Question

problem 23
clear
given: rectangle defg,
t is the midpoint
of \\( \overline { e f } \\).
prove: \\( \triangle d t g \\) is isosceles.

1. rectangle defg
2. given

2.

1. definition of a

rectangle
3.

1. all right angles are

congruent.

1. t is the midpoint of

\\( \overline { e f } \\).
4.

Explanation:

Step1: Write the property of rectangle

In a rectangle, opposite sides are equal. So, \(DE = FG\).

Step2: Write the property of angles in rectangle

\(\angle DEF=\angle GFE = 90^{\circ}\)

Step3: Write the property of mid - point

Since \(T\) is the mid - point of \(\overline{EF}\), \(ET=FT\)

Step4: Prove \(\triangle DET\cong\triangle FGT\)

By Side - Angle - Side (SAS) congruence criterion (\(DE = FG\), \(\angle DEF=\angle GFE\), \(ET = FT\)), \(\triangle DET\cong\triangle FGT\)

Step5: Write the property of congruent triangles

If \(\triangle DET\cong\triangle FGT\), then \(DT = GT\)

Step6: Use the definition of isosceles triangle

A triangle with two equal sides is isosceles. Since \(DT = GT\), \(\triangle DTG\) is isosceles.

Answer:

1. \(DE = FG\) (Opposite sides of a rectangle are equal)
2. \(\angle DEF=\angle GFE = 90^{\circ}\)
3. Given (as per the problem statement)