Math Problem Statement
Write a proof to show that triangle ABD is congruent to triangle CDB given AB || CD and angle CBD ≅ angle ADB.
Solution
To prove , we will use the given information and apply the appropriate congruence criteria. Here's the step-by-step proof:
Given:
- (AB is parallel to CD)
- (angles are congruent)
To Prove:
Proof:
-
Given Information:
- implies that these segments form parallel sides of a trapezoid.
- .
-
Angles formed by parallel lines:
- Since and is a transversal, the alternate interior angles and are congruent:
-
Shared side:
- is shared by both and , so:
-
Congruence Criterion:
- Now, we have:
- (alternate interior angles)
- (shared side)
- (given).
- Using the ASA (Angle-Side-Angle) criterion, .
- Now, we have:
Conclusion:
Let me know if you'd like further clarifications! Here are some related questions for practice:
- How can you prove two triangles are congruent using the SAS criterion?
- Why are alternate interior angles equal for parallel lines and a transversal?
- What is the reflexive property of congruence?
- Can this proof also work with other congruence criteria like AAS? Why or why not?
- How does proving congruence help in solving real-world geometry problems?
Tip: Always identify shared sides or angles in geometric figures—they are key to many proofs.
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Math Problem Analysis
Mathematical Concepts
Triangle Congruence
Parallel Lines and Transversals
Geometry Proofs
Formulas
-
Theorems
Angle-Side-Angle (ASA) Congruence Theorem
Alternate Interior Angles Theorem
Reflexive Property of Congruence
Suitable Grade Level
Grades 9-10