Math Problem Statement

Find the measure of the missing angles in the given geometric diagram where angles 51°, 90° (right angle), and unknown angles are shown.

Solution

Let's analyze the given diagram to find the missing angles.

Key observations:

  1. The angle given between the two diagonal lines is 5151^\circ.
  2. There is a right angle dd at the bottom formed by two perpendicular lines.
  3. The lines intersect, forming a vertical angle relationship, which means angles directly opposite to each other are congruent.

Steps to find the missing angles:

  1. Angle ee:

    • ee and 5151^\circ form a linear pair, meaning they are supplementary (sum to 180180^\circ).
    • So, e=18051=129e = 180^\circ - 51^\circ = 129^\circ.
  2. Angle ff:

    • Angle ff is vertically opposite to 5151^\circ, so it is also 5151^\circ (since vertical angles are congruent).
  3. Angle dd:

    • dd is a right angle, already marked as 9090^\circ, so no further calculation is needed.

Final values:

  • Angle e=129e = 129^\circ
  • Angle f=51f = 51^\circ
  • Angle d=90d = 90^\circ

Let me know if you would like more details or explanations.


Here are 5 related questions:

  1. What are vertical angles, and how do they relate to congruence?
  2. Can you explain what supplementary angles are and give more examples?
  3. What is the sum of angles on a straight line, and why does this rule hold?
  4. How would you calculate angles if the diagram involved parallel lines?
  5. What are complementary angles, and how are they different from supplementary angles?

Tip: When solving geometry problems involving intersecting lines, always check for vertical and supplementary angles. They often provide quick solutions!

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Math Problem Analysis

Mathematical Concepts

Geometry
Linear Pair
Vertical Angles
Supplementary Angles

Formulas

Supplementary angles: a + b = 180°
Vertical angles are congruent

Theorems

Linear Pair Theorem
Vertical Angle Theorem

Suitable Grade Level

Grades 6-8