Identifying Congruent TrianglesIdentifying Congruent Triangles

- Triangle classification by parts- Angle Sum Theorem & Exterior Angle Theorem- CPTC,SSS,SAS,ASA,and AAS Theorems- Problem solving by eliminating possibilities -Equilateral & Isosceles triangles

OBJECTIVES

Classifying trianglesBy side lengthsEquilateral 3 sidesIsosceles 2 sidesScaleneno lengths same

By angle measuresRight one angle 90°Obtuseone angle obtuseAcuteall angles acuteequiangularall 3 's

Parts of an Isosceles TriangleVertex Angle

2 sides leg leg

Base angle Base angleBase is side opposite vertex

Measuring angles in trianglesThe sum of the measures of the 3 angles of a triangle

always equals 180º

• If 2• In a right triangle, the 2 acute angles are complementary• There can be at most 1 obtuse or 1 right angle in a Δ• The measure of an exterior angle = the measure of

the two remote interior angles:

1 2' 2 ' 3 '_ _ _ _s of s of the rd s are

Exterior angle

Remote interior angles

Congruent triangles: CPCTC

Two Δ’s are congruent if and only if their corresponding parts are congruent (all sides & all angles)

1 2 corr parts

Congruence of triangles is:

reflexive (parts to self) symmetric

transitive

A A

if , then B AA B

if A B and B , then A CC

If 2 triangles are congruent:

The congruence statement statement tells which parts of triangle 1 ‘match up’ or correspond to the parts of triangle 2.

1 2

ABC DEF Means

A D, B E, C F

and

AB DE, BC EF,andAC DF

ORDER IS VERY IMPORTANT

Proving Δ’s congruent: SSS & SAS & ASA

Given: 2 Δ’s (match up sides/angles that are alike) If 3 sides 2 sides & an included angle OR 2 angles & an included side

are congruent THEN the 2 Δ’s are congruent

** remember-- in two column proofs the ‘if’ part matches what you know & goes in the left column. The ‘then’ part goes in the right column & gives direction towards the statement to be proven.

Congruent triangles: AAS‘Read around’ the vertices of a triangle: if an angle &

another angle & a side not between them are congruent to the corresponding parts of another triangle, THEN the triangles are congruent

Mark the given parts on your triangles to see which theorem or postulate to use. There WILL be a clue to get you started

A

B

C

D

EF

A D, B E, BC EF

ABC DEF

Isosceles triangles

If 2 sides are , then angles opposite them areIf 2 angles of a triangle are , then the sides opposite are

A triangle is equilateral if and only if it is equiangular

Each angle of an equiangular triangle measures 60°

A

B

C

Example:

Given: ΔTEN is an isosceles triangle with base TN

Prove: ΔTEA ΔNEC

1. ΔTEN is an isosceles triangle with base TN 1. Given2. 2.Def Isosceles3. 3.Given4. ΔTEA ΔNEC 4.AAS

T

E

N1 2 3 4

C A

2 3, T N

TE NE

2 3, T N

End with what you are to prove