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

5-Minute Check

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 E D F Each pair of segments forms an angle of the triangle. The vertex of each angle is a vertex of the triangle. 1 - 7

Classifying Triangles Triangles are named by the letters at their vertices. Triangle DEF, written ______, is shown below. ΔDEF vertex The sides are: EF, FD, and DE. E D F angle The vertices are: D, E, and F. The angles are: E, F, and D. side 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 _____. acute obtuse right

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

Classifying Triangles Classified by Sides scalene isosceles equilateral ___ sides congruent no __________ sides congruent at least two ___ sides congruent all 8 - 17

What You'll Learn You will learn to use the Angle Sum Theorem. 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. 1 - 7

The sum of the measures of the angles of a triangle is 180. 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

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

The measure of each angle of an equiangular triangle is 60. Angles of a Triangle Theorem 5-3 The measure of each angle of an equiangular triangle is 60. x° 3x = 180 x = 60 8 - 12

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 ________. translation reflection rotation 1 - 5

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

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

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

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

The symbol → is used to indicate a mapping. Geometry in Motion A B C D E F Each point on the original figure is called a _________. Its matching point on the corresponding figure is called its ______. 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.

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

Translations, reflections, and rotations are all __________. 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. 8 - 11

ΔABC  ΔXYZ What You'll Learn 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 ______________. congruent parts The order of the ________ indicates the corresponding parts! ΔABC  ΔXYZ vertices

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

If the _________________ of two triangles are congruent, then 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 ΔRST  ΔXYZ. Find the value of n. Congruent Triangles ΔRST  ΔXYZ. Find the value of n. T S R Z X Y 40° (2n + 10)° 50° 90° ΔRST  ΔXYZ identify the corresponding parts corresponding parts are congruent S  Y 50 = 2n + 10 subtract 10 from both sides 1 - 28 40 = 2n divide both sides by 2 20 = n

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

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

Triangles are congruent. sides three corresponding SSS and SAS 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 and BC  ST then ΔABC  ΔRST

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

In a triangle, the angle formed by two given sides is called the SSS and SAS In a triangle, the angle formed by two given sides is called the ____________ of the sides. included angle C is the included angle of CA and CB A B C A is the included angle of AB and AC 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 by using two sides and the ____________. included angle

If ________ and the ____________ of one triangle are SSS and SAS 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

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

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

It is the one side common to both angles. 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. 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

If _________ and the ___________ of one triangle are ASA and AAS 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 A B C R S T If A  R and AC  RT and C  T then ΔABC  ΔRST

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

If _________ and a ______________ of one triangle are ASA and AAS 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 A B C R S T If A  R and C  T and CB  TS then ΔABC  ΔRST

ΔDEF and ΔLNM have one pair of sides and one pair of angles marked to 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 F E L M N

ASA and AAS