Chapter 4.5, Problem 14CE

a.

To determine

To find which of the methods (SSS, SAS, ASA, AAS or HL) could be used instead of the method listed.

Answer to Problem 14CE

Side-Angle-Side (SAS)

Explanation of Solution

Given:

Two congruent triangles congruent by Leg-Leg Method (LL).

The given triangles $\Delta ABC$ and $\Delta XYZ$ are congruent by Leg-Leg Method (LL).

  $\begin{array}{l}AC\cong XZ\\ CB\cong ZY\end{array}$

Leg-Leg Method: If two legs of each right angled triangle are congruent to the legs of other right triangle then the two triangles are congruent.

The other congruency criteria that can be used is Side-Angle-Side (SAS).

  $AC\cong XZ$ (given)

  $\angle ACB=\angle XZY={90}^0$

  $CB\cong ZY$ (given)

Therefore, $\Delta ABC\cong \Delta XYZ$ by Side-Angle-Side (SAS) congruency criteria.

Conclusion:

Therefore, the other congruency criteria that can be used is Side-Angle-Side (SAS) congruency criteria.

b.

To determine

To find which of the methods (SSS, SAS, ASA, AAS or HL) could be used instead of the method listed.

Answer to Problem 14CE

Angle-Angle-Side (AAS)

Explanation of Solution

Given:

Two congruent triangles congruent by Hypotenuse-Acute Angle Method (HA).

The given triangles $\Delta ABC$ and $\Delta XYZ$ are congruent by Hypotenuse-Acute Angle Method (HA).

  $\begin{array}{l}\angle CBA\cong \angle ZYX\\ AB\cong XY\end{array}$

Hypotenuse-Angle Method (HA): If the hypotenuse and an acute angle of one right triangle is congruent to the hypotenuse and an acute angle of the other right triangle, then the triangles are said to be congruent.

The other congruency criteria that can be used is Angle-Angle-Side (AAS).

  $\angle CBA\cong \angle ZYX$ (given)

  $\angle ACB=\angle XZY={90}^0$

  $AB\cong XY$ (given)

Therefore, $\Delta ABC\cong \Delta XYZ$ by Angle-Angle-Side (AAS) congruency criteria.

Conclusion:

Therefore, the other congruency criteria that can be used is Angle-Angle-Side (AAS) congruency criteria.

c.

To determine

To find which of the methods (SSS, SAS, ASA, AAS or HL) could be used instead of the method listed.

Answer to Problem 14CE

Angle-Angle-Side (AAS)

Explanation of Solution

Given:

Two congruent triangles congruent by Leg-Leg Method (LL).

The given triangles $\Delta ABC$ and $\Delta XYZ$ are congruent by Leg-Acute Angle Method (LA).

  $\begin{array}{l}CB\cong ZY\\ \angle ABC\cong \angle XYZ\end{array}$

Leg-Acute Angle Method (LA): If a leg and an acute angle of one right triangle are congruent to the corresponding parts in the other right triangle, then the triangles are said to be congruent .

The other congruency criteria that can be used is Angle-Angle-Side (AAS).

  $CB\cong ZY$ (given)

  $\angle ACB=\angle XZY={90}^0$

  $\angle ABC\cong \angle XYZ$ (given)

Therefore, $\Delta ABC\cong \Delta XYZ$ by Angle-Angle-Side (AAS) congruency criteria.

Conclusion:

Therefore, the other congruency criteria that can be used is Angle-Angle-Side (AAS) congruency criteria.