Chapter 4.4, Problem 1CYU

a.

To determine

Toexplain:

The number of different sized triangles are used to create the illusion.

Answer to Problem 1CYU

Two different sized triangles are used to create the illusion.

Explanation of Solution

Given information:

  $ABCD$ is a square with $AB=CD$ and $DA=BC$ . $ABCD$ is congruent to the three other squares that from the pattern .

  

Two different sized triangles are used to create the illusion. One type is congruent to $△ABC$ and the other type is congruent the smaller triangle in the interior of $△ABC$ .

Hence, the answer is two.

b.

To determine

To prove:

  $△ABC\cong △CDA$ using the side-side-side congruence postulate.

Explanation of Solution

Given information:

  $ABCD$ is a square with $AB=CD$ and $DA=BC$ . $ABCD$ is congruent to the three other squares that from the pattern .

  

Proof:

If three sides of one triangle are congruent to three sides of a second triangle then the triangles are congruent.

A two column proof is a formal proof that contains statements and reasons organized in two columns.

As given $ABCD$ is a square.

Statements	Reasons
$ABCD$ is a square	Given
$\overline{AB}\cong \overline{CD},\overline{BC}\cong \overline{DA}$	Definition of a square
$\overline{AC}\cong \overline{CA},$	Reflexive property of congruence
$△ABC\cong △CDA$	SSS Congruence postulate

c.

To determine

To explain:

The relationship between the lines formed by the bases of the triangles.

Answer to Problem 1CYU

The relationship between the lines formed by the bases of the triangles is explained below.

Explanation of Solution

Given information:

  $ABCD$ is a square with $AB=CD$ and $DA=BC$ . $ABCD$ is congruent to the three other squares that from the pattern .

  

Since, $△ABC\cong △CDA$ , one obtains $\angle CAB\cong \angle ACD$ by CPCTC. Thus, $\overleftrightarrow{AB}\left|\right|\overleftrightarrow{CD},$ because $\angle CAB$ and $\angle ACD$ are alternate interior angles form by $\overleftrightarrow{AB},\overleftrightarrow{CD}$ and a transversal $\overleftrightarrow{AC}$ .