Question

A boat’s sail is in the shape of an isosceles triangle. If the largest angle of the sail is 90°, what is the measure of the smallest angle in degrees?

Short answer

The smallest angle in the sail's isosceles triangle is 45 degrees.

Step-by-step solution

Analyze the given triangle and identify the known values.

We know that the sail is in the shape of an isosceles triangle, which means that two sides of the triangle are equal in length. Additionally, one angle of the triangle is given as 90 degrees.

Determine the sum of the unknown angles.

Since we know the sum of the interior angles of any triangle, we can subtract the known 90-degree angle from 180 degrees to find the sum of the remaining two angles. \[Sum\ of\ remaining\ angles = 180° - 90°\] \[Sum\ of\ remaining\ angles = 90°\]

Determine the smallest angle of the isosceles triangle.

Since the sail has the shape of an isosceles triangle, the other two angles are equal in size. Therefore, to find the smallest angle, we will divide the sum of the unknown angles by 2. \[Smallest\ angle = \frac{90°}{2}\] \[Smallest\ angle = 45°\] The smallest angle in the sail's isosceles triangle is 45 degrees.

Key concepts

Triangle Interior Angles

Understanding the triangle interior angles is crucial in geometry, especially when dealing with exercises like finding the smallest angle in an isosceles triangle. An essential principle to grasp is that the sum of all interior angles in any triangle always adds up to 180 degrees. This rule can be particularly helpful when you only know one or two angles of a triangle.

In the case of an isosceles triangle, two of its sides are of equal length, and significantly, the angles opposite these sides are also equal. So, if you know one angle, you immediately know another. For example, in the given exercise, the largest angle is 90 degrees, and since the triangle is isosceles, the other two angles must be equal. To find their measures, you subtract the 90 degrees from the total of 180 degrees and then divide by 2 because the two remaining angles are the same in measure.

Geometry Problems

When solving geometry problems, especially when they involve shapes like triangles, it's important to be familiar with definitions and properties. For instance, the isosceles triangle property states that two sides are equal, and so are the angles opposite those sides. Another key aspect is visualizing or sketching the geometric shape as described in the problem.

This approach can aid in recognizing the relationships between angles and sides. In our boat sail problem, visualizing the isosceles triangle could help remind the solver that the base angles are equal. Furthermore, implementing the triangle's interior angle sum facilitates arriving at the correct solution. Applying these properties systematically can transform complex geometry problems into more manageable steps.

GED Math Practice

For students preparing for the GED, math practice is essential in becoming proficient at solving a wide range of problems, including geometry-related questions. Working with isosceles triangles is frequent, thus understanding their specific properties can build a strong geometry foundation. To excel in the GED math exam, it is not only important to learn the procedures for solving problems but also to understand the underlying concepts.

Practice problems like the one discussed can improve problem-solving strategies and reinforce concepts like the interior angles of a triangle. The GED math section will test on concepts such as these, so mastering them can lead to a higher score. Engaging with different problem types and remembering core principles like the interior angles sum can make GED math practice more effective and less daunting.