4 4 Proving Triangles are Congruent ASA and
![4. 4 Proving Triangles are Congruent: ASA and AAS 4. 4 Proving Triangles are Congruent: ASA and AAS](https://slidetodoc.com/presentation_image_h/1d98be367ebae288ce47380255959094/image-1.jpg)
![Objectives: 1. Prove that triangles are congruent using the ASA Congruence Postulate and the Objectives: 1. Prove that triangles are congruent using the ASA Congruence Postulate and the](https://slidetodoc.com/presentation_image_h/1d98be367ebae288ce47380255959094/image-2.jpg)
![Postulate 21: Angle-Side-Angle (ASA) Congruence Postulate • If two angles and the included side Postulate 21: Angle-Side-Angle (ASA) Congruence Postulate • If two angles and the included side](https://slidetodoc.com/presentation_image_h/1d98be367ebae288ce47380255959094/image-3.jpg)
![Theorem 4. 5: Angle-Side (AAS) Congruence Theorem • If two angles and a non-included Theorem 4. 5: Angle-Side (AAS) Congruence Theorem • If two angles and a non-included](https://slidetodoc.com/presentation_image_h/1d98be367ebae288ce47380255959094/image-4.jpg)
![Theorem 4. 5: Angle-Side (AAS) Congruence Theorem Given: A D, C F, BC EF Theorem 4. 5: Angle-Side (AAS) Congruence Theorem Given: A D, C F, BC EF](https://slidetodoc.com/presentation_image_h/1d98be367ebae288ce47380255959094/image-5.jpg)
![Theorem 4. 5: Angle-Side (AAS) Congruence Theorem You are given that two angles of Theorem 4. 5: Angle-Side (AAS) Congruence Theorem You are given that two angles of](https://slidetodoc.com/presentation_image_h/1d98be367ebae288ce47380255959094/image-6.jpg)
![Ex. 1 Developing Proof Is it possible to prove the triangles are congruent? If Ex. 1 Developing Proof Is it possible to prove the triangles are congruent? If](https://slidetodoc.com/presentation_image_h/1d98be367ebae288ce47380255959094/image-7.jpg)
![Ex. 1 Developing Proof A. In addition to the angles and segments that are Ex. 1 Developing Proof A. In addition to the angles and segments that are](https://slidetodoc.com/presentation_image_h/1d98be367ebae288ce47380255959094/image-8.jpg)
![Ex. 1 Developing Proof Is it possible to prove the triangles are congruent? If Ex. 1 Developing Proof Is it possible to prove the triangles are congruent? If](https://slidetodoc.com/presentation_image_h/1d98be367ebae288ce47380255959094/image-9.jpg)
![Ex. 1 Developing Proof B. In addition to the congruent segments that are marked, Ex. 1 Developing Proof B. In addition to the congruent segments that are marked,](https://slidetodoc.com/presentation_image_h/1d98be367ebae288ce47380255959094/image-10.jpg)
![Ex. 1 Developing Proof Is it possible to prove the triangles are congruent? If Ex. 1 Developing Proof Is it possible to prove the triangles are congruent? If](https://slidetodoc.com/presentation_image_h/1d98be367ebae288ce47380255959094/image-11.jpg)
![Ex. 1 Developing Proof The two pairs of parallel sides can be used to Ex. 1 Developing Proof The two pairs of parallel sides can be used to](https://slidetodoc.com/presentation_image_h/1d98be367ebae288ce47380255959094/image-12.jpg)
![Ex. 2 Proving Triangles are Congruent Given: AD ║EC, BD BC Prove: ∆ABD ∆EBC Ex. 2 Proving Triangles are Congruent Given: AD ║EC, BD BC Prove: ∆ABD ∆EBC](https://slidetodoc.com/presentation_image_h/1d98be367ebae288ce47380255959094/image-13.jpg)
![Proof: Statements: 1. BD BC 2. AD ║ EC 3. D C 4. ABD Proof: Statements: 1. BD BC 2. AD ║ EC 3. D C 4. ABD](https://slidetodoc.com/presentation_image_h/1d98be367ebae288ce47380255959094/image-14.jpg)
![Proof: Statements: 1. BD BC 2. AD ║ EC 3. D C 4. ABD Proof: Statements: 1. BD BC 2. AD ║ EC 3. D C 4. ABD](https://slidetodoc.com/presentation_image_h/1d98be367ebae288ce47380255959094/image-15.jpg)
![Proof: Statements: 1. BD BC 2. AD ║ EC 3. D C 4. ABD Proof: Statements: 1. BD BC 2. AD ║ EC 3. D C 4. ABD](https://slidetodoc.com/presentation_image_h/1d98be367ebae288ce47380255959094/image-16.jpg)
![Proof: Statements: 1. BD BC 2. AD ║ EC 3. D C 4. ABD Proof: Statements: 1. BD BC 2. AD ║ EC 3. D C 4. ABD](https://slidetodoc.com/presentation_image_h/1d98be367ebae288ce47380255959094/image-17.jpg)
![Proof: Statements: 1. BD BC 2. AD ║ EC 3. D C 4. ABD Proof: Statements: 1. BD BC 2. AD ║ EC 3. D C 4. ABD](https://slidetodoc.com/presentation_image_h/1d98be367ebae288ce47380255959094/image-18.jpg)
![Proof: Statements: 1. BD BC 2. AD ║ EC 3. D C 4. ABD Proof: Statements: 1. BD BC 2. AD ║ EC 3. D C 4. ABD](https://slidetodoc.com/presentation_image_h/1d98be367ebae288ce47380255959094/image-19.jpg)
![Note: • You can often use more than one method to prove a statement. Note: • You can often use more than one method to prove a statement.](https://slidetodoc.com/presentation_image_h/1d98be367ebae288ce47380255959094/image-20.jpg)
- Slides: 20
![4 4 Proving Triangles are Congruent ASA and AAS 4. 4 Proving Triangles are Congruent: ASA and AAS](https://slidetodoc.com/presentation_image_h/1d98be367ebae288ce47380255959094/image-1.jpg)
4. 4 Proving Triangles are Congruent: ASA and AAS
![Objectives 1 Prove that triangles are congruent using the ASA Congruence Postulate and the Objectives: 1. Prove that triangles are congruent using the ASA Congruence Postulate and the](https://slidetodoc.com/presentation_image_h/1d98be367ebae288ce47380255959094/image-2.jpg)
Objectives: 1. Prove that triangles are congruent using the ASA Congruence Postulate and the AAS Congruence Theorem 2. Use congruence postulates and theorems in real-life problems.
![Postulate 21 AngleSideAngle ASA Congruence Postulate If two angles and the included side Postulate 21: Angle-Side-Angle (ASA) Congruence Postulate • If two angles and the included side](https://slidetodoc.com/presentation_image_h/1d98be367ebae288ce47380255959094/image-3.jpg)
Postulate 21: Angle-Side-Angle (ASA) Congruence Postulate • If two angles and the included side of one triangle are congruent to two angles and the included side of a second triangle, then the triangles are congruent.
![Theorem 4 5 AngleSide AAS Congruence Theorem If two angles and a nonincluded Theorem 4. 5: Angle-Side (AAS) Congruence Theorem • If two angles and a non-included](https://slidetodoc.com/presentation_image_h/1d98be367ebae288ce47380255959094/image-4.jpg)
Theorem 4. 5: Angle-Side (AAS) Congruence Theorem • If two angles and a non-included side of one triangle are congruent to two angles and the corresponding nonincluded side of a second triangle, then the triangles are congruent.
![Theorem 4 5 AngleSide AAS Congruence Theorem Given A D C F BC EF Theorem 4. 5: Angle-Side (AAS) Congruence Theorem Given: A D, C F, BC EF](https://slidetodoc.com/presentation_image_h/1d98be367ebae288ce47380255959094/image-5.jpg)
Theorem 4. 5: Angle-Side (AAS) Congruence Theorem Given: A D, C F, BC EF Prove: ∆ABC ∆DEF
![Theorem 4 5 AngleSide AAS Congruence Theorem You are given that two angles of Theorem 4. 5: Angle-Side (AAS) Congruence Theorem You are given that two angles of](https://slidetodoc.com/presentation_image_h/1d98be367ebae288ce47380255959094/image-6.jpg)
Theorem 4. 5: Angle-Side (AAS) Congruence Theorem You are given that two angles of ∆ABC are congruent to two angles of ∆DEF. By the Third Angles Theorem, the third angles are also congruent. That is, B E. Notice that BC is the side included between B and C, and EF is the side included between E and F. You can apply the ASA Congruence Postulate to conclude that ∆ABC ∆DEF.
![Ex 1 Developing Proof Is it possible to prove the triangles are congruent If Ex. 1 Developing Proof Is it possible to prove the triangles are congruent? If](https://slidetodoc.com/presentation_image_h/1d98be367ebae288ce47380255959094/image-7.jpg)
Ex. 1 Developing Proof Is it possible to prove the triangles are congruent? If so, state the postulate or theorem you would use. Explain your reasoning.
![Ex 1 Developing Proof A In addition to the angles and segments that are Ex. 1 Developing Proof A. In addition to the angles and segments that are](https://slidetodoc.com/presentation_image_h/1d98be367ebae288ce47380255959094/image-8.jpg)
Ex. 1 Developing Proof A. In addition to the angles and segments that are marked, EGF JGH by the Vertical Angles Theorem. Two pairs of corresponding angles and one pair of corresponding sides are congruent. You can use the AAS Congruence Theorem to prove that ∆EFG ∆JHG.
![Ex 1 Developing Proof Is it possible to prove the triangles are congruent If Ex. 1 Developing Proof Is it possible to prove the triangles are congruent? If](https://slidetodoc.com/presentation_image_h/1d98be367ebae288ce47380255959094/image-9.jpg)
Ex. 1 Developing Proof Is it possible to prove the triangles are congruent? If so, state the postulate or theorem you would use. Explain your reasoning.
![Ex 1 Developing Proof B In addition to the congruent segments that are marked Ex. 1 Developing Proof B. In addition to the congruent segments that are marked,](https://slidetodoc.com/presentation_image_h/1d98be367ebae288ce47380255959094/image-10.jpg)
Ex. 1 Developing Proof B. In addition to the congruent segments that are marked, NP NP. Two pairs of corresponding sides are congruent. This is not enough information to prove the triangles are congruent.
![Ex 1 Developing Proof Is it possible to prove the triangles are congruent If Ex. 1 Developing Proof Is it possible to prove the triangles are congruent? If](https://slidetodoc.com/presentation_image_h/1d98be367ebae288ce47380255959094/image-11.jpg)
Ex. 1 Developing Proof Is it possible to prove the triangles are congruent? If so, state the postulate or theorem you would use. Explain your reasoning. UZ ║WX AND UW ║WX. 1 2 3 4
![Ex 1 Developing Proof The two pairs of parallel sides can be used to Ex. 1 Developing Proof The two pairs of parallel sides can be used to](https://slidetodoc.com/presentation_image_h/1d98be367ebae288ce47380255959094/image-12.jpg)
Ex. 1 Developing Proof The two pairs of parallel sides can be used to show 1 3 and 2 4. Because the included side WZ is congruent to itself, ∆WUZ ∆ZXW by the ASA Congruence Postulate. 1 2 3 4
![Ex 2 Proving Triangles are Congruent Given AD EC BD BC Prove ABD EBC Ex. 2 Proving Triangles are Congruent Given: AD ║EC, BD BC Prove: ∆ABD ∆EBC](https://slidetodoc.com/presentation_image_h/1d98be367ebae288ce47380255959094/image-13.jpg)
Ex. 2 Proving Triangles are Congruent Given: AD ║EC, BD BC Prove: ∆ABD ∆EBC Plan for proof: Notice that ABD and EBC are congruent. You are given that BD BC. Use the fact that AD ║EC to identify a pair of congruent angles.
![Proof Statements 1 BD BC 2 AD EC 3 D C 4 ABD Proof: Statements: 1. BD BC 2. AD ║ EC 3. D C 4. ABD](https://slidetodoc.com/presentation_image_h/1d98be367ebae288ce47380255959094/image-14.jpg)
Proof: Statements: 1. BD BC 2. AD ║ EC 3. D C 4. ABD EBC 5. ∆ABD ∆EBC Reasons: 1.
![Proof Statements 1 BD BC 2 AD EC 3 D C 4 ABD Proof: Statements: 1. BD BC 2. AD ║ EC 3. D C 4. ABD](https://slidetodoc.com/presentation_image_h/1d98be367ebae288ce47380255959094/image-15.jpg)
Proof: Statements: 1. BD BC 2. AD ║ EC 3. D C 4. ABD EBC 5. ∆ABD ∆EBC Reasons: 1. Given
![Proof Statements 1 BD BC 2 AD EC 3 D C 4 ABD Proof: Statements: 1. BD BC 2. AD ║ EC 3. D C 4. ABD](https://slidetodoc.com/presentation_image_h/1d98be367ebae288ce47380255959094/image-16.jpg)
Proof: Statements: 1. BD BC 2. AD ║ EC 3. D C 4. ABD EBC 5. ∆ABD ∆EBC Reasons: 1. Given 2. Given
![Proof Statements 1 BD BC 2 AD EC 3 D C 4 ABD Proof: Statements: 1. BD BC 2. AD ║ EC 3. D C 4. ABD](https://slidetodoc.com/presentation_image_h/1d98be367ebae288ce47380255959094/image-17.jpg)
Proof: Statements: 1. BD BC 2. AD ║ EC 3. D C 4. ABD EBC 5. ∆ABD ∆EBC Reasons: 1. Given 2. Given 3. Alternate Interior Angles
![Proof Statements 1 BD BC 2 AD EC 3 D C 4 ABD Proof: Statements: 1. BD BC 2. AD ║ EC 3. D C 4. ABD](https://slidetodoc.com/presentation_image_h/1d98be367ebae288ce47380255959094/image-18.jpg)
Proof: Statements: 1. BD BC 2. AD ║ EC 3. D C 4. ABD EBC 5. ∆ABD ∆EBC Reasons: 1. Given 2. Given 3. Alternate Interior Angles 4. Vertical Angles Theorem
![Proof Statements 1 BD BC 2 AD EC 3 D C 4 ABD Proof: Statements: 1. BD BC 2. AD ║ EC 3. D C 4. ABD](https://slidetodoc.com/presentation_image_h/1d98be367ebae288ce47380255959094/image-19.jpg)
Proof: Statements: 1. BD BC 2. AD ║ EC 3. D C 4. ABD EBC 5. ∆ABD ∆EBC Reasons: 1. Given 2. Given 3. Alternate Interior Angles 4. Vertical Angles Theorem 5. ASA Congruence Theorem
![Note You can often use more than one method to prove a statement Note: • You can often use more than one method to prove a statement.](https://slidetodoc.com/presentation_image_h/1d98be367ebae288ce47380255959094/image-20.jpg)
Note: • You can often use more than one method to prove a statement. In Example 2, you can use the parallel segments to show that D C and A E. Then you can use the AAS Congruence Theorem to prove that the triangles are congruent.
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