Leg Leg Congruence Theorem and LEg Acute angle Congruence Theorem

Proving Two Right Triangles are congruent (LL and LA Congruence Theorem)

1. Classroom Rules : 1. Listen to the teacher when she speaks . 2. Raise your hand before speaking . 3. Actively cooperate with others . 4. Work diligently and enjoy the learning process.

2. Motivation (Warm Up Activity) I am Me Instructions: Guess the term that will be form out from the given pictures.

3. + + hippopotamus tea nose HYPOTENUSE + +

4. + right angel + Right angle

5. + + cong grow hens + + CONGRUENCE

6. + eggs L + legs

7. As we go through the lesson today, you should say to yourself that : 1. I can state LL and LA congruence theorem . 2. I can illustrate LL and LA congruence theorem by putting markings . 3. I can apply these theorems to prove two right triangles are congruent .

8. Guide Questions : What kind of triangle shown below ? What kind of angle can be seen in this triangle ? What is the measure of the right angle ? Right triangle Right angle Exactly 90  

9. A right triangle is a triangle with one right angle . hypotenuse leg leg

10. Here are the parts of a right triangle:   a. hypotenuse – is the longest side and a side opposite the right angle. b. legs - perpendicular sides of a triangle .

11. Right Triangle Congruence Theorem

12. Leg – Leg (LL) Congruence Theorem - If the legs of one right triangle are congruent to the legs of another right triangle, then the triangles are congruent.

13. Example 1: Consider the right triangles FRI and DAY with right angles at R and A, respectively, such that , and . Prove: .   Y A F I R D

14. Guide Questions : 1. Are the two right triangles congruent? 2. What are your bases for identifying the congruent parts of the two right triangle? 3. Which sides of the right triangles are congruent to each other? 4. What congruence theorem did you use to prove ?  

15. Guide Questions : 1. Are the two right triangles congruent ? YES Y A F I R D

16. 2. What are your bases for identifying the congruent parts of the two right triangle? Y A F I R D By comparing the identical tick marks on those parts. The tick marks signifies that the corresponding sides of right triangles are congruent.

17. Y A F I R D 3. Which sides of the right triangles are congruent to each other?   4. What congruence theorem did you use to prove ?   The theorem use to prove is the LL Congruence Theorem.  

18. Leg - Acute angle (LA) Congruence Theorem - If a leg and an acute angle of one right triangle are congruent to a leg and an acute angle of another right triangle, then the triangles are congruent.

19. Example 2: Consider the right triangles TRY and LIE with right angles at R and I, respectively, such that , and Prove:   T   R   Y   L   E   I  

20. Guide Questions : 1. Are the two right triangles congruent? 2. What are your bases for identifying the congruent parts of the two right triangle? 3. Which side and angle of the right triangles are congruent to each other? 4. What congruence theorem did you use to prove ?   T   R   Y   L   E   I  

21. Guide Questions : 1. Are the two right triangles congruent ? YES T   R   Y   L   E   I  

22. 2. What are your bases for identifying the congruent parts of the two right triangle? By comparing the identical tick marks on those parts. The tick marks signifies that the corresponding sides of right triangles are congruent. T   R   Y   L   E   I  

23. 3. Which sides of the right triangles are congruent to each other?   4. What congruence theorem did you use to prove ?   The theorem use to prove is the LA Congruence Theorem.   T   R   Y   L   E   I  

24. Scenario: Teacher Cherrie helped her third-quarter art class participants create a doormat by employing right triangles with distinct styles in a cooperative fashion. The spirit of Central Asian artifacts and things served as inspiration for them as they incorporated components to create a cohesive sculpture. Arts 8 Integration MELC Arts 8: Incorporates the design, form, and spirit of South, West and Central Asian artifacts and objects to one’s creation (A8PL-lllh-3)

25. Sample of Doormat

26. In proving we use two-column proofs.   A two-column proof is one common way to organize a proof in geometry. Two columns: one for statements and one for reasons . It consists of a list of statements and the reasons why those statements are true. The statements are written in the left column and the reasons are written in the right column.

27. Situation 1 :   Arkin , an eighth-grade student suggested to apply one of the theorem they learned which is the LL congruence theorem, their lesson in Math 8, specifically recalling the concept of right triangles. As the group leader, he led a discussion to ensure everyone grasped the concept. Recognizing that one member had missed the Math lesson on LL congruence theorem, Arkin illustrated and explained the process of identifying corresponding sides of right triangles by drawing two congruent right triangles.

28. M D I S U P

29. 1. Given: Prove:   Statements Reasons Statements Reasons M D I S U P GIVEN GIVEN LL CONGRUENCE THEOREM

30. Situation 2: Erich , who was absent from math class, requested clarification on congruent right triangles specifically the other theorem which is LA Congruence theorem to ensure their project scored well. Arkin drew two right triangles and asked, “If we assume these are congruent based on given statements, how do we prove that this two right triangles are congruent.

31. F W P U C B

32. 2. Given: Prove:   Statements Reasons Statements Reasons F W P U C B GIVEN GIVEN LA CONGRUENCE THEOREM

33. Situation 3:   As the team talks about making a wall decor, they find some colorful pieces of cloth with patterns made of right triangles. Luna thinks they could use these patterns in their design. They decide to put the right triangles together like a puzzle to make interesting shapes and pictures on the décor. Working together, they use the right triangles to create a colorful and exciting wall décor that shows how cool right triangles can be!

34. EQUAL Rules (promote fairness, respect, and care ) 1. Encourage Respect : Listen actively when others are speaking and avoid interrupting Value diversity and different perspectives . 2. Quiet Voices: Maintain a noise level that allows everyone to focus and participate effectively. Keep conversations at an appropriate volume during activities . 3. Use Kind Words: Avoid using hurtful language or making negative comments. Offer constructive feedback and support your classmates . 4. Act Responsibly: Take ownership of your actions and their consequences. Follow classroom procedures and complete assignments on time. Respect classroom materials and property . 5 . Listen Attentively: Pay attention to instructions and directions provided by the teacher. Raise your hand to contribute and ask questions when needed .

35. Group Activity:   The class will be tasked with creating their own designs using cutouts of right triangles. Each group will be provided a set of right triangle cutouts in various colors. They will then have the opportunity to design a unique pattern or image using these triangles, showcasing their understanding of angles, side lengths, and geometric concepts.  

36. Criteria Exemplary (3) Proficient (2) Needs Improvement (1) Originality The pattern demonstrates exceptional creativity and originality, featuring innovative designs and unique concepts. The pattern shows some creativity and originality, with elements of personal style or interpretation evident. The pattern lacks originality, relying heavily on familiar designs or reproductions without adding personal insight. Captivating The pattern captivates the viewer with its engaging design, compelling composition, and visually striking elements. The pattern is visually appealing and holds the viewer's interest to some extent but may lack a wow factor or strong focal point. The pattern fails to captivate the viewer, with a bland or uninspired design that lacks visual interest. Neatness The pattern is impeccably neat, with clean lines, precise measurements, and meticulous attention to detail throughout. The pattern is generally neat, with some minor imperfections or inconsistencies in construction or presentation. The pattern is messy or sloppy, with noticeable errors, smudges, or untidy elements detracting from the overall appearance.

37. Guide Questions:   1. Did you find the activity enjoyable? 2. What congruence theorems can you use to prove that right triangles are congruent? 3. Can you differentiate LL Congruence Theorem from LA Congruence Theorem?

38. Individual Work:   Scenario : In Ms. Orpilla’s geometry class, student’s have been learning about right triangle theorems particularly the Leg – Leg congruence theorem and Leg – Acute angle congruence theorem. As a creative task, Ms. Orpilla decides to challenge her students to prove the following given.

39. 1. Given : Prove:   S S U N E T

40. Statements Reasons Statements Reasons GIVEN GIVEN LL CONGRUENCE THEOREM S S U N E T

41. 2. Given: Prove:   U T P U S H

42. Statements Reasons Statements Reasons U T P U S H GIVEN GIVEN LA CONGRUENCE THEOREM

43. What specific situation from your daily life allows you to see and apply the LL congruence theorem as well as LA congruence theorem ?

44. Complete Me! Direction: Complete the sentence given below.   If the 1 )___ of one right triangle are congruent to the legs of another 2 )____ triangle, then the triangles are 3 )_______.   If a leg and an 4 )___ angle of one right triangle are congruent to a 5 ) ___ and an acute angle of another right triangle, then the triangles are congruent.   leg acute right congruent leg

45. Evaluation: Directions: Read carefully each question and write the letter that corresponds to the correct answer in the space provided before each number. Do it within 5 minutes only.   1. Which of the following is true about LL Congruence Theorem? A. If a leg and an acute angle of one right triangle are congruent respectively to a leg and an acute angle of another triangle, then the triangles are congruent. B. If the hypotenuse and a leg of one right triangle are congruent respectively to the hypotenuse and a leg of another right triangle, then the triangles are congruent. C. If the legs of one right triangle are congruent to the legs of another right triangle, then the triangles are congruent. D. If the hypotenuse and an acute angle of one right triangle are congruent respectively to the hypotenuse and an acute angle of another right triangle, then the triangles are congruent.

46. C

47. 2. Which of the following illustration show LL Congruence Theorem ? A. B. C. D.

48. A

49. 3. Which of the following is true about LA Congruence Theorem? A. If a leg and an acute angle of one right triangle are congruent respectively to a leg and an acute angle of another triangle, then the triangles are congruent. B. If the hypotenuse and a leg of one right triangle are congruent respectively to the hypotenuse and a leg of another right triangle, then the triangles are congruent. C. If the legs of one right triangle are congruent to the legs of another right triangle, then the triangles are congruent. D. If the hypotenuse and an acute angle of one right triangle are congruent respectively to the hypotenuse and an acute angle of another right triangle, then the triangles are congruent.

50. A

51. 4. Which of the following illustration show LA Congruence Theorem ? A. B. C. D.

52. B

53. P X Y Z R Q 5. Given and , which theorem proves that are congruent right triangles? A. HyL Congruence Theorem B. LA Congruence Theorem C. HyA Congruence Theorem D. LL Congruence Theorem  

54. D

55. Performance Task : Tessellation: Exploring LL and LA Congruence Theorem with Right Triangles Objective: To explore tessellation using right triangles and understand the application of the LL and LA Congruence Theorem in creating tessellations . Directions: Create your own tessellation using LL and LA Congruence Theorem with Right Triangles . Note: Tessellation – is the process of covering a surface with repeating patterns without any gaps or overlaps.

56. Scoring Rubric: Criteria 1-2 3-4 5 Creativity No evidence of creativity or effort in the design. Minimal attempt at creating a tessellation with some basic patterns. Demonstrates moderate creativity with interesting patterns and shapes. Shows exceptional creativity with innovative designs and intricate patterns. Application of LL and LA Congruence Theorem   No evidence of applying LL and LA Congruence Theorem. Limited application of the theorems, resulting in incorrect or inconsistent tessellation. Adequate application of the theorems, resulting in mostly correct tessellation with some minor inconsistencies. Thorough application of the theorems, resulting in a flawless tessellation with correct angles and side lengths. Completeness Tessellation is incomplete or significantly lacking in detail. Tessellation is partially complete but lacks refinement or coherence. Tessellation is mostly complete but may have some minor gaps or overlaps. Tessellation is fully complete with no gaps or overlaps, demonstrating attention to detail.