Math Problem Statement

ABC and EDC are straight lines. AE and BD are parallel. Angle ABD = 125°, angle BCD = 30°. Work out the size of the angle marked x. Give reasons for your answer.

Solution

Let’s solve the problem step by step.

Problem Overview:

  • Given:
    • ABAB and EDED are straight lines.
    • AEBDAE \parallel BD.
    • Angle ABD=125ABD = 125^\circ.
    • Angle BCD=30BCD = 30^\circ.
  • Find: The angle marked xx and justify the reasoning.

Step-by-Step Solution:

Step 1: Consider the geometry and relationships between angles.

  • Since AEBDAE \parallel BD, and ABAB acts as a transversal:
    • ABD+CBD=180\angle ABD + \angle CBD = 180^\circ (co-interior angles on parallel lines).
    • ABD=125\angle ABD = 125^\circ, so: CBD=180125=55\angle CBD = 180^\circ - 125^\circ = 55^\circ

Step 2: Focus on triangle BCDBCD.

  • In BCD\triangle BCD:
    • BCD=30\angle BCD = 30^\circ,
    • CBD=55\angle CBD = 55^\circ,
    • Use the triangle angle sum property (BCD+CBD+DBC=180\angle BCD + \angle CBD + \angle DBC = 180^\circ) to find DBC\angle DBC: DBC=180(30+55)=95\angle DBC = 180^\circ - (30^\circ + 55^\circ) = 95^\circ

Step 3: Consider EDB\angle EDB and angle xx.

  • Since EDBDED \parallel BD, DBC\angle DBC and xx are corresponding angles. Thus: x=DBC=95x = \angle DBC = 95^\circ

Final Answer:

x=95x = 95^\circ

Reasoning:

  • xx is equal to DBC\angle DBC due to corresponding angles formed by parallel lines AEBDAE \parallel BD and the transversal DEDE.

Would you like a detailed diagram explanation or further clarification?


Related Questions:

  1. What are the other types of angle relationships formed by parallel lines and transversals?
  2. How does the triangle angle sum property apply in different types of triangles?
  3. Can the concept of corresponding angles be extended to polygons with more than three sides?
  4. What are the properties of co-interior and alternate angles on parallel lines?
  5. How would the calculation change if one of the given angles was different?

Tip: Always look for parallel lines and transversals to identify useful angle relationships in geometry problems.

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Math Problem Analysis

Mathematical Concepts

Geometry
Angles in Parallel Lines
Triangle Angle Sum Property

Formulas

Co-interior angles: Angle 1 + Angle 2 = 180°
Triangle angle sum property: Sum of angles in a triangle = 180°

Theorems

Co-interior angles theorem
Corresponding angles theorem
Triangle angle sum property

Suitable Grade Level

Grades 8-10