- An isosceles triangle has two congruent base angles \(b\) and a vertex angle \(v\), satisfying the equation \(2b + v = 180^{\circ}\).
- Given one base angle \(b\), the second base angle is also \(b\).
- The vertex angle is found using the formula \(v = 180^{\circ} - 2b\).
- Ashley is correct, as all three angles can be determined from one base angle using this method.
- Final answer: \(\boxed{\text{She is correct. Take the given base angle, multiply by 2 and subtract from 180}^{\circ}\text{ to get the vertex angle. The other base angle matches the given base angle.}}\)
Explanation
- Analyze the properties of an isosceles triangle
An isosceles triangle is defined by having at least two sides of equal length. A fundamental property of an isosceles triangle is that the angles opposite to these equal sides are also congruent. These two congruent angles are referred to as the 'base angles' of the triangle. The third angle, which is not necessarily congruent to the base angles, is called the 'vertex angle'. Let the measure of each base angle be \(b\) and the measure of the vertex angle be \(v\). The sum of all interior angles in any triangle must always equal \(180^{\circ}\). Therefore, the relationship between these angles can be expressed as:
\[2b + v = 180^{\circ}\]
- Evaluate Ashley's claim and explain the reasoning
If we are given the measure of one base angle, \(b\), we immediately know the measure of the second base angle because it must be equal to the first base angle. Thus, the second base angle is also \(b\). Knowing the two base angles allows us to find the third angle, \(v\), by substituting the known value of \(b\) into the sum of angles equation. Rearranging the equation \(2b + v = 180^{\circ}\) to solve for \(v\), we get:
\[v = 180^{\circ} - 2b\]
Since both the second base angle and the vertex angle can be determined using only the value of the given base angle \(b\), Ashley's claim is correct. The procedure is to take the given base angle, multiply it by 2, and subtract the result from \(180^{\circ}\) to find the vertex angle. The other base angle is identical to the one given.
- Conclusion
Based on the analysis, Ashley is correct. The correct description of the process is: "She is correct. Take the given base angle, multiply by \(2\) and subtract from \(180^{\circ}\) to get the vertex angle. The other base angle matches the given base angle." The final answer is:
\(\boxed{\text{She is correct. Take the given base angle, multiply by 2 and subtract from 180}^{\circ}\text{ to get the vertex angle. The other base angle matches the given base angle.}}\)
Examples
Imagine you are building a triangular roof truss for a garden shed where the two rafters are equal in length. If you know that one of the angles where the rafter meets the flat bottom frame is \(40^{\circ}\), you can easily determine the other base angle (which is also \(40^{\circ}\)) and the top peak angle (the vertex angle) as \(180^{\circ} - (40^{\circ} + 40^{\circ}) = 100^{\circ}\). This property is incredibly useful in carpentry and construction, allowing you to quickly cut materials at the correct angles with minimal measurements.