Question

In the figure, AB is parallel to CD, XY is the perpendicular bisector of AB, and E is the midpoint of XY. Prove that △AEB ≅ △DEC by matching each mathematical statement with its reason.
*(select) = A, B, C, D, E, F, G, H, I, J

A. XY is perpendicular to AB.
B. XY ⊥ CD
C. m∠AXE = 90°, m∠DYE = 90°.
D. ∠AXE ≅ ∠DYE.
E. XE ≅ YE
F. ∠A ≅ ∠D
G. △AEX ≅ △DEY
H. AE ≅ DE
I. ∠AEB ≅ ∠DEC
J. △AEB ≅ △DEC

Statements:
(select) Definition of a perpendicular bisector
(select) ASA Triangle Congruence
(select) Definition of a midpoint
(select) In a plane, if a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other.
(select) Vertical Angles Theorem
(select) Definition of perpendicular lines
(select) Right angles are congruent.
(select) AAS Triangle Congruence
(select) Alternate Interior Angles Theorem
(select) Corresponding parts of congruent triangles are congruent.

Answer 1

Answer:

From top to bottom:

A, J, E, B, I, C, D, G, F, H

See below for more clarification.

Step-by-step explanation:

We are given that AB is parallel to CD, XY is the perpendicular bisector of AB, and E is the midpoint of XY. And we want to prove that ΔAEB ≅ ΔDEC.

Statements:

1) XY is perpendicular to AB.

Definition of perpendicular bisector.

2) XY ⊥ CD.

In a plane, if a transveral is perpendicular to one of the two parallel lines, then it is perpendicular to the other.

3) m∠AXE = 90°, m∠DYE = 90°.

Definition of perpendicular lines.

4) ∠AXE ≅ ∠DYE.

Right angles are congruent.

5) XE ≅ YE

Definition of a midpoint.

6) ∠A ≅ ∠D.

Alternate Interior Angles Theorem

7) ΔAEX ≅ ΔDEY

AAS Triangle Congruence*

(*∠A ≅ ∠D, ∠AXE ≅ ∠DYE, and XE ≅ YE)

8) AE ≅ DE

Corresponding parts of congruent triangles are congruent (CPCTC).

9) ∠AEB ≅ ∠DEC

Vertical Angles Theorem

10) ΔAEB ≅ ΔDEC

ASA Triangle Congruence**

(**∠A ≅ ∠D, AE ≅ DE, and ∠AEB ≅ ∠DEC)