Congruent Triangles
What are Congruent Triangles?
Congruent triangles occur if and only if the corresponding parts of those congruent triangles are congruent.
Hint: The parts with the same color are corresponding parts .
This is also called CPCTC.
Congruent triangles occur if and only if the corresponding parts of those congruent triangles are congruent.
Hint: The parts with the same color are corresponding parts .
This is also called CPCTC.
Example 1:
Which parts are corresponding?
Answers:
<c & <f. <a & <d. <b & <e. line ca & line fd. line cb & line fe. line ab & line de.
<c & <f. <a & <d. <b & <e. line ca & line fd. line cb & line fe. line ab & line de.
How do you name triangles?
When naming 2 congruent triangles, you use a congruence statement. A congruence statement is simply something that shows the order in which the triangles are congruent. For instance, in example 1 the congruence statement would be triangle CAB = triangle FDE, triangle ABC = triangle.
Try to name these triangles on your own:
Answer: Triangle ABC and Triangle XWY
Properties of Congruent Triangles
Reflexive Property Of Congruent Triangles
For any triangle ABC, Triangle ABC is congruent to Triangle ABC
In other words...a triangle is congruent to itself.
Symmetric Property of congruent triangles
If triangle ABC is congruent to Triangle DEF,then Triangle DEF is congruent to Triangle ABC
In other words...the order of the triangles doesn't matter when writing a congruence statement.
Transitive Property of Congruent Triangles
If triangle ABC is congruent to Triangle DEF and triangle DEF is congruent to Triangle JKL,then triangle ABC is congruent to Triangle JKL
For any triangle ABC, Triangle ABC is congruent to Triangle ABC
In other words...a triangle is congruent to itself.
Symmetric Property of congruent triangles
If triangle ABC is congruent to Triangle DEF,then Triangle DEF is congruent to Triangle ABC
In other words...the order of the triangles doesn't matter when writing a congruence statement.
Transitive Property of Congruent Triangles
If triangle ABC is congruent to Triangle DEF and triangle DEF is congruent to Triangle JKL,then triangle ABC is congruent to Triangle JKL
Third Angle Theorem
What is the third angle theorem?
If two angles of one triangle are congruent to two angles of another triangle, then the third angles are also congruent.
If two angles of one triangle are congruent to two angles of another triangle, then the third angles are also congruent.
In the example to the right, you can assume that the two triangles are congruent because if you add the two angles and subtract the sum from 180 degrees the differences are the same.
Examples:
Using third angle theorem determine whether or not the triangles on the right are congruent
Answer: Yes, the two triangles are congruent because two angles are congruent and the third angle theorem only needs to show two angles being congruent in order to prove that the two triangles are congruent.