Chapter 12, Problem 1NT

Assume $\Delta ABC\cong \Delta DEF$.

a. List the congruent angles and congruent sides.

b. List all possible ways that the congruence can be symbolized.

To determine

(a)

To list:

The congruent angles and congruent sides if $\Delta ABC\cong \Delta DEF$.

Answer to Problem 1NT

Solution:

The congruent angles are $\angle A$ corresponds to $\angle D$. $\angle B$ corresponds to $\angle E$. $\angle C$ corresponds to $\angle F$. The congruent sides are $\overline{AB}$ corresponds to $\overline{DE}$. $\overline{BC}$ corresponds to $\overline{EF}$. $\overline{AC}$ corresponds to $\overline{DF}$.

Explanation of Solution

Definition used:

The triangles are congruent or corresponding parts if they have exactly the same three sides and exactly the same three angles.

Rules for two triangles are congruent:

1. SSS: If three sides of the triangles are equal to the sides of another triangle, then the two triangles are congruent.

2. SAS: If two sides and included angle of one triangle are equal to corresponding sides and angle of another triangle, then the two triangles are congruent.

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

4. AAS: If two angles and non-included side of one triangle are equal to corresponding angles and side of another triangle, then the two triangles are congruent.

5. HL: If the hypotenuse and one leg of one right-angled triangle are equal to corresponding hypotenuse and leg of another right-angled triangle, then the two triangles are congruent.

Calculation:

Draw the traingles $\Delta ABC\text{and}\Delta DEF$ is as follows.

From the above triangle, it is observed that $\Delta ABC$ and $\Delta DEF$ are scalene triangle.

Here, the two sides and included angle of one triangle are equal to corresponding sides and angle of another triangle.

Check the congruent by finding all the six corresponding parts of the two triangles $\Delta ABC$ and $\Delta DEF$.

Compute the corresponding sides of $\Delta ABC$ and $\Delta DEF$ by using the given triangles.

$\overline{AB}$ corresponds to $\overline{DE}$.

$\overline{BC}$ corresponds to $\overline{EF}$.

$\overline{AC}$ corresponds to $\overline{DF}$.

Compute the corresponding angles of $\Delta ABC$ and $\Delta DEF$ by using the given triangles.

$\angle A$ corresponds to $\angle D$.

$\angle B$ corresponds to $\angle E$.

$\angle C$ corresponds to $\angle F$.

Therefore, all the six corresponding parts are congruent.

Hence, the triangles are congruent $\Delta ABC\simeq \Delta DEF$ by using the rule SAS by using the rule (2).

Final Statement:

The corresponding parts of the triangles are obtained.

To determine

(b)

To list:

All the possible ways to symbolize the congruence.

Explanation of Solution

Calculation:

All the possible ways to symbolize the congruence is as follows.

1. $\Delta ABC\cong \Delta DEF$

2. $\Delta ACB\cong \Delta DFE$

3. $\Delta BAC\cong \Delta EDF$

4. $\Delta BCA\cong \Delta EFD$

5. $\Delta CAB\cong \Delta FDE$

6. $\Delta CBA\cong \Delta FED$