Let's begin by analyzing the desired outcome of our proof. We want to show that AB ≅ CD. Notice that we could conclude it if △ AEB was congruent to △ CFD, as corresponding parts of congruent triangles are congruent. We are given that BE ≅ DF and AF ≅ CE. This is how we will begin our proof!
Statement1)& BE ≅ DF and AF ≅ CE Reason1)& Given
We are also given that BE and DF are both perpendicular to AC. Therefore, by the definition of perpendicular lines, ∠ AEB and ∠ CFD are right angles, and hence they are congruent.
Statement2)& ∠ AEB ≅ ∠ CFD Reason2)& Definition of perpendicular lines
Knowing that AF ≅ CE, we can conclude that AF=CE by the definition of congruent segments.
Statement3)& AF=CE Reason3)& Definition of congruent segments
Notice that the segment EF is part of both AF and CE. Additionally, AF has the side AE, and CE has the side CF. Thus, we can write the following equations.
AF=EF+AE and CE= EF+ CF
Let's list this as the next step in our proof!
Statement4)& AF=EF+AE & CE= EF+ CF Reason4)& Segment Addition Postulate
Now, we can substitute these expression in AF=CE by the Substitution Property of Equality. This gives us EF+AE=EF+CF.
Statement5)& EF+AE=EF+CF Reason5)& Substitution Property & of Equality
Next, we can subtract EF from both sides of our equation by the Subtraction Property of Equality. Then, we are left with AE=CF.
Statement6)& AE=EF Reason6)& Subtraction Property of Equality
By the definition of congruent segments, we can conclude that AE ≅ CF.
Statement7)& AE ≅ CF Reason7)& Definition of congruent & segments.
Notice that now we know that two sides and the included angle of △ AEB are congruent to two sides and the included angle of △ CFD. Thus, by the Side-Angle-Side (SAS) Theorem, △ AEB ≅ △ CFD.
Statement8)& △ AEB ≅ △ CFD Reason8)& SAS Theorem
Corresponding parts of congruent triangles are congruent. Therefore, AB ≅ CD.
Statement9)& AB ≅ CD Reason9)& Corresponding parts of & congruent triangles are & congruent.
Completed Proof
Statements
Reasons
1.
BE ≅ DF and AF ≅ CE
1.
Given
2.
∠ AEB ≅ ∠ CFD
2.
Definition of perpendicular lines
3.
AF=CE
3.
Definition of congruent segments
4.
AF=EF+AE, CE=EF+CF
4.
Segment Addition Postulate
5.
EF+AE=EF+CF
5.
Substitution Property of Equality
6.
AE=EF
6.
Subtraction Property of Equality
7.
AE ≅ CF
7.
Definition of congruent segments
8.
△ AEB ≅ △ CFD
8.
SAS Theorem
9.
AB ≅ CD
9.
Corresponding parts of congruent triangles are congruent.
