1. Congruent Triangles
Geometry Chapter 4

2. 4.1 Triangles and Angles
Classification by Sides:

3. Triangles and Angles
Classification by Angles

4. Parts of Triangles leg hypotenuse leg leg leg base Exterior angle
Interior
angle
Vertex angle
leg
hypotenuse
leg
leg
Base angle
Base angle
leg
base

5. 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. B C A 3 1 2

6. 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.

7. Examples Sides of lengths 2mm, 3mm and 5mm.

8. Examples Angles of measures 90, 25, 65. Angles of measures 60, 60, 60.

9. Examples
A triangle has angles that measure x, 7x, and x. Find x.

10. Examples
A right triangle has angle measures of x and (2x-21). Find x.

11. Examples
Find the measure of the exterior angle shown.

12. 4.2 Congruence and TrianglesB
Congruent – same size, same shape
Congruent Polygons(Triangles) – Two polygons (triangles) are congruent iff their corresponding sides and corresponding angles are congruent A C F D
If ΔABC  ΔDEF,
then
A  D AB  DE
B  E BC  EF
C  F AC  DF

13. 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 S N R T M O

14. Examples H G L M F N E O If LMNO  EFGH, find x and y. (2x +3)m 110°
87°
72° F N E O
10m
If LMNO  EFGH, find
x and y.

15. Examples

16. 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

17. 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

18. 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

19. 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

20. HL D
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. A B E C F
If ABC,DEF Right s,
AB  DE, AC  DF,
then
ABC  DEF.

21. 4.5 Using Congruent Triangles
Definition of Congruent Triangles (rewritten)
Corresponding Parts of Congruent
Triangles are Congruent
CPCTC is used often in proofs involving congruent triangles.

22. A is the midpoint of MT. A is the midpoint of SR. MS ll TR 1.
1. Given

23. UR ll ST R and T are right angles 1. UR ll ST R and T are right angles
1. Given

24. 4.6 Isosceles, Equilateral and Right TrianglesB
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 C
If BA  BC, then A  C.
If A  C, then BA  BC.

25. More Corollaries If a triangle is equilateral, then it is equiangular.
If a triangle is equiangular then it is equilateral. B A C

26. Examples
Find x and y. y 35 x

27. Examples
Find the unknown measures. ? 50 ?

28. Examples
Find x.
(x-11) in
33 in

29. Examples
Find x and y. y 40 x