1. Geometry 1 Unit 4Congruent Triangles Casa Grande Union High School Fall 2008

2. Geometry 1Unit 4 4.1 Triangles and Angles

3. Equilateral- all 3 sides are congruent Isosceles- at least 2 sides are congruent Scalene- No sides congruent Classifying Triangles by Sides

4. Classifying Triangles by Angles Acute Triangle- All angles are less than 90° Obtuse Triangle- 1 angle greater than 90 ° Right Triangle- 1 angle measuring 90°

5. Classifying Triangles • Example 1 • Name each triangle by its sides and angles A. B. C.

6. A C B Parts of triangles • Vertex (plural vertices) • The points joining the sides of a triangle • Adjacent sides • Sides sharing a common vertex • Side AB is adjacent to side BC

7. Exterior angle A Interior angle C B • Interior angle • Angle on the inside of a triangle • Exterior angle • Angle outside the triangle that is formed by extending one side

8. Triangle Sum Theorem • The sum of the three interior angles of a triangle is 180º

9. B 1 A C • Exterior Angle Theorem • The measure of the exterior angle of a triangle is equal to the sum of the two nonadjacent interior angles. • Example: m∠1=m∠A+ m∠B

10. Example 2 Find the measure of each angle. 2x + 10 x x + 2

11. B A C D Example 3 Given that ∠ A is 50º and ∠B is 34º, what is the measure of ∠BCD? What is the measure of ∠ACB?

12. Hypotenuse Legs Right triangle vocabulary • Legs • Sides that form the right angle • Hypotenuse • Side opposite the right angle

13. Corollary to the Triangle Sum Theorem • The acute angles of a right triangle are complementary. m∠A+ m∠B = 90°

14. 13 12 5 Example 4 A. Given the following triangle, what is the length of the hypotenuse? B. What are the length of the legs? C. If one of the acute angle measures is 32°, what is the other acute angle’s measurement?

15. Legs • The two congruent sides of an isosceles triangle. • Base • The noncongruent side of an isosceles triangle. • Base Angles • The two angles that contain the base of an isosceles triangle. • Vertex Angle • The noncongruent angle in an isosceles triangle.

16. Vertex Angle Legs Base Angles Isosceles triangle vocabulary Base

17. A 15 75º B C 7 Example 5 A. Given the following isosceles triangle, what is the measurement of segment AC? B. What is the measurement of angle A?

18. 80° 53° Example 6 Find the missing measures

19. Example 7 Given: ∆ABC with mC = 90° Prove: mA + mB = 90°

20. Geometry 1Unit 4 4.2 Congruence and Triangles

21. Congruent Figures • Congruent Figures • Figures are congruent if corresponding sides and angles have equal measures. • Corresponding Angles of Congruent Figures • When two figures are congruent, the angles that are in corresponding positions are congruent. • Corresponding Sides of Congruent Figures • When two figures are congruent, the sides that are in corresponding positions are congruent.

22. B Q A P C R Congruent Figures • For the triangles below, ∆ABC ≅ ∆PQR • The notation shows congruence and correspondence. • When writing congruence statements, be sure to list corresponding angles in the same order.

23. Example 1 S E V D F T Complete the congruence statement for the two given triangles: DEF What side corresponds with DE? What angle corresponds with E?

24. L K A B 9 cm 91° (4x – 3) cm (5y – 12)° 86° 113° H D 6 cm C J Example 2 In the diagram, ABCD ≅ KJHL • Find the value of x. • Find the value of y.

25. Third Angles Theorem • If 2 angles of 1 triangle are congruent to 2 angles of another triangle, then the third angles are also congruent.

26. (6y – 4)° Q R A 85° (10x + 5)° P 50° C B Example 3 Given ABC  PQR, find the values of x and y.

27. H E 58° G 58° F J Example 4 • Decide whether the triangles are congruent. Justify your answer.

28. M N O P Q Example 5 Given: MN ≅ QP, MN || PQ, O is the midpoint of MQ and PN. Prove: ∆MNO ≅ ∆QPO

29. Geometry 1Unit 4 4.3 Proving Triangles are Congruent: SSS and SAS

30. B A I T R G Warm-Up Complete the following statement BIG 

31. Definitions • Included Angle • An angle that is between two given sides. • Included Side • A side that is between two given angles.

32. L J K P Example 1 • Use the diagram. Name the included angle between the pair of given sides.

33. Triangle Congruence Shortcut • SSS • If the three sides of one triangle are congruent to the three sides of another triangle, then the triangles are congruent.

34. Triangle Congruence Shortcuts • SAS • If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent.

35. U S T V STW   W Example 2 • Complete the congruence statement. • Name the congruence shortcut used.

36. H L M I N J HIJ  LMN Example 3 • Determine if the following are congruent. • Name the congruence shortcut used.

37. A B C O R X XBO   Example 4 • Complete the congruence statement. • Name the congruence shortcut used.

38. P T S Q Example 5 • Complete the congruence statement. • Name the congruence shortcut used. SPQ  

39. Constructing Congruent Triangles C A B • Construct segment DE as a segment congruent to AB • Open your compass to the length of AC. Place the point of your compass on point D and strike an arc. • Open the compass to the width of BC. Place the point of your compass on E and strike an arc. Label the point where the arcs intersect as F.

40. M A B P Example 6 Given: AB ≅ PB, MB ⊥ AP Prove: ∆MBA ≅ ∆MBP

41. Example 7 Use SSS to show that ∆NPM ≅∆DFE N(-5, 1) P(-1, 6) M(-1, 1) D(6, 1) F(2, 6) E(2, 1)

42. Geometry 1Unit 4 4.4 Proving Triangles are Congruent: ASA and AAS

43. Triangle Congruence Shortcuts • ASA • If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent.

44. Triangle Congruence Shortcuts • AAS • If two angles and a non-included side of one triangle are congruent to the corresponding angles and side of another triangle, then the two triangles are congruent.

45. Q U QUA   D A Example 1 • Complete the congruence statement. • Name the congruence shortcut used.

46. R M N P Q RMQ   Example 2 • Complete the congruence statement. • Name the congruence shortcut used.

47. F B E A C D ABC  FED Example 3 • Determine if the following are congruent. • Name the congruence shortcut used.

49. C B F D M Example 4 Given: B ≅C, D ≅F; M is the midpoint of DF. Prove: ∆BDM ≅∆CFM

50. Geometry 1Unit 4 4.5 Using Congruent Triangles