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Transcript
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 Classifying Triangles 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 _____ three noncollinear points are connected by segments. D E F Each pair of segments forms an angle of the triangle. The vertex of each angle is a vertex of the triangle. Classifying Triangles Triangles are named by the letters at their vertices. ΔDEF is shown below. Triangle DEF, written ______, vertex angle D The sides are: EF, FD, and DE. The vertices are: D, E, and F. E side F 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 angles. All triangles have at least two _____ right obtuse or _____. The third angle is either _____, acute ______, Classifying Triangles acute triangle 60° Triangles Classified by Angles obtuse right triangle 120° triangle 17° 30° 80° 60° 43° 40° 3rd angle is 3rd angle is 3rd angle is acute _____ obtuse ______ right ____ Classifying Triangles scalene isosceles equilateral no ___ sides congruent at least two __________ sides congruent all ___ sides congruent Triangles Classified by Sides Classifying Triangles The angle formed by the congruent sides is called the vertex angle ___________. The two angles formed by the base and one of the congruent sides are called base angles ___________. The congruent sides are called legs. leg leg The side opposite the vertex angle is called the _____. base Angles of a Triangle 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. Angles of a Triangle The sum of the measures of the angles of a triangle is 180. x° Theorem 5-1 Angle Sum Theorem x + y + z = 180 y° z° Angles of a Triangle The acute angles of a right triangle are complementary. Theorem 5-2 x° x + y = 90 y° Angles of a Triangle The measure of each angle of an equiangular triangle is 60. x° 3x = 180 Theorem 5-3 x = 60 x° x° Geometry in Motion 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: reflection and ________. translation _________, rotation __________, Geometry in Motion In a translation, you slide a figure from one position to another without turning it. Translations are sometimes called ______. slides Geometry in Motion line of reflection In a reflection, you flip a figure over a line. The new figure is a mirror image. flips Reflections are sometimes called ____. Geometry in Motion In a rotation, you rotate a figure around a fixed point. turns Rotations are sometimes called _____. 30° Geometry in Motion A Each point on the original figure is called preimage a _________. D B C E Its matching point on the corresponding figure is called image its ______. F 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 _______. Geometry in Motion A Each point on the original figure is called preimage a _________. B C The symbol → D E Its matching point on the corresponding figure is called image its ______. F 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. Geometry in Motion A Each point on the original figure is called preimage a _________. D B C E F Image Preimage D AB B → → E C → F Preimage A Its matching point on the corresponding figure is called image its ______. Image DE BC → → CA → FD This mapping is called a _____________. transformation EF Geometry in Motion isometries Translations, reflections, and rotations are all __________. 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. Congruent Triangles You will learn to identify corresponding parts of congruent triangles If a triangle can be translated, rotated, or reflected onto another triangle, so congruent triangles that all of the vertices correspond, the triangles are _________________. The parts of congruent triangles that “match” are called corresponding parts __________________. vertices indicates the corresponding parts! The order of the ________ ΔABC ΔXYZ Congruent Triangles In the figure, ΔABC ΔFDE. A As in a mapping, the order of the vertices _______ indicates the corresponding parts. C Congruent Angles B Congruent Sides A F AB FD B D BC DE C E AC FE F E These relationships help define the congruent triangles. D Congruent Triangles Definition of Congruent Triangles corresponding parts of two triangles are congruent, then If the _________________ the two triangles are congruent. congruent then the corresponding parts If two triangles are _________, of the two triangles are congruent. Congruent Triangles ΔRST ΔXYZ. Find the value of n. S Z 50° R 40° 90° T ΔRST ΔXYZ S Y Y identify the corresponding parts corresponding parts are congruent 50 = 2n + 10 subtract 10 from both sides 40 = 2n divide both sides by 2 20 = n (2n + 10)° X SSS and SAS You will learn to use the SSS and SAS tests for congruency. SSS and SAS 4) CB. 5) theacute intersection F.triangle on 3) Label AB. 1) Draw an scalene a piece of paper. Label its 6) DF and EF. congruent 2) Construct a segment to AC. Label the endpoints of vertices the A, B, and C, on the segment D and E. interior of each angle. B A C F D E This activity suggests the following postulate. SSS and SAS three sides of one triangle are congruent to _____ If three _____ corresponding sides of another triangle, then the two _____________ Triangles are congruent. S B Postulate 5-1 SSS Postulate A C If AC RT and AB RS and then ΔABC ΔRST T R BC ST SSS and SAS In two triangles, ZY FE, XY DE, and XZ DF. Write a congruence statement for the two triangles. D X Z Y Sample Answer: ΔZXY ΔFDE F E SSS and SAS In a triangle, the angle formed by two given sides is called the included angle of the sides. ____________ C is the included angle of CA and CB C A A is the included angle of AB and AC B B is the included angle of BA and BC Using the SSS Postulate, you can show that two triangles are congruent if their corresponding sides are congruent. You can also show their congruence included angle by using two sides and the ____________. SSS and SAS included angle of one triangle are sides and the ____________ If two ________ congruent to the corresponding sides and included angle of another triangle, then the triangles are congruent. S B Postulate 5-2 SAS Postulate A C If AC RT and A R and then ΔABC ΔRST T R AB RS 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. Q D R P F NO! D is not the included angle for DF and EF. E ASA and AAS You will learn to use the ASA and AAS tests for congruency. ASA and AAS The side of a triangle that falls between two given angles is called the included side of the angles. It is the one side common to both angles. ___________ C AC is the included side of A and C CB is the included side of C and B A B AB is the included side of A and B You can show that two triangles are congruent by using _________ two angles and the included side of the triangles. ___________ ASA and AAS included side of one triangle are angles and the ___________ If two _________ congruent to the corresponding angles and included side of another triangle, then the triangles are congruent. S B Postulate 5-3 ASA Postulate C A If A R and AC RT and then ΔABC ΔRST T R C T ASA and AAS CA and CB are the nonincluded sides of A and B C A B You can show that two triangles are congruent by using _________ two angles and a nonincluded side ______________. ASA and AAS nonincluded side of one triangle are angles and a ______________ If two _________ congruent to the corresponding two angles and nonincluded side of another triangle, then the triangles are congruent. S B Theorem 5-4 AAS Theorem C A If A R and C T then ΔABC ΔRST R and CB TS T 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? If F and M are marked congruent, then FE and MN would be included sides. However, AAS requires the nonincluded sides. Therefore, D and L must be marked. D L M F E N