Chapter 12.2, Problem 22E

(a)

To determine

To explain:

The given triangle $\overline{BE}\cong \overline{DE}\text{and}\angle ABE\cong \angle CDE$ .

Explanation of Solution

Given:

  $ABEF\cong CDEF$

Concept used:

Two geometric figures are congruent if a rigid motion or a composition of rigid motions maps one of the figures onto other.

Calculation:

If two triangles are congruent.

So all the corresponding parts are congruent.

If all corresponding parts of two triangles are congruent.

So the triangle are congruent.

The pairs of congruent corresponding angles

  $\begin{array}{l}\angle E=\angle F\\ \angle A=\angle C\\ \overline{BE}\cong \overline{DE}\\ \angle ABE\cong \angle CDE\end{array}$

Therefore,

The corresponding parts of congruent figures are congruent.

(a)

To determine

To explain:

The given triangle $\overline{BE}\cong \overline{DE}\text{and}\angle ABE\cong \angle CDE$ .

Explanation of Solution

Given:

  $ABEF\cong CDEF$

Concept used:

Two geometric figures are congruent if a rigid motion or a composition of rigid motions maps one of the figures onto other.

Calculation:

If two triangles are congruent.

So all the corresponding parts are congruent.

If all corresponding parts of two triangles are congruent.

So the triangle are congruent.

The pairs of congruent corresponding angles

  $\begin{array}{l}\angle E=\angle F\\ \angle A=\angle C\\ \overline{BE}\cong \overline{DE}\\ \angle ABE\cong \angle CDE\end{array}$

Therefore,

The corresponding parts of congruent figures are congruent.

(b)

To determine

To explain:

The given triangle $\angle GBE\cong \angle GDE$ .

Explanation of Solution

Given:

  $ABEF\cong CDEF$

Concept used:

Two geometric figures are congruent if a rigid motion or a composition of rigid motions maps one of the figures onto other.

Calculation:

If two triangles are congruent.

So all the corresponding parts are congruent.

If all corresponding parts of two triangles are congruent.

So the triangle are congruent.

The pairs of congruent corresponding angles

  $\begin{array}{l}\angle E=\angle F\\ \angle A=\angle C\\ \overline{BE}\cong \overline{DE}\\ \angle ABE\cong \angle CDE\\ \angle GBE\cong \angle GDE\end{array}$

If two angles are supplymentry to congruent angles

Therefore,

The corresponding parts of congruent figures are congruent.

(c)

To determine

To explain:

The given triangle $\angle GEB\cong \angle GED$ .

Explanation of Solution

Given:

  $ABEF\cong CDEF$

Concept used:

Two geometric figures are congruent if a rigid motion or a composition of rigid motions maps one of the figures onto other.

Calculation:

If two triangles are congruent.

So all the corresponding parts are congruent.

If all corresponding parts of two triangles are congruent.

So the triangle are congruent.

The pairs of congruent corresponding angles

  $\begin{array}{l}\angle E=\angle F\\ \angle A=\angle C\\ \overline{BE}\cong \overline{DE}\\ \angle ABE\cong \angle CDE\\ \angle GBE\cong \angle GDE\\ \angle GEB\cong \angle GED\end{array}$

If two angles are supplymentry to congruent angles

The corresponding parts of congruent figures are congruent.

Therefore,

  $\angle GEB$ isa also a right angle and all right angles are congruent.

(d)

To determine

To prove:

The given triangle $\Delta BEG\cong \Delta DEG$ .

Explanation of Solution

Given:

  $ABEF\cong CDEF$

Concept used:

Two geometric figures are congruent if a rigid motion or a composition of rigid motions maps one of the figures onto other.

Calculation:

If two triangles are congruent.

So all the corresponding parts are congruent.

If all corresponding parts of two triangles are congruent.

So the triangle are congruent.

By reflexive property

  $\overline{GE}\cong \overline{GE}$

The pairs of congruent corresponding angles

  $\begin{array}{l}\angle E=\angle F\\ \angle A=\angle C\\ \overline{BE}\cong \overline{DE}\\ \angle ABE\cong \angle CDE\\ \angle GBE\cong \angle GDE\\ \angle GEB\cong \angle GED\\ \Delta BEG\cong \Delta DEG\end{array}$

Therefore,

If two angles anglesof one triangle are congruent to two angles of another triangle

So, the third angles are also congruent.

The corresponding parts of congruent figures are congruent.