b Draw segments perpendicular to the bases through K and L. Then, use the Hypotenuse-Leg Congruence Theorem and the Side-Angle-Side (SAS) Congruence Theorem.
A
aProof:
Statements
Reasons
1.
JKLM is an isosceles trapezoid, KL∥ JM, JK ≅ LM
1.
Given
2.
∠ JKL ≅ ∠ KLM
2.
Isosceles Trapezoid Base Angles Theorem
3.
KL ≅ KL
3.
Reflexive Property of Congruence
4.
△ JKL ≅ △ MLK
4.
Side-Angle-Side (SAS) Congruence Theorem
5.
JL ≅ KM
5.
Definition of congruent triangles
B
bConditional Statement: If the diagonals of a trapezoid are congruent, then the trapezoid is an isosceles trapezoid.
Proof:
By the Isosceles Trapezoid Base Angles Theorem, each pair of base angles is congruent, so ∠ JKL ≅ ∠ KLM. Next, let's separate the two triangles △ JKL and △ KLM.
Notice that KL is common for both triangles. Let's list the congruent parts between these two triangles.
ccc
JK ≅ ML & & Side
∠ JKL ≅ ∠ KLM & & Angle
KL ≅ KL & & Side
By the Side-Angle-Side (SAS) Congruence Theorem, we have that △ JKL ≅ △ MLK. Then, JL ≅ MK, which is what we wanted to prove.
Two-Column Proof
Given: & JKLM is an isosceles trapezoid
& KL∥ JM, JK ≅ LM
Prove: & JL ≅ KM
The proof we did above is summarized in the following two-column table.
Statements
Reasons
1.
JKLM is an isosceles trapezoid, KL∥ JM, JK ≅ LM
1.
Given
2.
∠ JKL ≅ ∠ KLM
2.
Isosceles Trapezoid Base Angles Theorem
3.
KL ≅ KL
3.
Reflexive Property of Congruence
4.
△ JKL ≅ △ MLK
4.
Side-Angle-Side (SAS) Congruence Theorem
5.
JL ≅ KM
5.
Definition of congruent triangles
b Let's begin by writing the other part of Isosceles Trapezoid Diagonals Theorem as a conditional.
If the diagonals of a trapezoid are congruent, then the trapezoid is an isosceles trapezoid.
Now, let JKLM be a trapezoid such that its diagonals are congruent.
Let's construct line segments through K and L that are perpendicular to JM. Notice that since JM∥ KL, we have that ABLK is a rectangle which implies that AK ≅ BL.
Notice that △ KAM and △ LBJ are right triangles with one pair of congruent legs (AK≅ LB), and their hypotenuses are also congruent. Thus, by the Hypotenuse-Leg Congruence Theorem, △ KAM ≅ △ LBJ.
△ KAM ≅ △ LBJ
⇕
∠ KMA ≅ ∠ LJB
Next, from the diagram, let's separate △ KJM and △ LMJ and let's mark the congruent parts between them.
As we can see, JM≅ JM, ∠ JMK ≅ ∠ MJL, and MK ≅ JL. Then, by the Side-Angle-Side (SAS) Congruence Theorem, we get △ JKM ≅ △ MLJ.
△ JKM ≅ △ MLJ
⇒
JK ≅ LM
By definition, the latter congruence relation implies that the trapezoid JKLM is isosceles.
Two-Column Proof
Given: & JKLM is a trapezoid
& KL∥ JM, JL ≅ KM
Prove: & JKLM is an isosceles trapezoid
The proof we did above is summarized in the following two-column table.
