Triangle Congruence by ASA & AAS

This page introduces two important concepts in triangle congruence: the Angle-Side-Angle (ASA) Postulate and the Angle-Angle-Side (AAS) Theorem. These are fundamental tools for proving triangles congruent.

Definition: The ASA Postulate states that if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent.

Definition: The AAS Theorem states that if two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of another triangle, then the triangles are congruent.

The page provides visual representations of both ASA and AAS congruence, showing the corresponding parts that need to be congruent for each theorem to apply.

Example: An example is given asking students to identify which two triangles are congruent by ASA, emphasizing the importance of the included side in ASA congruence.

Highlight: Understanding the difference between included and non-included sides is crucial for correctly applying ASA and AAS theorems.

Writing a Proof Using ASA

This page focuses on applying the ASA theorem in practical scenarios and writing formal proofs. It presents a real-world example involving the design of a miniature golf course.

Example: A teen organization is building a miniature golf course with triangular bumpers. Students are asked to prove that the bumpers meet the design conditions using the ASA theorem.

The proof is structured step-by-step, demonstrating how to use given information to establish congruence between two triangles. This example shows the practical application of geometric theorems in real-world design problems.

Highlight: The proof emphasizes the importance of identifying the given information, stating what needs to be proved, and logically applying the ASA theorem to reach the conclusion.

The page also includes a "Got It?" section, providing another example for students to practice applying the ASA theorem in a proof.

Vocabulary: Right angles are introduced in the proof, reminding students of the importance of recognizing special angle relationships in geometric proofs.

Determining Whether Triangles Are Congruent

This page focuses on applying the knowledge of triangle congruence theorems to determine whether given triangles are congruent. It emphasizes critical thinking and the ability to justify answers.

Example: A diagram is presented with a question asking whether two triangles are congruent. Students are required to analyze the given information and select the best justification for their answer.

The example highlights the importance of understanding corresponding sides in triangle congruence, demonstrating that congruence cannot be established if the sides do not correspond correctly.

Highlight: This section emphasizes the importance of not just identifying congruence but also explaining why triangles are or are not congruent.

The page includes a "Got It?" section with more complex examples, requiring students to determine congruence and justify their answers using the appropriate theorem or postulate.

Vocabulary: Terms like "midpoint" and "vertical angles" are introduced, showing how these concepts can be used in conjunction with congruence theorems.