1. Congruent Triangles 4-1: Classifying Triangles
4-2: Angle Measure in Triangles
4-3: Congruent Triangles
4-4 & 4-5: Tests for Congruent Triangles
4-7: Isosceles Triangles
Home
Next

2. 4-1: Classifying Triangles
The Parts of a Triangle
Triangle ABC can be written as ABC and is made up of the following parts.
Sides: AB, BC, AC are line segments
Vertices: A, B, and C are points
Angles: BAC or A, ABC or B, and BCA or C
Back
Next

3. 4-1: Classifying Triangles
Classifying by Angles
Acute Triangle
Obtuse Triangle
In an Acute Triangle all the angles are acute, less than 90.
In an Obtuse Triangle there is only one obtuse angle, an angle greater than 90.
Right Triangle
A Right Triangle has one 90 degree angle.
Back
Next

4. 4-1: Classifying Triangles
Right Triangles
In Right Triangle XYZ, the sides XY and YZ are called the legs, and side XZ is called the hypotenuse.
Leg
Hypotenuse
Equiangular Triangles
Leg
In an Equiangular Triangle all three angles are congruent to one another.
Back
Next

5. 4-1: Classifying Triangles
Classifying by Sides
Isosceles Triangle
Scalene Triangle
In an Isosceles Triangle at least two of the sides are congruent.
In an Scalene Triangle all the sides have different lengths
Equilateral Triangle
In an Equilateral Triangle all the sides are congruent to each other
Back
Next

6. 4-1: Classifying Triangles
Isosceles Triangles
Leg
Base
Vertex Angle
Base Angles
In an Isosceles Triangle the two congruent sides are called legs and form the vertex angle. The side opposite the vertex angle is called the base. The base and each leg form two base angles.
* The base angles of an isosceles triangle are congruent.
Back
Section 4-2

7. 4-2: Angle Measures in Triangles
Theorem:
The sum of the measures of the angles of a triangle is 180.
Given: EFG
mE + mF + mG = 180.
Topic 4
Next

8. 4-2: Angle Measures in Triangles
Theorem:
If two angles of one triangle are congruent to two angles of another triangle, then the third angles of the triangles are congruent.
Given EFG and PQR:
If E  R, and F  Q, then G  P
Back
Next

9. 4-2: Angle Measures in Triangles
Definitions:
An Exterior Angle is formed by one side of a triangle and another side extended. The angles of the triangle not adjacent to a given exterior angle are called Remote Interior Angles.
Given: ZXY an interior angle of XYZ. WXZ is the corresponding exterior angle. Z and Y are the resulting remote interior angles
Given: ZXY
Exterior Angle
Remote Interior Angles
Back
Next

10. 4-2: Angle Measures in Triangles
Theorem:
The measure of an exterior angle of a triangle is equal to the sum of the measures of its two remote interior angles. 
mWXZ = mZ + mY
Back
Next

11. 4-2: Angle Measures in Triangles
* Corollary
Given Right Triangle ABC
The acute angles of a right triangle are complementary.
* Corollary
There can be at most one right or obtuse angle in a triangle.
mA + mB = 90.
* A Corollary is a statement that follows directly from another theorem and that can be easily proved from that theorem.
Back
Next

12. 4-3: Congruent Triangles
* Definition:
Two triangles are congruent if and only if their corresponding parts are congruent.
ABC  XYZ if and only if the three corresponding sides are congruent and the three corresponding angles are congruent
Sides
AB  XY
BC  YZ
AC  XZ
Angles
A  X
B  Y
C  Z
*This definition can be abbreviated CPCTC which means Corresponding Parts of Congruent Triangles are Congruent.
Topic 4
Next

13. 4-3: Congruent Triangles
Theorem
Congruence of Triangles is Reflexive, Symmetric and Transitive.
Reflexive Property
Symmetric Property
Transitive Property
ABC  ABC
If ABC  DEF, then DEF  ABC
If ABC  DEF, and DEF  GHI, then ABC  GHI
Section 4-4
Back

14. 4-4 & 4-5: Tests for Congruent Triangles
Side-Side-Side (SSS) Postulate
If the three sides of one triangle are congruent to the three sides of another triangle, then the triangles are congruent.
If AB  XY, BC  YZ, and AC  XZ, then ABC  XYZ.
Topic 4
Next

15. 4-4 & 4-5: Tests for Congruent Triangles
Side-Angle-Side (SAS) Postulate
If two sides and an included angle of one triangle are congruent to two sides and an included angle of another triangle, then the triangles are congruent.  
If PQ  XY, QR  YZ, and PQR  XYZ, then PQR  XYZ.
Back
Next

16. 4-4 & 4-5: Tests for Congruent Triangles
Angle-Side-Angle (ASA) Postulate
If two angles and an included side of one triangle are congruent to two angles and an included side of another triangle, then the triangles are congruent.    
If DE  PQ, D  P, and E  Q, then DEF  PQR.
Back
Next

17. 4-4 & 4-5: Tests for Congruent Triangles
Angle-Angle-Side (AAS) Postulate
If two angles and an non-included side of one triangle are congruent to the corresponding two angles and side of another triangle, then the triangles are congruent.    
If D  P, F  R, and DE  PQ, then DEF  PQR.
Back
Next

18. 4-4 & 4-5: Tests for Congruent Triangles
Sample Proof
Given: R and T are Right Angles. 1  2
Prove: RV  TV
RV  TV
1. R and T are Right Angles. 1  2
1. Given
2. R  T
2. Right Angles are Congruent
3. SV  SV
3. Reflexive
4. RSV  TSV
4. AAS
5. CPCTC
5. RV  TV
Back
Section 4-7

19. 4-7: Isosceles Triangles
Isosceles Triangle Theorem:
If two sides of a triangle are congruent, then the angles opposite those sides are congruent.
If AB  AC, then B  C 
Theorem:
If two angles of a triangle are congruent, then the sides opposite those angles are congruent.
If B  C, then AB  AC
Topic 4
Next

20. 4-7: Isosceles Triangles
Corollary
A triangle is equilateral if and only if it is equiangular
If XY  YZ  ZX, then X  Y  Z
If X  Y  Z, then XY  YZ  ZX.
Corollary
Each angle of an equilateral triangle measures 60 degrees.
If XYZ is equilateral then, mX = mY = mZ = 60.
Back
Applying Congruent Triangles