To solve for [tex]\( x \)[/tex] and determine the measure of the exterior angle, follow these steps:
1. Identify the Given Information:
- The included (interior) angle of the triangle is [tex]\( 90^\circ \)[/tex].
- The measure of the exterior angle is given by the expression [tex]\( 2x + 50 \)[/tex].
2. Understand the Relationship Between Interior and Exterior Angles:
- The sum of an interior angle and its corresponding exterior angle of a triangle is always [tex]\( 180^\circ \)[/tex].
3. Set Up the Equation:
- Using the fact that the interior angle and the exterior angle sum to [tex]\( 180^\circ \)[/tex]:
[tex]\[
\text{interior angle} + \text{exterior angle} = 180^\circ
\][/tex]
- Substituting the values:
[tex]\[
90^\circ + (2x + 50) = 180^\circ
\][/tex]
4. Combine Like Terms:
- Simplify the equation by combining like terms on the left side:
[tex]\[
90 + 2x + 50 = 180
\][/tex]
- This simplifies further to:
[tex]\[
140 + 2x = 180
\][/tex]
5. Solve for [tex]\( x \)[/tex]:
- Subtract 140 from both sides of the equation to isolate the term with [tex]\( x \)[/tex]:
[tex]\[
2x = 40
\][/tex]
- Divide both sides by 2 to solve for [tex]\( x \)[/tex]:
[tex]\[
x = 20
\][/tex]
6. Determine the Measure of the Exterior Angle:
- Substitute the value of [tex]\( x \)[/tex] back into the expression for the exterior angle:
[tex]\[
\text{Exterior angle} = 2x + 50
\][/tex]
[tex]\[
\text{Exterior angle} = 2 \times 20 + 50
\][/tex]
- Calculate the value:
[tex]\[
\text{Exterior angle} = 40 + 50 = 90^\circ
\][/tex]
So, the value of [tex]\( x \)[/tex] is [tex]\( 20 \)[/tex] and the measure of the exterior angle is [tex]\( 90^\circ \)[/tex].
