Question

Match the reasons with the statements in the proof to prove AB || DC , given that AD is parallel to BC and AD = CB .

Answer 1

Answer:

In triangle ACD and triangle CAB

$\overline {AD} || \overline {BC}$ and

$\overline {AD} =\overline {CB}$      [Side]                {Given}

Reflexive property of equality simply states that a value is equal to itself.

$\overline {AC} =\overline {AC}$   {side]      {Relexive property of equality}

Alternate interior angle states that if the two lines being crossed are Parallel Lines then the Alternate Interior Angles are equal.

$\angle 2 = \angle 3$       [Angle]

SAS(Side-Angle-Side) theorem states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent.

Therefore, by SAS theorem,

$\triangle ACD \cong \triangle CAB$

Then by CPCT  [Corresponding part of congruent triangle] are equal.

$\angle 1 = \angle 4$

By alternate interior Angles, if the alternate interior angles are congruent then the lines are congruent.

$AB || DC$               proved!

Answer 2

Given: $AD \parallel BC$ and AD=CB

To Prove: $AB \parallel DC$

1. $AD \parallel BC$, AD=CB

Reason: Given

2. AC = AC

Reason: Reflexive Property of Equality

3. $\angle 2 = \angle 3$

Reason: If Lines are parallel, then Alternate Interior Angles are Equal).

4. $\Delta ACD \cong \Delta CAB$

Reason: SAS

5. $\angle 1 = \angle 4$

Reason: CPCTE

6. $AB \parallel DC$

Reason: If Alternate Interior Angles are Congruent, then Lines are Parallel.