Congruent Figures
Standard
G.CO.7 ...Show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.
≅ this is the symbol for congruent
Congruent means to have the same size (segments are equal) and shape (angles are equal). The figures may be oriented differently however (meaning that one of them went through a series of transformations).
Order matters!
The figures look reflected
What we CAN do
- We can write AB ≅ EF
- We can write BA ≅ FE
- Order is preserved!
- We can write ABCD ≅ EFGH
- The letters/segments match up!
- AB ≅ EF
- BC ≅ FG
- CD ≅ GH
- DA ≅ HE
- We can also write ADCB ≅ EHGF
- Starting at A, we go counterclockwise around the figure.
What we CANNOT do
- We cannot write AB ≅ FE
- The letters do not correspond with each other!
- We cannot write ADBC
- The are no segments connecting DB.
- When writing the names, follow the segments/lines!
Third Angles Theorem
The three angles of a triangle add up to 180
∠A + ∠B + ∠C = 180
Theorem:
- If two angles of one triangle are congruent to two angles of another triangle, then the third angles are congruent.
- Note: This does not mean that the triangles are congruent! It's just that their angles are congruent.
If...
∠A ≅ ∠D and ∠B ≅∠E
Then...
∠C ≅ ∠F
Proving triangles are congruent
Given:
- AE ≅ DC
- EB ≅ CB
- BA ≅ BD
- ∠A ≅ ∠D
Prove ΔAEB ≅ ΔDCB.
Remember, congruent means to have the same size and same angles.
Statement:
- AE ≅ DC, EB ≅ CB, BA ≅ BD
Reason:
- Given
This statement implies that the shapes have the same size.
Statement:
2. ∠A ≅ ∠D
3. ∠B ≅ ∠B
4. Since ∠A ≅ ∠D, ∠B ≅ ∠B, then ∠E ≅ ∠C
Reason:
2. Given
3. Vertical angles
4. Third Angles Theorem
This is all we need to write. The two triangles have the same size and the same angles. They are congruent.