Warm-Up: Perspective drawing To draw parallel lines use perspective drawing. Draw your own perspective drawing to describe your real world .

4.1 Congruence and Transformations Objective: Draw, identify, and describe transformations in the coordinate plane. Use the properties of rigid motions to determine whether figures are congruent and prove figures congruent.

Discover Transformations Graph each set of points, color code each set. Determine the transformation: Translation, Dilation, Rotation or Reflection Analyze any patterns Examine if the shapes are congruent. Explain.

Practice: P. 219 # 1, 2, 3- 6, 14-18 even. Determine whether the polygons are congruent. Support your answer by describing a transformation.

Discuss & Draw Triangles 1. Draw 3 real world examples of triangles. A. B. C.

Practice: P. 219 # 1, 2, 3- 6, 14-18 even. Determine whether the polygons are congruent. Support your answer by describing a transformation.

By their ANGLES and SIDES Classify Triangles By their ANGLES and SIDES

Real World Example

Classifying Triangles Polygon: 3-sided close figure Name 3 sides: AB, BC, & AC Name 3 angles: < A or < CAB < B or < ABC < C or < BCA Name 3 vertices: A, B, C

Classifying triangles Angle Sum Addition Equal to 180º (<1 + <2 + <3 = 180º)

Classify Triangles by “Angles”

Acute triangle ALL Angles are acute.

Obtuse triangle ONE Angle is obtuse.

Right Triangle ONE right angle

Equiangular triangle ACUTE Triangle ALL Angles are congruent

Classify Triangles By “SIDES”

Scalene triangle NO Sides are congruent

Isosceles triangles 2 sides are congruent

Equilateral triangle ALL Sides are congruent

Real World Example

Practice Worksheet Part 1 # 1-8 Classification Part 2  #1-12 Missing Measure Practice p. 227 #1-17

Warm-up: 1. M: (x, y)  ( x + 5, y – 4) G(4, -1), H(7, 3), I(7, -1) Name coordinates of image Graph: Transformation: Congruent or not? Explain. 2. Classify by sides and angles; find x. X°

Real world example

Classify Triangles Practice

Identify Triangles A. Equilateral B. Isosceles C. Scalene D. Right Answer: C

Identify Triangles A. Equilateral B. Isosceles C. Scalene D. Right Answer: A

Identify Triangles A. Equilateral and acute B. Isosceles and right C. Obtuse and Scalene D. Right and Equilateral Answer: B

Identify Triangles A. Equilateral and right B. Isosceles and obtuse C. Scalene and Right D. Scalene and obtuse Answer: D

Identify Triangles A. Acute B. Obtuse C. Equiangular D. Right Answer: B

Identify Triangles A. Equilateral B. Acute C. Scalene D. Right Answer: D

Identify Triangles A. Equilateral B. Isosceles C. Scalene D. Obtuse Answer: C

Identify Triangles A. Equilateral B. Isosceles C. Acute D. Right Answer: C

Identify Triangles A. Equilateral & Obtuse B. Isosceles & right C. Acute & Equiangular D. Right & scalene Answer: D

Identify Triangles A. Equilateral B. Obtuse C. Acute D. Isosceles Answer: B

Practice: Name The triangle by sides and angles 1. 45, 90, 45 Answer: Right, Isosceles 2. 100, 30, 50 Answer: Obtuse, Scalene 3. 60, 60, 60 Answer: Acute, Equilateral, Equiangular 4. 33, 77, 70 Answer: Acute, Scalene

Practice Check p. 227 #1-17

Measuring Angles in triangles: Trio Activity Return questions and answers & check work

Warm-Up: 1. Find x. 65º 50º xº 80º 2. Find x. 50º 53º xº 62º 80º

Practice: Name The triangle 1. 35, 60, 85 Answer: Acute, Scalene 2. 110, 25, 45 Answer: Obtuse, Scalene 3. 45, 90, 45 Answer: Right, Isosceles 4. 60, 60, 60 Answer: Acute, Equilateral, Equiangular

Check: Trio Activity

corollary interior exterior interior angle exterior angle Vocabulary corollary interior exterior interior angle exterior angle remote interior angle

4.3 Angle Relationships in Triangle

Angle Relationships in Triangles A corollary is a theorem whose proof follows directly from another theorem. Here are two corollaries to the Triangle Sum Theorem.

Angle Relationships in Triangles The interior is the set of all points inside the figure. The exterior is the set of all points outside the figure. 4 is an exterior angle. The remote interior angles of 4 are 1 and 2. Exterior Interior 3 is an interior angle.

Angle Relationships in Triangles

Angle Relationships in Triangles

Angle Relationships in Triangles Practice: Pairs: P. 235 # 4, 5, 6, 10. 12. 14 Draw figures Substitute measures Analyze given info Compute measures

Quiz: Classify triangles and Measure Angles No talking during Quiz Take your time Good LUCK! 14. M: (x, y) (y, -x) L (3, 1) M (3, 4) N (5, 4) O (5, 1) Questions 14 & 15: a) Name coordinates of image, b) Name Transformation, c) Determine if figures are congruent, explain. 15. M: (x, y) (x-1, y+1) N (1, -2) O (0, 4) P (2, 4)

cont  Quiz: Classify triangle and Measure Angles No talking during Quiz Take your time Good LUCK! 14. M: (x, y) (y, -x) L (3, 1) M (3, 4) N (5, 4) O (5, 1) Questions 14 & 15: a) Name coordinates of image, b) Name Transformation, c) Determine if figures are congruent, explain. 15. M: (x, y) (x-1, y+1) N (1, -2) O (0, 4) P (2, 4)

Warm-Up: 1. After an accident, the positions of cars are measured by law enforcement to investigate the collision. Use the diagram drawn from the information collected to find mYWZ. 2.

check Quiz: Classify triangles and Measure Angles

Congruent triangles ABC = NOP Sketch, label, & mark all congruent parts. 4.4 Stand Up: Form Student Triangles Triangle have to be congruent Each student represent one unit Cannot Talk Side 1 vs Side 2 JKL = XYZ ABC = NOP

Congruent triangles Define: Triangles that are the same size and shape. Each triangle have 6 corresponding parts. 3 corresponding angles corresponding sides

SKETCH, LABEL, & MARK all congruent parts. ABC = JKL B K A C J L Angles Sides

Practice Activity Worksheet Page 242 #1-10 Page 246 Congruent Triangles forms Origami

Warm-up: 1. ∆ABC  ∆JKL and AB = 2x + 12. JK = 4x – 50. Find x and AB. Given that polygon MNOP  polygon QRST, identify the congruent corresponding part. 2. NO  ____ 3. T  ____

Complete & check P. 242 #1-10 Activity Page 246 Congruent Triangles forms Origami

Prove triangles are Congruent List postulates

objectives Apply SSS, SAS, ASA, and AAS to construct triangles and solve problems. Prove triangles congruent by using SSS, SAS, ASA, and AAS. Use CPCTC to prove parts of triangles are congruent.

VOCABULARY triangle rigidity included angle included side CPCTC

Prove Triangles Are Congruent In Lesson 4-3, you proved triangles congruent by showing that all six pairs of corresponding parts were congruent. The property of triangle rigidity gives you a shortcut for proving two triangles congruent. It states that if the side lengths of a triangle are given, the triangle can have only one shape.

Prove Triangles Are Congruent Triangle Congruence: SSS For example, you only need to know that two triangles have three pairs of congruent corresponding sides. This can be expressed as the following postulate.

Triangle Congruence: SSS Adjacent triangles share a side, so you can apply the Reflexive Property to get a pair of congruent parts. Remember!

Triangle Congruence: SSS It is given that AC  DC and that AB  DB. By the Reflexive Property of Congruence, BC  BC. Therefore ∆ABC  ∆DBC by SSS Use SSS to explain why ∆ABC  ∆DBC.

Triangle Congruence: SSS Use SSS to explain why ∆ABC  ∆CDA. It is given that AB  CD and BC  DA. By the Reflexive Property of Congruence, AC  CA. So ∆ABC  ∆CDA by SSS.

Triangle Congruence: SAS An included angle is an angle formed by two adjacent sides of a polygon. B is the included angle between sides AB and BC.

Triangle Congruence: SAS It can also be shown that only two pairs of congruent corresponding sides are needed to prove the congruence of two triangles if the included angles are also congruent.

Prove Triangles Are Congruent Triangle Congruence: SAS

Triangle Congruence: SAS The letters SAS are written in that order because the congruent angles must be between pairs of congruent corresponding sides. Caution

Triangle Congruence: SAS The diagram shows part of the support structure for a tower. Use SAS to explain why ∆XYZ  ∆VWZ. It is given that XZ  VZ and that YZ  WZ. By the Vertical s Theorem. XZY  VZW. Therefore ∆XYZ  ∆VWZ by SAS.

Triangle Congruence: SAS Use SAS to explain why ∆ABC  ∆DBC. It is given that BA  BD and ABC  DBC. By the Reflexive Property of , BC  BC. So ∆ABC  ∆DBC by SAS.

Triangle Congruence: SAS The SAS Postulate guarantees that if you are given the lengths of two sides and the measure of the included angles, you can construct one and only one triangle.

Triangle Congruence: SAS Show that the triangles are congruent for the given value of the variable. ∆STU  ∆VWX, when y = 4. ST  VW, TU  WX, and T  W. ∆STU  ∆VWX by SAS.

Triangle Congruence: SSS & SAS Which postulate, if any, can be used to prove the triangles congruent? 1. 2.

Triangle Congruence: ASA An included side is the common side of two consecutive angles in a polygon. The following postulate uses the idea of an included side.

Prove Triangles Are Congruent Triangle Congruence: ASA

Triangle Congruence: ASA Determine if you can use ASA to prove NKL  LMN. Explain. By the Alternate Interior Angles Theorem. KLN  MNL. NL  LN by the Reflexive Property. No other congruence relationships can be determined, so ASA cannot be applied

Prove Triangles Are Congruent Triangle Congruence: AAS You can use the Third Angles Theorem to prove another congruence relationship based on ASA. This theorem is Angle- Angle-Side (AAS).

Triangle Congruence: sss, sas, asa, and AAS Identify the postulate or theorem that proves the triangles congruent.

Prove Triangles Are Congruent Triangle Congruence: CPCTC CPCTC is an abbreviation for the phrase “Corresponding Parts of Congruent Triangles are Congruent.” It can be used as a justification in a proof after you have proven two triangles congruent.

Triangle Congruence: CPCTC SSS, SAS, ASA, and AAS use corresponding parts to prove triangles congruent. CPCTC uses congruent triangles to prove corresponding parts congruent. Remember!

Triangle Congruence Practice 3.5 – 3.7 SSS SAS ASA AAS CPCTC

Warm-Up Isosceles powerpoint

Practice:Triangle Congruence-sss, sas, asa, AAS, and CPCTC Analyze the congruent parts Match congruent triangles Write the congruent statements Identify the postulate or theorem that proves the triangles congruent

Check: Triangle Congruence Practice 3.5 – 3.7 SSS SAS ASA AAS CPCTC

Analyze isosceles triangles Identify congruent parts of triangles Online powerpoint

Isosceles and Equilateral Triangles Practice Worksheet P. 288 #1-10

Warm-up: 1. If XYZ is congruent to PRS, then write the 6 congruence statements. 2. Name theorem. 3. Find x & y 32 x y

Isosceles and Equilateral Triangles Check Practice Worksheet P. 288 #1-10

Quiz: Take your Time Label all parts NO talking DO YOUR BEST! When finish, complete study guide. TEST: Monday

Warm-Up: solve 1. In XYZ, m< X = 3x – 31, m< Y = 5x+8, and m< Z = 4x -13, find each angle. 2. A ACD is equilateral. m< B = 30º. Find m< BAC. B C D

Practice Chapter 4 Study Guide Ask Questions Label congruent parts on triangles Analyze figures

Focused on finding measurements. Practice before Test Focused on finding measurements.

Find the angle measures. yº answer: x= 40º y= 100º xº 40º

Find the angle measures. answer: x = 6 and y =16 16 y 7xº 42º

Find the angle measures. answer: x =12 and y = 60º 5xº yº

Find the angle measures. 72º 45º xº Answer: 72 + 45 = 117

Find the angle measures. 54 3xº Answer: x = 18

Find the angle measures. xº answer: x = 80 52º 48º

Find the angle measures. 8xº 7xº Answer: 15x = 90…. X = 6

Find the angle measures. answer: x = 120º xº

Find the angle measures. 40º xº Answer: x = 110º

Find x & the angle measures. B C Answer: x = 20 <A = 70 <B = 90 <C = 20

Chap 4 test: Identifying congruent triangles NO talking Take your time LABEL, LABEL, LABEL Show all your work Read, read, & reread each question GOOD LUCK! #32 Construction For full credit, Classify triangle List the angle measurements Use protractor and straightedge For additional credit, construct a 2nd triangle using compass and straightedge