Chapter 4.4, Problem 15WE

To determine

To find: the true statement.

Answer to Problem 15WE

The proof given above concludes that only (1) and (3) statements given in prove are true.

Explanation of Solution

Given information:

Given figure,

Given,

  $\overline{PQ}\cong \overline{PR};\overline{TR}\cong \overline{TS}$

Given options are,

  $\overline{ST}\parallel \overline{QP}$

  $\overline{ST}\cong \overline{QP}$

  $\angle T\cong \angle P$

Calculation:

The Isosceles Triangle Theorem: If two sides of triangle are congruent, then the angles opposite to those sides are congruent.

Given: $\overline{PQ}\cong \overline{PR};\overline{TR}\cong \overline{TS}$

Prove: $\overline{ST}\parallel \overline{QP}$

  $\overline{ST}\cong \overline{QP}$

  $\angle T\cong \angle P$

Proof:

Statements	Reasons
1. $\overline{PQ}\cong \overline{PR}$	Given
2. $\angle Q\cong \angle PRQ$	Isosceles theorem
3. $\overline{TR}\cong \overline{TS}$	Given
$\angle TRS\cong \angle S$	Isosceles theorem
4. $\angle PRQ\cong \angle TRS$	Vertically opposite angles
5. $\angle Q\cong \angle S$	Transitive property
6. $\overline{ST}\parallel \overline{QP}$	If two lines are cut by a transversal and alternate interior angles are   $\cong$ then the lines are parallel
7. $\angle T\cong \angle P$	Alternate interior angles

From the proof given above conclude that only (1) and (3) statements given in prove are true.