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Triangles and Congruence

Triangles and Congruence. § 5.1 Classifying Triangles. § 5.2 Angles of a Triangle. § 5.3 Geometry in Motion. § 5.4 Congruent Triangles. § 5.5 SSS and SAS. § 5.6 ASA and AAS. D. E. F. Classifying Triangles. What You'll Learn.

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Triangles and Congruence

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  1. Triangles and Congruence • § 5.1 Classifying Triangles • § 5.2 Angles of a Triangle • § 5.3 Geometry in Motion • § 5.4 Congruent Triangles • § 5.5 SSS and SAS • § 5.6 ASA and AAS

  2. D E F Classifying Triangles What You'll Learn You will learn to identify the parts of triangles and to classify triangles by their parts. In geometry, a triangle is a figure formed when _____ noncollinear points are connected by segments. three Each pair of segments forms an angle of the triangle. The vertex of each angle is a vertex of the triangle.

  3. vertex EF, FD, and DE. angle D side E F Classifying Triangles Triangles are named by the letters at their vertices. ΔDEF Triangle DEF, written ______, is shown below. The sides are: D, E, and F. The vertices are: The angles are: E, F, and D. In Chapter 3, you classified angles as acute, obtuse, or right. Triangles can also be classified by their angles. acute All triangles have at least two _____ angles. The third angle is either _____, ______, or _____. right obtuse acute

  4. 30° 17° 60° 80° 43° 60° 120° 40° Classifying Triangles acute triangle obtuse triangle right triangle 3rd angle is _____ 3rd angle is ______ 3rd angle is ____ acute obtuse right

  5. Classifying Triangles scalene isosceles equilateral no all at least two ___ sides congruent __________ sides congruent ___ sides congruent

  6. The angle formed by the congruent sides is called the ___________. Classifying Triangles vertex angle The two angles formed by the base and one of the congruent sides are called ___________. The congruent sides are called legs. base angles leg leg The side opposite the vertex angle is called the _____. base

  7. End of Section 5.1

  8. Angles of a Triangle What You'll Learn You will learn to use the Angle Sum Theorem. 1) On a piece of paper, draw a triangle. 2) Place a dot close to the center (interior) of the triangle. 3) After marking all of the angles, tear the triangle into three pieces. then rotate them, laying the marked angles next to each other. 4) Make a conjecture about the sum of the angle measures of the triangle.

  9. x° y° z° Angles of a Triangle The sum of the measures of the angles of a triangle is 180. x + y + z = 180

  10. x° y° Angles of a Triangle The acute angles of a right triangle are complementary. x + y = 90

  11. x° x° x° Angles of a Triangle The measure of each angle of an equiangular triangle is 60. 3x = 180 x = 60

  12. End of Section 5.2

  13. Geometry in Motion What You'll Learn You will learn to identify translations, reflections, and rotations and their corresponding parts. We live in a world of motion. Geometry helps us define and describe that motion. In geometry, there are three fundamental types of motion: __________, _________, and ________. reflection translation rotation

  14. Geometry in Motion Translation In a translation, you slide a figure from one position to another without turning it. Translations are sometimes called ______. slides

  15. line of reflection Geometry in Motion Reflection In a reflection, you flip a figure over a line. The new figure is a mirror image. flips Reflections are sometimes called ____.

  16. Geometry in Motion Rotation In a rotation, you rotate a figure around a fixed point. turns Rotations are sometimes called _____. 30°

  17. A D B E C F Geometry in Motion Its matching point on the corresponding figure is called its ______. Each point on the original figure is called a _________. preimage image Each point on the preimage can be paired with exactly one point on its image, and each point on the image can be paired with exactly one point on its preimage. mapping This one-to-one correspondence is an example of a _______.

  18. A D B E C F Geometry in Motion Its matching point on the corresponding figure is called its ______. Each point on the original figure is called a _________. preimage image The symbol → is used to indicate a mapping. In the figure, ΔABC→ΔDEF. (ΔABC maps to ΔDEF). In naming the triangles, the order of the vertices indicates the corresponding points.

  19. A D B E C F AB DE BC EF CA FD Geometry in Motion Its matching point on the corresponding figure is called its ______. Each point on the original figure is called a _________. preimage image Image Image Preimage Preimage → A D → → B E → → C F → This mapping is called a _____________. transformation

  20. Geometry in Motion Translations, reflections, and rotations are all __________. isometries An isometry is a movement that does not change the size or shape of the figure being moved. When a figure is translated, reflected, or rotated, the lengths of the sides of the figure DO NOT CHANGE.

  21. End of Section 5.3

  22. The order of the ________ indicates the corresponding parts! ΔABC ΔXYZ Congruent Triangles What You'll Learn You will learn to identify corresponding parts of congruent triangles If a triangle can be translated, rotated, or reflected onto another triangle, so that all of the vertices correspond, the triangles are _________________. congruent triangles The parts of congruent triangles that “match” are called__________________. corresponding parts vertices

  23. DE FE FD AB  BC  AC  F E D Congruent Triangles In the figure, ΔABC  ΔFDE. A vertices As in a mapping, the order of the _______ indicates the corresponding parts. C B Congruent Sides Congruent Angles A  F B  D C  E These relationships help define the congruent triangles.

  24. Congruent Triangles corresponding parts If the _________________ of two triangles are congruent, then the two triangles are congruent. congruent If two triangles are _________, then the corresponding parts of the two triangles are congruent.

  25. S Z 50° 40° R (2n + 10)° 90° Y X T Congruent Triangles ΔRST ΔXYZ. Find the value of n. ΔRST ΔXYZ identify the corresponding parts corresponding parts are congruent S  Y subtract 10 from both sides 50= 2n + 10 40= 2n divide both sides by 2 20= n

  26. End of Section 5.4

  27. SSS and SAS What You'll Learn You will learn to use the SSS and SAS tests for congruency.

  28. B A C 2) Construct a segment congruent to AC. Label the endpoints of the segment D and E. 3) Construct a segment congruent to AB. 4) Construct a segment congruent to CB. F 6) Draw DF and EF. D E SSS and SAS 5) Label the intersection F. 1) Draw an acute scalene triangle on a piece of paper. Label its vertices A, B, and C, on the interior of each angle. This activity suggests the following postulate.

  29. S B ST RS and RT and If AC  T C R A BC  AB  SSS and SAS three sides corresponding then ΔABC ΔRST

  30. In two triangles, ZY  FE, XY  DE, and XZ  DF. X D Y E Z F SSS and SAS Write a congruence statement for the two triangles. Sample Answer: ΔZXY ΔFDE

  31. C is the included angle of CA and CB C A B B is the included angle of BA and BC A is the included angle of AB and AC SSS and SAS In a triangle, the angle formed by two given sides is called the ____________ of the sides. included angle Using the SSS Postulate, you can show that two triangles are congruent if theircorresponding sides are congruent. You can also show their congruence by using two sides and the ____________. included angle

  32. S B RS RT and If AC  T C R A AB  SSS and SAS included angle two sides A  R and then ΔABC ΔRST

  33. Q D R F E P D is not the included angle for DF and EF. SSS and SAS On a piece of paper, write your response to the following: Determine whether the triangles are congruent by SAS. • If so, write a statement of congruence and tell why they are congruent. • If not, explain your reasoning. NO!

  34. End of Section 5.5

  35. ASA and AAS What You'll Learn You will learn to use the ASA and AAS tests for congruency.

  36. C AC is the included side of A and C CB is the included side of C and B AB is the included side of A and B A B ASA and AAS The side of a triangle that falls between two given angles is called the___________ of the angles. included side It is the one side common to both angles. You can show that two triangles are congruent by using _________ and the ___________ of the triangles. two angles included side

  37. S B RT and T C R A AC  ASA and AAS included side two angles C  T If A  R and then ΔABC ΔRST

  38. CA and CB are the nonincluded sides of A and B C A B ASA and AAS You can show that two triangles are congruent by using _________ and a ______________. two angles nonincluded side

  39. S B TS T C R A CB  ASA and AAS nonincluded side two angles C  T and If A  R and then ΔABC ΔRST

  40. L D If F and M are marked congruent, then FE and MN would be included sides. M F N E ASA and AAS ΔDEF and ΔLNM have one pair of sides and one pair of angles marked to show congruence. What other pair of angles must be marked so that the two triangles are congruent by AAS? However, AAS requires the nonincluded sides. Therefore, D and L must be marked.

  41. End of Section 5.6

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