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Chapter 2: Problem 6

Name the angle included between sides \(X Y\) and \(Y Z\) in \(\Delta X Y Z\).

Short Answer

Expert verified
The included angle is \(\angle XYZ\).

Step by step solution

01

Identify the triangle

We are given a triangle, denoted as \(\Delta XYZ\), which consists of the vertices \(X\), \(Y\), and \(Z\). Each pair of these vertices forms one side of the triangle.
02

Locate the sides

The sides of the triangle are named based on the vertices they connect. The sides are \(XY\), \(YZ\), and \(XZ\). We need to focus on the sides \(XY\) and \(YZ\) as the question specifies these.
03

Determine the included angle

The included angle is the angle formed at the vertex where two given sides meet. Since the sides in question are \(XY\) and \(YZ\), the angle between these two sides is located at the vertex \(Y\). Thus, the included angle is \(\angle XYZ\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Triangle
A triangle is a fundamental shape in geometry, consisting of three sides and three angles. It's defined by three points called vertices, connected by line segments. These line segments are called sides. In our given problem, the triangle is named \( \Delta XYZ \), suggesting a specific order in which the vertices are connected. All the sides and angles reside within the plane defined by these three points. Understanding the triangle's structure is key to solving problems about angles and side relationships.
Vertices
Vertices are the distinct points where two sides of a triangle meet. In the case of \( \Delta XYZ \), the vertices are labeled \( X \), \( Y \), and \( Z \). Each serves as a "corner" of the triangle. How these vertices are arranged affects which angle is formed between any two sides. For example, when discussing the angle included between sides \( XY \) and \( YZ \), focus on the vertex where these sides meet, which is \( Y \). This understanding helps in identifying angles formed in a triangle.
Sides
The sides of a triangle are the line segments between its vertices. They are key components that help in defining the shape and size of the triangle. In \( \Delta XYZ \), the sides are identified as \( XY \), \( YZ \), and \( ZX \). Each side is named after the vertices it connects. Sides are important as they influence the measurement of angles and overall triangle properties. The length of the sides can also determine the type of triangle, such as equilateral, isosceles, or scalene.
Angle Measurement
Angle measurement is central in problems involving triangles. Each triangle contains three angles, one at each vertex. The term "included angle" refers to the angle formed between two specific sides of a triangle at a given vertex. In \( \Delta XYZ \), the included angle between the sides \( XY \) and \( YZ \) is located at vertex \( Y \), marked as \( \angle XYZ \). Understanding how to identify and measure the included angles is crucial for correctly solving geometric problems involving triangles.

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Most popular questions from this chapter

Name the angle included between sides RS and TS in \(\triangle\) RST.

Name the angle included between sides \(A B\) and \(B C\) in \(\triangle A B C\).

Name the side included between the angles: \(\angle X\) and \(\angle Y\) in \(\triangle X Y Z\).

Draw the triangle with the two sides and the included angle and (2) measure the remaining side and angles: \(A B=2\) inches, \(A C=1\) inch, \(\angle A=60^{\circ}\).

Name the side included between the angles: \(\angle \mathrm{A}\) and \(\angle \mathrm{B}\) in \(\triangle \mathrm{ABC}\).
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