Let's begin by analyzing the desired outcome of our proof. We want to show that S is the midpoint of BK, which means that BS and KS are congruent. This will be true if △ ASB and △ ASK are congruent, as corresponding parts of congruent triangles are congruent.
We are given that BA ≅ KA and BE ≅ KE. This is how we will begin our proof!
Statement1)& BA ≅ KA and BE ≅ KE Reason1)& Given
From the diagram we can tell that △ ABE and △ AKE share the side AE. By the Reflexive Property of Congruence we know that AE ≅ EA.
Statement 2)& AE ≅ EA Reason 2)& Reflexive Property & of Congruence
Notice that the three sides of △ ABE are congruent to the three sides of △ AKE. Thus, by the Side-Side-Side (SSS) Theorem, △ ABE ≅ △ AKE.
Statement3)& △ ABE ≅ △ AKE Reason3)& SSS Theorem
We know that corresponding parts of congruent triangles are congruent. Therefore, the corresponding parts of △ ABE and △ AKE are congruent and hence ∠ BAS ≅ ∠ KAS.
Statement4)& ∠ BAS ≅ ∠ KAS Reason4)& Corresponding parts of & congruent triangles are & congruent.
Now let's have a look at △ ASB and △ ASK again. We know that BA ≅ KA and that ∠ KAS ≅ ∠ BAS. From the diagram we can also tell that the triangles share the side AS. Hence, by the Reflexive Property of Congruence AS ≅ SA.
Statement 5)& AS ≅ SA Reason 5)& Reflexive Property & of Congruence
Notice that we know that two sides and the included angle of △ ASB are congruent to two sides and the included angle of △ ASK. Thus, by the Side-Angle-Side (SAS) Theorem, △ ASB ≅ △ ASK.
Statement6)& △ ASB ≅ △ ASK Reason6)& SAS Theorem
Corresponding parts of congruent triangles are congruent. Therefore, ∠ ASB ≅ ∠ ASK.
Statement7)& ∠ ASB ≅ ∠ ASK Reason7)& Corresponding parts of & congruent triangles are & congruent.
Now, notice that ∠ ADB and ∠ CDB are supplementary angles as they form a linear pair. If two angles are congruent and supplementary, then each is a right angle.
Statement8)& ∠ ASB and ∠ ASK are & right angles Reason8)& If two angles are congruent & and supplementary, then each & is a right angle.
We know that BK and AE intersect at a right angle. Thus, by the definition of perpendicular segments, BK ⊥ AE.
Statement9)& BK ⊥ AE Reason9)& Definition of perpendicular & segments.
Completed Proof
Statements
Reasons
1.
BA ≅ KA and BE ≅ KE
1.
Given
2.
AE ≅ EA
2.
Reflexive Property of Congruence
3.
△ ABE ≅ △ AKE
3.
SSS Theorem
4.
∠ BAS ≅ ∠ KAS
4.
Corresponding parts of congruent triangles are congruent.
5.
AS ≅ SA
5.
Reflexive Property of Congruence
6.
△ ASB ≅ △ ASK
6.
SAS Theorem
7.
∠ ASB ≅ ∠ ASK
7.
Corresponding parts of congruent triangles are congruent.
8.
∠ ASB and ∠ ASK are right angles
8.
If two angles are congruent and supplementary, then each is a right angle.
