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Congruent Triangles
Geometry Chapter 4
4.1 Triangles and Angles
Classification by Sides:
Triangles and Angles
Classification by Angles
Parts of Triangles
leg leg
base
leg
leg
hypotenuse
Vertex angle
Base angle Base angle
ExteriorangleInterior
angle
Theorems Involving Triangles
The sum of the measures of the angles of a triangle = 180°
The measure of the exterior angle of a triangle = the sum of the two remote interior angles. 12
3
A
B
C
Corollaries to Triangle Theorems
The acute angles of a right triangle are complementary.
Each angle of an equiangular triangle has a measure of 60°.
In a triangle, there can be at most one right angle or one obtuse angle.
¬
Examples
Sides of lengths 2mm, 3mm and 5mm.
Sides of lengths 3m, 3m, 3m.
Sides of lengths 8m, 8m, 5m.
Examples
Angles of measures 90, 25, 65.
Angles of measures 60, 60, 60.
Angles of measures 80, 70, 30.
Angles of measures 140, 30, 10.
Examples
A triangle has angles that measure x, 7x, and x. Find x.
Examples
A right triangle has angle measures of x and (2x-21). Find x.
Examples
Find the measure of the exterior angle shown.
4.2 Congruence and Triangles
Congruent – same size, same shape
Congruent Polygons(Triangles) – Two polygons (triangles) are congruent iff their corresponding sides and corresponding angles are congruent
A C
B
D
E
F
If ΔABC ΔDEF,thenA D AB DEB E BC EFC F AC DF
Theorems about Congruent Figures
If two angles of one triangle are congruent to two angles of another triangle, then the third angles are congruent.
If R M and S N, then T O
R
S
T M
N
O
Examples
110°
87°
10m
72°
L M
NO
If LMNO EFGH, find x and y.
E F
GH
(7y + 9)°
(2x +3)m
Examples
4.3-4.3 Proving Triangles Congruent
SSS – Side Side Side – If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent.
If AB DE BC EF AC DF, then
ABC DEF
SAS
SAS – Side Angle Side – If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent.
If AB DE BC EF B E, then ABC DEF
ASA
ASA – Angle Side Angle – If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent.
If A D C F AC DF,then ABC DEF
AAS
AAS – Angle Angle Side – If two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of another triangle, then the triangles are congruent.
If A D C F AB DE,then ABC DEF
HL
HL – Hypotenuse Leg – If the hypotenuse and leg of one RIGHT triangle are congruent to the hypotenuse and leg of another RIGHT triangle then the triangles are congruent.
C
A
B F
D
E
If ABC,DEF Right s,AB DE, AC DF, thenABC DEF.
4.5 Using Congruent Triangles
Definition of Congruent Triangles (rewritten)
Corresponding Parts of Congruent
Triangles are CongruentCPCTC is used often in proofs involving
congruent triangles.
1.
M
TS
R
AA is the midpoint of MT.A is the midpoint of SR.
MS ll TR
A is the midpoint of MT.A is the midpoint of SR.
1. Given
UR ll ST R and T are right angles
UR ll ST R and T are right angles
1. 1. Given
U
R S
T
4.6 Isosceles, Equilateral and Right Triangles
If two sides of a triangle are congruent, then the angles opposite are congruent. (Base angles of an isosceles triangle are congruent.
Converse – If two angles of a triangle are congruent, then the sides opposite are congruent.
A
B
C
If BA BC, then A C.
If A C, then BA BC.
More Corollaries
If a triangle is equilateral, then it is equiangular.
If a triangle is equiangular then it is equilateral.
A
B
C
Examples
Find x and y.
35 x
y
Examples
Find the unknown measures.
50 ?
?
Examples
Find x.
33 in(x-11) in
Examples
Find x and y.
y
40
x