Question 45 - Triangle Congruence

Given:
C and D lie on segment BF

AB = DE AC = EF BD = CF

a. Prove that the triangles are congruent and write the congruence

Step 1: Finding a third pair of equal sides

On the segment B-C-D-F we have:

BD = BC + CD and also CF = CD + DF

From the given BD = CF we get:

BC + CD = CD + DF → BC =

Correct! BC = DF

Step 2: List of equal sides

Now we have three pairs of equal sides:

AB = DE (given)

AC = EF (given)

BC = DF (proved)

Excellent! Three pairs of equal sides!

Step 3: Choosing the congruence theorem

Which congruence theorem applies when there are three pairs of equal sides?

Correct! SSS - three pairs of equal sides!

Step 4: Writing the congruence

Write the congruence (pay attention to vertex order!):

AB=DE, AC=EF, BC=DF → A corresponds to ?, B corresponds to ?, C corresponds to ?

ΔABC ≅ Δ

Excellent! ΔABC ≅ ΔEDF

Proof summary:
From the given BD = CF we obtained: BC = DF
We have three pairs of equal sides:
AB = DE, AC = EF, BC = DF
Therefore by SSS: ΔABC ≅ ΔEDF
(The correspondence: A↔E, B↔D, C↔F)

b. Which angles are equal to angle B?

Step 5: Identifying equal angles

From the congruence ΔABC ≅ ΔEDF, which angle equals angle B?

Hint: B is the second vertex in ABC; who is the second in EDF?

Correct! ∠B = ∠D

From the congruence ΔABC ≅ ΔEDF:
The correspondence is: A↔E, B↔D, C↔F
Therefore ∠B = ∠D
(and also: ∠A = ∠E, ∠C = ∠F)

Bonus: Additional angles

Step 6: Matching all angles

Match all pairs of equal angles:

∠A = ∠ ∠C = ∠

Excellent! You understood all the correspondences!

From the congruence ΔABC ≅ ΔEDF:
∠A = ∠E (first vertex)
∠B = ∠D (second vertex)
∠C = ∠F (third vertex)

Well done! You finished the question!