 § 5.1 Classifying Triangles Classifying TrianglesClassifying Triangles  § 5.4 Congruent Triangles Congruent TrianglesCongruent Triangles  § 5.3 Geometry in Motion Geometry in MotionGeometry in Motion  § 5.2 Angles of a Triangle Angles of a TriangleAngles of a Triangle  § 5.6 ASA and AAS ASA and AASASA and AAS  § 5.5 SSS and SAS SSS and SASSSS and SAS

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. E D F The vertex of each angle is a vertex of the triangle.

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

Triangles Classified by Angles 60° 80° 40° acute triangle 3 rd angle is _____ acute obtuse triangle right triangle 3 rd angle is ______ obtuse 3 rd angle is ____ right 17° 43° 120° 30° 60°

Triangles Classified by Sides scaleneisoscelesequilateral no ___ sides congruent __________ sides congruent ___ sides congruent at least two all

leg The side opposite the vertex angle is called the _____. The congruent sides are called legs. base leg 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

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.

Theorem 5-1 Angle Sum Theorem The sum of the measures of the angles of a triangle is 180. z° x° y° x + y + z = 180

Theorem 5-2 The acute angles of a right triangle are complementary. x + y = 90 x° y°

Theorem 5-3 The measure of each angle of an equiangular triangle is 60. 3x = 180 x° x = 60

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 ________. translation reflection rotation

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

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

30° In a rotation, you rotate a figure around a fixed point. Rotations are sometimes called _____. turns

Each point on the original figure is called a _________. preimage Its matching point on the corresponding figure is called its ______. image A B C D E 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. This one-to-one correspondence is an example of a _______. mapping

Each point on the original figure is called a _________. preimage Its matching point on the corresponding figure is called its ______. image A B C D E F 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.

Each point on the original figure is called a _________. preimage Its matching point on the corresponding figure is called its ______. image A B C D E F → Preimage A B C Image D E F → → → Preimage Image → → ABDE BCEF CAFD This mapping is called a _____________. transformation

When a figure is translated, reflected, or rotated, the lengths of the sides of the figure DO NOT CHANGE. 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.

The order of the ________ indicates the corresponding parts! ΔABC  ΔXYZ 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

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

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

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

You will learn to use the SSS and SAS tests for congruency.

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

Postulate 5-1 SSS Postulate If three _____ of one triangle are congruent to _____ _____________ sides of another triangle, then the two Triangles are congruent. sides three corresponding A B C R S T If AC  RT and AB  RS andBC  ST then ΔABC  ΔRST

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

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

Postulate 5-2 SAS Postulate If ________ and the ____________ of one triangle are congruent to the corresponding sides and included angle of another triangle, then the triangles are congruent. two sides included angle A B C R S T If AC  RT and  A   R and AB  RS then ΔABC  ΔRST

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. On a piece of paper, write your response to the following: P R Q F E D NO!  D is not the included angle for DF and EF.

You will learn to use the ASA and AAS tests for congruency.

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. A B 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 You can show that two triangles are congruent by using _________ and the ___________ of the triangles. two angles included side

R S T A B C Postulate 5-3 ASA Postulate If _________ and the ___________ of one triangle are congruent to the corresponding angles and included side of another triangle, then the triangles are congruent. two angles included side If  A   R and AC  RT and then ΔABC  ΔRST  C  TT

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

R S T A B C Theorem 5-4 AAS Theorem If _________ and a ______________ of one triangle are congruent to the corresponding two angles and nonincluded side of another triangle, then the triangles are congruent. two angles nonincluded side If  A   R and CB  TS then ΔABC  ΔRST  C   T and

D F E L M N 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. If  F and  M are marked congruent, then FE and MN would be included sides.