Question

Camping When Kendra's tent is set up, the front of the tent is in the shape of an isosceles triangle. If each tent side makes a \(75^{\circ}\) angle with the ground, what is the measure of the angle at which the sides of the tent meet? (Lesson 6-5)

Short answer

The sides of the tent meet at a 30° angle.

Step-by-step solution

Identify Known Angles

We are given that each side of the isosceles triangle (representing the tent sides) makes a \(75^{\circ}\) angle with the ground. These angles are the base angles of the triangle.

Apply Triangle Angle Sum Theorem

According to the triangle angle sum theorem, the three interior angles of a triangle sum up to \(180^{\circ}\). In the isosceles triangle, there are two base angles of \(75^{\circ}\) each.

Calculate Angle at Apex

To find the angle at the apex (where the two sides meet), we subtract the sum of the two base angles from \(180^{\circ}\): \[180^{\circ} - 2 \times 75^{\circ} = 180^{\circ} - 150^{\circ} = 30^{\circ}\]

Key concepts

Isosceles Triangle

An isosceles triangle is a special type of triangle where two of its sides are of equal length, and consequently, the angles opposite those sides are equal. This unique property makes isosceles triangles a frequent subject of study in geometry.

In the context of the original exercise, the tent's shape is an isosceles triangle. This is because two of the tent's sides are equal in length, creating symmetry that is both aesthetically pleasing and structurally stable.

• The equal sides are known as the legs of the triangle.
• The unequal side is referred to as the base.
• The equal angles opposite these sides are called the base angles.

In a geometry problem involving isosceles triangles, identifying the equal sides and angles first can significantly simplify solving for unknown angles or sides.

Interior Angles

Interior angles are the angles inside a triangle. Understanding these angles is essential as they form the basis for many trigonometric calculations and constructions in geometry.

According to the Triangle Angle Sum Theorem, the sum of the three interior angles of a triangle is always \(180^{\circ}\). This theorem is a cornerstone of geometry and applies universally to all triangles.

• For an isosceles triangle, as in Kendra's tent, knowing two of the interior angles can help determine the third.
• The calculation involves summing the known angles and subtracting this sum from \(180^{\circ}\).

Hence, knowing about these interior angles and the sum theorem allows us to solve various geometry problems effectively.

Geometry Problem Solving

Solving geometry problems involves breaking down complex shapes into known elements and using established theorems and properties to find unknown values.

The problem with Kendra's tent can be solved effectively using these steps:

• First, identify what you know. In this example, we know the angles the tent sides make with the ground are each \(75^{\circ}\).
• Apply the Triangle Angle Sum Theorem to understand the relationship between the triangle's angles.
• Calculate the unknown angle by subtracting the sum of the known angles from \(180^{\circ}\).

By practicing these structured steps, solving geometric problems becomes more intuitive and less daunting.