Math Problem Statement

Find the measure of z given the angles in the diagram (80°, 60°, 115°, and 30°).

Solution

Let’s analyze the problem step by step.

Problem Overview

We are tasked with finding the value of zz, based on the angles provided in the diagram:

  1. The angles 8080^\circ, 6060^\circ, 115115^\circ, and 3030^\circ are labeled in the diagram.
  2. The diagram contains intersecting lines and triangles.

Step-by-Step Solution

  1. Identify the straight-line relationships:

    • The angle 115115^\circ forms a linear pair with the angle adjacent to it. Therefore, the adjacent angle is: 180115=65180^\circ - 115^\circ = 65^\circ
  2. Focus on the triangle containing zz:

    • In the triangle at the bottom where zz is one of the angles:
      • One of the angles is 6565^\circ (calculated above).
      • Another angle is 3030^\circ (given in the diagram).
      • Let zz be the third angle.
  3. Use the triangle sum property:

    • The sum of all angles in a triangle is 180180^\circ. Therefore: z+65+30=180z + 65^\circ + 30^\circ = 180^\circ
    • Simplify: z=1806530z = 180^\circ - 65^\circ - 30^\circ z=85z = 85^\circ

Final Answer

The measure of zz is: z=85z = 85^\circ


Would you like further clarification or a breakdown of the geometry concepts used here?

Related Questions:

  1. What is the definition of a linear pair of angles, and how is it applied here?
  2. How does the triangle sum property work in geometry?
  3. What are some techniques for solving angles in intersecting-line problems?
  4. Why is the calculation 180115=65180^\circ - 115^\circ = 65^\circ important in this diagram?
  5. How do you identify which angles belong to the same triangle in a complex diagram?

Tip:

When working with diagrams, always look for linear pairs and triangle angle sums—they are the key to unlocking most angle-based problems.

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Math Problem Analysis

Mathematical Concepts

Triangle Sum Property
Linear Pair of Angles
Angle Relationships

Formulas

Sum of angles in a triangle = 180°
Linear pair of angles: a + b = 180°

Theorems

Triangle Angle Sum Theorem
Linear Pair Theorem

Suitable Grade Level

Grades 8-10