Question

care is a rectangle. \\(rc = 28\\).

\\(m\angle rge =\\) \\(^{\circ}\\)
\\(m\angle gec =\\) \\(^{\circ}\\)

\\(\delta rge\\) is a(n) select triangle.
\\(\delta rga\\) is a(n) select triangle.

\\(rg =\\)
\\(ec =\\)
\\(re =\\)

Explanation:

Identify given properties

We are given rectangle \(CARE\) with diagonal \(RC = 28\) and angle \(m\angle REG = 30^\circ\).
In a rectangle, diagonals are equal and bisect each other:

$$ AE = RC = 28 $$

$$ RG = GC = AG = GE = 14 $$

Find requested angles

Using Isosceles Triangle Properties:
Since \(RG = GE = 14\), \(\triangle RGE\) is isosceles with \(m\angle ERG = m\angle REG = 30^\circ\).
The sum of angles in \(\triangle RGE\) is \(180^\circ\):

$$ m\angle RGE = 180^\circ - 30^\circ - 30^\circ = 120^\circ $$

Since each corner angle of a rectangle is \(90^\circ\):

$$ m\angle GEC = m\angle REC - m\angle REG = 90^\circ - 30^\circ = 60^\circ $$

Classify triangles

Since \(RG = GE = 14\) and the vertex angle \(m\angle RGE = 120^\circ\), \(\triangle RGE\) is an obtuse isosceles triangle.
Since \(RG = AG = 14\) and \(m\angle RGA = 180^\circ - m\angle RGE = 60^\circ\), \(\triangle RGA\) is an equilateral triangle (an isosceles triangle with a \(60^\circ\) angle).

Calculate segment lengths

We already established:

$$ RG = \frac{1}{2} RC = \frac{1}{2} (28) = 14 $$

In right triangle \(\triangle REC\) with \(m\angle REC = 90^\circ\) and \(m\angle ERG = 30^\circ\):

$$ EC = RC \cdot \sin(30^\circ) = 28 \cdot 0.5 = 14 $$

$$ RE = RC \cdot \cos(30^\circ) = 28 \cdot \frac{\sqrt{3}}{2} = 14\sqrt{3} \approx 24.25 $$

Answer:

\(CARE\) is a rectangle. \(RC = 28\).

\(m\angle RGE =\) <blank>120</blank>\(^\circ\)

\(m\angle GEC =\) <blank>60</blank>\(^\circ\)

\(\triangle RGE\) is a(n) <blank>isosceles</blank> triangle.

\(\triangle RGA\) is a(n) <blank>equilateral</blank> triangle.

\(RG =\) <blank>14</blank>

\(EC =\) <blank>14</blank>

\(RE =\) <blank>\(14\sqrt{3}\)</blank>