4-1 Classifying Triangles

I. Geometric ShapesWhat 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 tothe 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

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 

4-3 Congruent Triangles

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

3. Given triangle STU with S (2,3), T (4,3) and U (3,-2). Use the distance formula to prove it is isosceles.

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. Postulates4-1 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 NON-INCLUDED 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. ReviewWhat 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