Math Problem Statement

Jawablah

Diketahui ∠1 ≅ ∠4, ∠2 ≅ ∠3. Buktikan: ∠3 ≅ ∠4

Solution

To solve this problem, let’s analyze the given information and apply angle relationships in parallel lines and parallelograms:

Given:

  1. ∠1 ≅ ∠4
  2. ∠2 ≅ ∠3

To Prove:

∠3 ≅ ∠4

Explanation:

  1. Notice that the figure resembles a parallelogram (ABCD), where opposite sides are parallel. In a parallelogram, opposite angles are congruent.

  2. Since ∠1 ≅ ∠4, they are corresponding angles on parallel lines intersected by a transversal (line BD). This tells us that line AD is parallel to line BC.

  3. Similarly, ∠2 ≅ ∠3 are alternate interior angles created by transversal line DB across parallel lines AD and BC. This means ∠2 and ∠3 are equal because parallel lines produce equal alternate interior angles.

  4. Using transitivity, since ∠1 ≅ ∠4 and ∠2 ≅ ∠3, it follows that ∠3 ≅ ∠4 because they both equal the corresponding angles ∠1 and ∠2.

Conclusion:

Thus, ∠3 ≅ ∠4 is proven by properties of parallel lines and congruent angles in a parallelogram.

Would you like further details or explanations on any part of this solution?


Related Questions

  1. How do corresponding angles help determine parallel lines?
  2. What are the properties of a parallelogram regarding opposite angles?
  3. How can alternate interior angles prove parallelism?
  4. What other angle properties apply to parallelograms?
  5. How is transitivity used in geometric proofs?

Tip

In any quadrilateral with parallel opposite sides, checking for congruent angles can reveal if it’s a parallelogram and help in solving for unknown angles.

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Math Problem Analysis

Mathematical Concepts

Geometry
Angle Relationships
Parallel Lines

Formulas

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Theorems

Corresponding Angles
Alternate Interior Angles
Transitive Property of Equality

Suitable Grade Level

Grades 8-10