Chapter 4, Problem 17CR

To determine

To write: the proofs in two column method.

Explanation of Solution

Given information:

Given figure,

  

Give the reason for given key step of the proof.

Given:

  $\overline{AX}\cong \overline{CY};\angle A\cong \angle C$

  $\overline{BX}\perp \overline{AD};\overline{DY}\perp \overline{BC}$

Prove: $\overline{AD}\parallel \overline{BC}$

 1. $\Delta ABX\cong \Delta CDY$

 2. $\overline{BX}\cong \overline{DY}$

 3. $\Delta BDX\cong \Delta DBY$

 4. $\angle 1\cong \angle 2$

 5. $\overline{AD}\parallel \overline{BC}$

Calculation:

SSS Postulate: If three sides of a triangle are equal to the three corresponding sides of another triangle, then the triangles are congruent.

SAS Postulate: If two sides and the angle included between them of a triangle are equal to the two corresponding sides and the angle included between them of another triangle, then the triangles are congruent.

ASA Postulate: If two angles and the included side of a triangle are equal to the two corresponding angles and the included sides of another, then the triangles are congruent.

AAS Theorem: If two angles and non included side of a triangle is congruent to the corresponding parts of another triangle, then the triangles are said to be congruent.

HL Theorem: If the hypotenuse and a leg of one right triangle is congruent to the corresponding parts of another right triangle, then the triangles are said to be congruent

  

Statements	Reasons
1. $\Delta ABX\cong \Delta CDY$	ASA postulate
2. $\overline{BX}\cong \overline{DY}$	Corresponding parts of $\cong$ triangles are $\cong$
3. $\Delta BDX\cong \Delta DBY$	HL Theorem
4. $\angle 1\cong \angle 2$	Corresponding parts of $\cong$ triangles are $\cong$
5. $\overline{AD}\parallel \overline{BC}$	if two lines are cut by a transversal and alternate interior angles are $\cong$ , then the lines are parallel.