The angles of a triangle are x, x, and y where y > 90. What kind of triangle is it? Write down the inequalities in terms of x. Write down an inequality in terms of x and y. What is... The angles of a triangle are x, x, and y where y > 90. What kind of triangle is it? Write down the inequalities in terms of x. Write down an inequality in terms of x and y. What is the range of values of x?
Understand the Problem
The question describes a triangle with angles x, x, and y, where y is greater than 90 degrees (making it an obtuse angle). The task involves classifying the triangle, establishing inequalities based on the given information, and determining the range of possible values for angle x.
Answer
$0^\circ < x < 45^\circ$
Answer for screen readers
$0^\circ < x < 45^\circ$
Steps to Solve
- Classify the triangle
Since two angles of the triangle are equal (both are $x$), the triangle is an isosceles triangle. Also, since one angle $y > 90^\circ$, it is an obtuse triangle. Therefore, the triangle is an obtuse isosceles triangle.
- Use the triangle angle sum theorem
The sum of the angles in any triangle is $180^\circ$. In this case, we have: $$x + x + y = 180^\circ$$ $$2x + y = 180^\circ$$
- Express y in terms of x
From the equation above, we can express $y$ in terms of $x$: $$y = 180^\circ - 2x$$
- Apply the given inequality
We are given that $y > 90^\circ$. Substituting the expression for $y$ in terms of $x$, we get: $$180^\circ - 2x > 90^\circ$$
- Solve the inequality for x
Subtract $180^\circ$ from both sides of the inequality: $$-2x > 90^\circ - 180^\circ$$ $$-2x > -90^\circ$$ Divide both sides by $-2$. Remember to reverse the inequality sign when dividing by a negative number: $$x < \frac{-90^\circ}{-2}$$ $$x < 45^\circ$$
- Consider the constraint that x must be greater than 0
Since $x$ is an angle in a triangle, it must be greater than $0^\circ$. Therefore, we have: $$x > 0^\circ$$
- Combine the inequalities
Combining the inequalities $x < 45^\circ$ and $x > 0^\circ$, we get the range of possible values for $x$: $$0^\circ < x < 45^\circ$$
$0^\circ < x < 45^\circ$
More Information
The problem combines triangle properties with inequality manipulation to find the possible range of an angle. Knowing that the sum of angles in a triangle is always 180 degrees is crucial here.
Tips
A common mistake is forgetting to reverse the inequality sign when dividing by a negative number. Also overlooking the basic rule that angles in a triangle have to be greater than $0^\circ$. Failing to recognise the basic properties of isosceles and obtuse triangles could also lead to incorrect solutions.
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