Transcript 4-1 Classifying Triangles I. Geometric Shapes What is a triangle? A TRIANGLE is a three-sided polygon.

4-1 Classifying Triangles
I. Geometric Shapes
What is a triangle?
A TRIANGLE is a three-sided
polygon.
II. How is a triangle labeled?
III. Classification by Angles:
One way of classifying
triangles is by their angles.
Acute triangle
Obtuse triangle
Right triangle
An EQUIANGULAR triangle is an acute
triangle in which all angles are congruent.
IV. Classification by Sides:
Triangles can also be classified according to
the number of congruent sides they have.
4-2 Measuring Angles in
Triangles
I. Triangle Angle Sum Theorem:
4-1 The sum of the measures of the angles in a triangle is 180.
II. Third Angle Congruence:
Theorem 4-2 Third Angle Theorem
If two angles of one triangle are congruent to two angles
of a second triangle, then the third angles of the triangles
are congruent.
III.Triangle Exterior Angles & its
Corollaries:
Theorem 4-3 Exterior Angle Theorem
- The measure of an exterior angle of a
triangle is equal to
the sum of the measures of the two
remote interior angles.
By the way.... what is a corollary? A COROLLARY is a
statement that can be easily proved using a
theorem..... A better way of saying this... is that a
corollary is a fact or statement that directly falls from a
given theorem.
Corollary 4-1 The acute angles of a right
triangle are
complementary.
Corollary 4-2 –
There can be at most one
right or obtuse angle in a
triangle
4-3 Congruent Triangles
I. When two triangles are congruent
to each other then......
there are SIX pieces of information that must be true:
3 congruent corresponding sides
3 congruent corresponding angles
Even if you slide, turn, or flip
II. Definition of Congruent
Triangles (CPCTC)
Two triangles are congruent if and
only if their corresponding parts
are congruent.
IV. Examples
Triangle RST is isosceles with S as
the vertex angle. If ST = 3x - 11,
SR = x + 3, and RT = x - 2, find RT.
•
2. Draw and classify the triangle:
Triangle KLM
angle K= 90
KL=2.5,
KM=3
Given triangle STU with S (2,3), T
(4,3) and U (3,-2). Use the distance
formula to prove it is isosceles.
3.
Examples
1. Find the value of x.
2. What is the value of
angle W if
angle X is 59 and
angle XYZ is 137?
3. What is the value of angle B?
4. Find angle 1.
Theorem 4-4 - Congruence
of triangles is reflexive,
symmetric, and transitive.
III. Examples
1. Name the corresponding parts if
triangle PQR is congruent to triangle
STU.
2. Refer to the design shown. How
many of the triangles in the design
appear to be congruent to triangle A?
4-4 Proving Triangles are
Congruent
I. Postulates
4-1 Side-Side-Side SSS
4-2 Side-Angle-Side
ACB and
DCE, vertical angles.
4-3 Angle-Side-Angle
II. Examples
1. PQR with P(3,4) Q (2,2) R (7,2)
STU with S(6,-3) U (4,-7) T (4,-2)
Prove that PQR
SUT

2. Given: BE bisects AD and
angle A
angle D.


Prove: AB
CD

3. Prove: STR
PTR.
Given: angles STR and RTP right
and ST
TP
R

S
T
P
4-5 More Congruent Triangles
I. Modification of 4-3
Postulate 4-3 AAS (Angle - Angle Side) - If two angles and a NONINCLUDED side of one triangle are
congruent to the corresponding two
angles and side of a second
triangle, the two triangles are
congruent.
II. Examples
Worksheet
4-6 Isosceles Triangle Theorem
I. Review
What is an isosceles triangle?
II. Theorem 4-6 Isosceles
Triangle Theorem (ITT)
If two sides of a triangle are
congruent, then the angles opposite
those sides are congruent. Summary
- In other words if you have two
congruent sides, you have two
congruent base angles.
III. Theorem 4-7 Converse of
the ITT
If two angles of a triangle are
congruent, then the sides opposite
those angles are congruent.
Summary - If you have two
congruent angles, then you have two
congruent legs.
IV. Corollaries
Corollary 4-3 - A triangle is
equilateral if and only if it is
equiangular.
Corollary 4-4 - Each angle of an
equilateral triangle measures 60
degrees