1 / 108

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

rob
Download Presentation

Chapter 4

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related