Quadrilaterals and Polygons - Angles Worksheets - Free Printable

Problem Analysis:

The given image shows a quadrilateral \( STRU \) with the following properties:
- \( ST \parallel RU \), indicating that \( STRU \) is an isosceles trapezoid.
- The angles at vertices \( S \) and \( R \) are given as \( 70^\circ \) and \( 110^\circ \), respectively.
- The lengths of the non-parallel sides \( ST \) and \( RU \) are given as \( 23x - 5 \) and \( 14x \), respectively.

We need to solve for \( x \).

Step-by-Step Solution:

#### Step 1: Understand the properties of the isosceles trapezoid
In an isosceles trapezoid:
1. The non-parallel sides (legs) are equal in length.
2. The base angles are equal in pairs. Specifically, the angles adjacent to each parallel side are supplementary (sum to \( 180^\circ \)).

From the diagram:
- \( ST \parallel RU \)
- The non-parallel sides are \( SR \) and \( TU \).
- The base angles at \( S \) and \( T \) are equal, and the base angles at \( R \) and \( U \) are equal.

#### Step 2: Use the property of supplementary angles
The angles at \( S \) and \( R \) are given as \( 70^\circ \) and \( 110^\circ \), respectively. Since \( ST \parallel RU \), the angles at \( T \) and \( U \) must be:
- \( \angle T = \angle S = 70^\circ \) (opposite base angles)
- \( \angle U = \angle R = 110^\circ \) (opposite base angles)

This confirms that the trapezoid is indeed isosceles.

#### Step 3: Use the property of equal legs
In an isosceles trapezoid, the non-parallel sides (legs) are equal. Therefore:
\[ ST = RU \]

From the problem, the lengths of \( ST \) and \( RU \) are given as:
\[ ST = 23x - 5 \]
\[ RU = 14x \]

Setting these equal gives:
\[ 23x - 5 = 14x \]

#### Step 4: Solve for \( x \)
Rearrange the equation to isolate \( x \):
\[ 23x - 14x = 5 \]
\[ 9x = 5 \]
\[ x = \frac{5}{9} \]

#### Step 5: Verify the solution
Substitute \( x = \frac{5}{9} \) back into the expressions for \( ST \) and \( RU \):
\[ ST = 23x - 5 = 23\left(\frac{5}{9}\right) - 5 = \frac{115}{9} - \frac{45}{9} = \frac{70}{9} \]
\[ RU = 14x = 14\left(\frac{5}{9}\right) = \frac{70}{9} \]

Since \( ST = RU \), the solution is consistent with the properties of the isosceles trapezoid.

Final Answer:

\[
\boxed{\frac{5}{9}}
\]