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