways
SSS
SAS
ASA
AAS
HL
4.7 Triangles and Coordinate Proof
Place a 2 unit by 6 unit rectangle in a coordinate plane.
A right triangle has legs of 5 and 12 units. Place the triangle in a coordinate plane
and then find the length of the hypotenuse.
In the diagram ∆MLO = ∆KLO. Find the coordinate of L
Write a plan to prove that SQ bisects <PSR.
Can the two triangles be proved congruent? If so, which postulate or
theorem can be used?
1.
4.
2.
5.
6.
7.
10.
3.
8.
11.
9.
12.
13.
Write the SSS Congruence Postulate in your own words.
14.
Write the SAS Congruence Postulate in your own words.
15.
Write the ASA Congruence Postulate in your own words.
16.
Write the AAS Congruence Theorem in your own words.
Find the values of the missing angles or sides.
17.
18.
19.
20.
21.