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Chapter 4

Chapter 4 . Congruent Triangles. 4.1 . Triangles and Angles. Parts of Triangles. Vertex Points joining the sides of a triangle Adjacent Sides Sides that share a common vertex. Classification by Sides. Equilateral 3 congruent sides Isosceles At least 2 congruent sides Scalene

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Chapter 4

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  1. Chapter 4 Congruent Triangles

  2. 4.1 Triangles and Angles

  3. Parts of Triangles • Vertex • Points joining the sides of a triangle • Adjacent Sides • Sides that share a common vertex

  4. Classification by Sides • Equilateral • 3 congruent sides • Isosceles • At least 2 congruent sides • Scalene • No congruent sides

  5. Classification by Angles • Acute • 3 acute angles • Equiangular • 3 congruent angles • Right • 1 right angle • Obtuse • 1 obtuse angle

  6. Parts of Isosceles Triangles • Legs • The sides that are congruent. • Base • The non-congruent side.

  7. Vertex angle legs Base angles Isosceles triangles • Base angles are congruent. Base

  8. Parts of Right Triangles • Hypotenuse • The side that is opposite the right angle. It is always the longest side. • Legs • The sides that form the right angle

  9. Right Triangles hypotenuse leg leg

  10. Interior Angles • The angles on the inside of a triangle.

  11. Triangle Sum Conjecture • The sum of the measures of the angles in every triangle is 180.

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

  13. Exterior Angles • The angles that are adjacent to the interior angles • The exterior angles always add to equal 360°

  14. Definitions Exterior Angle Adjacent Interior Angle Remote Interior Angles

  15. B b x a c A C Exterior Angles of a Triangle • Use your straightedge to draw a triangle. • Extend one side out as shown

  16. B b x a a c A b C Exterior Angles of a Triangle • Trace angles a and b onto a transparency so that they are adjacent. • How does this compare to angle x?

  17. B b x = a + b a c A C Triangle Exterior Angle Conjecture • The measure of the exterior angle of a triangle is equal to the sum of the measures of the remote interior angles

  18. Example Find the missing measures 80° 53°

  19. Example Find the missing measures 120° 60°

  20. (2x – 8)° x° 31° Example • Page 199 #37

  21. 4.2 Congruence and Triangles

  22. Terms • Congruent • Figures that are exactly the same size and shape are congruent • Corresponding angles • The angles that are in corresponding positions are congruent • Corresponding sides • The sides that are in corresponding positions are congruent

  23. Naming Congruent Figures • When a congruence statement is made it is important to match up corresponding parts.

  24. Third Angle Theorem • If two angles in one triangle are equal to two angles in another triangle, then the third angles in each triangle are also equal.

  25. Q P 45° R Examples 1 (page 205) What is the measure of: P M R N Which side is congruent to segment QR Segment LN • ΔLMN ΔPQR N 105° L M

  26. Example 2 • Given ABC  PQR, find the values of x and y. (6y – 4)° Q R A 85° (10x + 5)° P 50° C B

  27. 4.3 Proving Triangles Congruent SSS and SAS

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

  29. Definitions • included angle • An angle that is between two given sides. • included side • A side that is between two given angles.

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

  31. 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.

  32. 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.

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

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

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

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

  37. 4.4 Proving Triangles Congruent ASA and AAS

  38. 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.

  39. Triangle Congruence Shortcuts • SAA • 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.

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

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

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

  43. 4.6 Isosceles, Equilateral, and Right Triangles

  44. Warm-Up 1 Find the measure of each angle. 60° a 90° b 90° 30°

  45. Warm-Up 2 Find the measure of each angle. 110 90 150

  46. Isosceles triangles • The base angles of an isosceles triangle are congruent. • If a triangle has at least two congruent angles, then it is an isosceles triangle. • If the sides are congruent then the base angles are congruent.

  47. Example 1 35° x

  48. Example 2 b a 15°

  49. Example 3 Find each missing measure m n 10 cm 63° p

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