The Two Congruent Angles In An Isosceles Right Triangle Measure \[$3x + 10\$\] Degrees And \[$5x - 16\$\] Degrees. What Value Of \[$x\$\] Makes This Relationship True?A. 13 Degrees B. 16 Degrees C. 18 Degrees D. 20 Degrees
Understanding the Problem
In an isosceles right triangle, two congruent angles are given as ${3x + 10\$} degrees and ${5x - 16\$} degrees. We need to find the value of {x$}$ that makes this relationship true.
Properties of an Isosceles Right Triangle
An isosceles right triangle has two congruent angles, which are the base angles. The sum of the base angles in a right triangle is always 90 degrees. Since the two base angles are congruent, we can set up an equation to represent this relationship.
Setting Up the Equation
Let's set up an equation to represent the sum of the base angles in the isosceles right triangle:
${3x + 10\$} + ${5x - 16\$} = 90
Simplifying the Equation
Now, let's simplify the equation by combining like terms:
${3x + 5x + 10 - 16\$} = 90
Combining Like Terms
Combine the like terms:
${8x - 6\$} = 90
Adding 6 to Both Sides
Add 6 to both sides of the equation to isolate the term with the variable:
${8x - 6 + 6\$} = 90 + 6
Simplifying the Equation
Simplify the equation:
${8x\$} = 96
Dividing Both Sides by 8
Divide both sides of the equation by 8 to solve for {x$}$:
${8x / 8\$} = 96 / 8
Simplifying the Equation
Simplify the equation:
{x$}$ = 12
Conclusion
The value of {x$}$ that makes the relationship true is 12. However, this is not among the given options. Let's re-examine the problem and see if we can find the correct value of {x$}$.
Re-Examining the Problem
Since the two base angles are congruent, we can set up an equation to represent this relationship:
${3x + 10\$} = ${5x - 16\$}
Simplifying the Equation
Now, let's simplify the equation by subtracting ${3x\$} from both sides:
${10\$} = ${5x - 3x - 16\$}
Simplifying the Equation
Simplify the equation:
${10\$} = ${2x - 16\$}
Adding 16 to Both Sides
Add 16 to both sides of the equation to isolate the term with the variable:
${10 + 16\$} = ${2x - 16 + 16\$}
Simplifying the Equation
Simplify the equation:
${26\$} = ${2x\$}
Dividing Both Sides by 2
Divide both sides of the equation by 2 to solve for {x$}$:
${26 / 2\$} = ${2x / 2\$}
Simplifying the Equation
Simplify the equation:
${13\$} = {x$}$
Conclusion
The value of {x$}$ that makes the relationship true is 13.
Final Answer
The final answer is 13.
Frequently Asked Questions
Q: What is an isosceles right triangle?
A: An isosceles right triangle is a type of triangle that has two congruent base angles and a right angle (90 degrees). The two base angles are equal in measure.
Q: What are the two congruent angles in an isosceles right triangle?
A: The two congruent angles in an isosceles right triangle are given as ${3x + 10\$} degrees and ${5x - 16\$} degrees.
Q: How do we find the value of {x$}$ that makes this relationship true?
A: To find the value of {x$}$, we need to set up an equation to represent the sum of the base angles in the isosceles right triangle. We can then solve for {x$}$ using algebraic methods.
Q: What is the sum of the base angles in a right triangle?
A: The sum of the base angles in a right triangle is always 90 degrees.
Q: How do we set up an equation to represent the sum of the base angles in the isosceles right triangle?
A: We can set up an equation by adding the two base angles together and setting the sum equal to 90 degrees.
Q: What is the equation that represents the sum of the base angles in the isosceles right triangle?
A: The equation that represents the sum of the base angles in the isosceles right triangle is:
${3x + 10\$} + ${5x - 16\$} = 90
Q: How do we simplify the equation?
A: We can simplify the equation by combining like terms.
Q: What is the simplified equation?
A: The simplified equation is:
${8x - 6\$} = 90
Q: How do we solve for {x$}$?
A: We can solve for {x$}$ by adding 6 to both sides of the equation and then dividing both sides by 8.
Q: What is the value of {x$}$ that makes the relationship true?
A: The value of {x$}$ that makes the relationship true is 13.
Q: Why is the value of {x$}$ 13?
A: The value of {x$}$ is 13 because when we substitute 13 into the equation ${3x + 10\$} and ${5x - 16\$}, we get two congruent angles that add up to 90 degrees.
Q: What are the two congruent angles when {x$}$ is 13?
A: The two congruent angles when {x$}$ is 13 are ${3(13) + 10\$} = 49 degrees and ${5(13) - 16\$} = 41 degrees.
Q: Why are the two angles congruent?
A: The two angles are congruent because they have the same measure.
Q: What is the relationship between the two congruent angles?
A: The two congruent angles are related by the equation ${3x + 10\$} = ${5x - 16\$}.
Q: How do we verify that the two angles are congruent?
A: We can verify that the two angles are congruent by substituting the value of {x$}$ into the equation and checking that the two angles have the same measure.
Q: What is the final answer?
A: The final answer is 13.