Lesson 4.1 Classifying Triangles Today, you will learn to… * classify triangles by their sides and angles * find measures in triangles

A B C  ABC

Equilateral Triangle 3 congruent sides

Isosceles Triangle 2 congruent sides

Scalene Triangle no congruent sides

Equiangular Triangle 3 congruent angles

Acute Triangle 3 acute angles 60° 70° 50°

Obtuse Triangle 1 obtuse angle 95° 25° 60°

Right Triangle 1 right angle 60° 30°

We classify triangles by their sides and angles. SIDES ANGLES Equilateral Isosceles Scalene Equiangular Acute Obtuse Right

A B C _____ is opposite A. CB _____ is opposite B. AC _____ is opposite C. AB Identify the side opposite the given angle.

hypotenuse leg leg ? Leg?

base leg leg Leg? ?

Theorem 4.1 Triangle Sum Theorem The sum of the measures of the interior angles of a triangle is ________ 180°

1. Find m  X. 61º 75º Y m  X = Z X 44˚

If the sum of the interior angles is 180º, what do you know about 1 and 2? 1 2

Corollary to the Triangle Sum Theorem The acute angles of a right triangle are _________________. complementary

2. Find m  F. 54˚ F D E m  F = 54˚ + m  F = 90˚ 36˚

3. Find m  1 and m  º 70º exterior angle adjacent angles 1 m  1 = m  2 = 60˚ 120˚

Theorem 4.2 Exterior Angle Theorem The measure of an exterior angle of a triangle is equal to the sum of the measures of the 2 nonadjacent interior angles m  1 + m  2 = m  4

m  1 + m  2 + m  3 =180˚ m  4 + m  3 = 180˚ m  1 + m  2 m  4 m  1 + m  2 = m  4 Sum of nonadjacent interior  s = ext. 

E 60˚ F 4. Find m  E. m  E = 110˚ D G m  E + 60˚ = 110˚ 50˚

E 60˚ F 5. Find x. x = (3x + 10)˚ D G x + 60 = 3x x˚x˚

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Lesson 4.2 Congruence and Triangles Today, you will learn to… * identify congruent figures and corresponding parts * prove that 2 triangles are congruent

Figures are congruent if and only if all pairs of corresponding angles and sides are congruent. Def. of Congruent Figures

Statement of Congruence Δ ABC  Δ XYZ vertices are written in corresponding order    A B C  X  Z  Y XZ YZ XY

E FD B A C 1. Mark ΔDEF to show that Δ ABC  Δ DEF.

2. Find all missing measures. B E F D B AC 35˚ ˚ 35˚ 55˚ ? ? ? ? ? ? ?

3. In the diagram, ABCD  KJHL. Find x and y. A C L K J H B D 9 cm (4x – 3) cm (3y)˚ 85˚ 93˚ 75˚ x = 3 y = 25 4x-3 = 3y = 9 75˚

4. ΔABC  ΔDEF. Find x. A C B 93˚ 30˚ F (4x + 15)˚ D E x = x + 15 = 

70˚ Theorem 4.3 Third Angles Theorem If 2 angles of one triangle are congruent to 2 angles of another triangle, then… the third angles are also congruent. D O G C A T 70˚ 60˚ ? ?

E F G J H 58° 5. Decide if the triangles are congruent. Justify your reasoning. ΔEFG  Δ______ J HG Vertical Angles Theorem Third Angles Theorem

WX Y Z M 6. 1) WX  YZ, WX | | YZ, M is the midpoint of WY and XZ 2) 3) 4) 5) ΔWXM  ΔYZM  1   62) Alt. Int.  s Theorem 1) Given  3   4 3) Vertical Angles Th. WM  MY and ZM  MX 4) Def. of midpoint 5) Def. of  figures and  2   5

7. Identify any figures you can prove congruent & write a congruence statement. A B C D Reflexive Property Alt. Int.  Th. Third Angle Th.  ACD   C AB

Theorem 4.4 Properties of Congruent Triangles Reflexive Symmetric Transitive  ABC   ABC If  ABC   XYZ, then  XYZ   ABC If  ABC   XYZ and  XYZ   MNO then  ABC   MNO

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Lesson 4.3 Proving Triangles are Congruent Today, you will learn to… * prove that triangles are congruent * use congruence postulates to solve problems

SSS ExperimentExperiment Using 3 segments, can you ONLY create 2 triangles that are congruent?

Side-Side-Side Congruence Postulate X Y Z A C B If Side AB  XY Side AC  XZ Side BC  YZ, then ΔABC  ΔXYZ by SSS If 3 pairs of sides are congruent, then the two triangles are congruent.

1. Does the diagram give enough info to use SSS Congruence? A B C J K L no

Given:LN  NP and M is the midpoint of LP Prove: ΔNLM  ΔNPM N L M P 2. Def of midpoint LM  MP Reflexive Property NM  NM  NLM   NPM SSS Congruence

3. Show that ΔNPM  ΔDFE by SSS if N(-5,1), P (-1,6), M (-1,1), D (6,1), F (2,6), and E (2,1). N P M D F E NM = MP = NP = DE = EF = DF = ( - 5 – - 1) 2 + (1 – 6) 2 (6 – 2) 2 + (1 – 6) 2

Using 2 congruent segments and 1 included angle, can you ONLY create 2 triangles that are congruent? SAS ExperimentExperiment

Side-Angle-Side Congruence Postulate If Side AB  XY Angle  B   Y Side BC  YZ, then ΔABC  ΔXYZ by SAS X Y Z A C B If 2 pairs of sides and their included angle are congruent, then the two triangles are congruent.

SAS? SAS NO!

8. Does the diagram give enough info to use SAS Congruence? A B C D AC D

9. Does the diagram give enough info to use SAS Congruence? V W X ZY no

10. Does the diagram give enough info to use SAS Congruence? A E C D B no

Given:W is the midpoint of VY and the midpoint of ZX Prove: ΔVWZ  ΔYWX 11. X Z W Y V VW  WY and ZW  WX Def. of midpoint  VWZ  YWX Vertical Angles Th  VWZ   YWX SAS Congruence

12. Given:AB  PB, MB  AP Prove: ΔMBA  ΔMBP M P B A MB  MB Reflexive Property  ABM &  PBM are right  s Def of  lines  MBA   MBP SAS Congruence  ABM   PBMAll right  s are 

What is the best way to get better at proofs?

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Lesson 4.4 Proving Triangles are Congruent Today, you will learn to… * prove that triangles are congruent * use congruence postulates to solve problems

Using 2 angles connected by 1 segment, can you ONLY create two triangles that are congruent? ASA ExperimentExperiment

Angle-Side-Angle Congruence Postulate If Angle  B   Y, Side BC  YZ, Angle  C   Z then ΔABC  ΔXYZ by ASA X Y Z A C B If 2 pairs of angles and the included sides are congruent, then the two triangles are congruent.

A B C Included side? The included side between  A and  B is _____ AB

The included side between  B and  C is _____ A B C CB Included side?

A B C AC The included side between  A and  C is _____ Included side?

ASA? ASA NO!

5. Does the diagram give enough info to use ASA Congruence? A B C D Δ ABD  Δ by ASA Third Angles Theorem Reflexive Property A CD

6. Does the diagram give enough info to use ASA Congruence? A B C D yes, Δ ACB  ______ by ASA Δ CΔ C A D Alt. Int. Angles Theorem Reflexive Property

7. Does the diagram give enough info to use ASA Congruence? A B C D no Reflexive Property

8. Does the diagram give enough info to use ASA Congruence? A B C J K L A CB

9. Determine whether the triangles are congruent by ASA. L K GH J Vertical Angles Theorem Alt. Int. Angles Theorem K JL

Angle-Angle-Side Congruence Theorem If Angle  B   Y Angle  C   Z Side AB  XY then ΔABC  ΔXYZ by AAS X Y Z A C B If 2 pairs of angles and a pair of nonincluded sides are congruent, then the two triangles are congruent.

AAS? NO! AAS

14. Does the diagram give enough info to use AAS Congruence? A B C D Reflexive Property

15. Does the diagram give enough info to use AAS Congruence? A B C J K L

16. Determine whether the triangles are congruent by AAS. L K GH J Vertical Angles Theorem Alt. Int. Angles Theorem K JL

SSA ExperimentExperiment Using 2 sides and 1 angle that is NOT included, can you ONLY create two triangles that are congruent? NO

AAA Experiment Using 3 angles, can you ONLY create two triangles that are congruent? NO All of the angles are , but the  s are NOT 

Triangle Congruence? SSS AAA SSA SAS ASA AAS

Mark the given information on the triangles. What additional congruence would you need to show  ABC   XYZ? 17. CB  ZY, AC  XZ SAS Congruence C A B X YZ  C   Z

Mark the given information on the triangles. What additional congruence would you need to show  ABC   XYZ? 18.CB  ZY, AC  XZ SSS Congruence C A B X YZ AB  XY

Mark the given information on the triangles. What additional congruence would you need to show  ABC   XYZ? 19.CB  ZY,  C   Z SAA Congruence C A B X YZ  A   X

What is the best way to get better at proofs?

Lesson 4.5 Corresponding Parts of Congruent Triangles are Congruent Today, you will learn to… * use congruence postulates to solve problems CPCTC

1. Given: AB || CD, BC || DA Prove: AB  CD B C D A  1   2 Alt. Int. Angles Theorem,  3   Reflexive Property BD  BD ASA  ABD   C D B AB  CD CPCTC

C A D B Given:  1   2,  3   4 Prove: CD  CB Reflexive Property CA  CA ASA  ABC   A D C CD  CB CPCTC

A B D C 3. Given: AC  AD, BC  BD Prove:  C   D  C   D Reflexive Property AB  AB SSS  ABC   ABD CPCTC

4. Given: A is the midpoint of MT A is the midpoint of SR Prove: MS || TR M S A T R Def. of midpoint Vertical Angles Theorem MA  AT SAS  SAM   RA T MS | | TR CPCTC  SAM   RAT SA  AR Alt. Int. Angles Converse  M  TT

Triangle Congruence? SAS SSA AAS ASA SSS AAA 2 angles & 1 side? 2 sides & 1 angle? 3 sides or 3 angles? You can ONLY use CPCTC after you use one of these!

Does the quilt design have vertical, horizontal, or diagonal symmetry?

Lesson 4.6 Isosceles, Equilateral, and Right Triangles Today, you will learn to… * use properties of isosceles, equilateral, and right triangles Students need rulers and protractors.

Use a ruler to draw two congruent segments that share one endpoint.

Connect the endpoints to create a triangle.

Measure each interior angle. What do you notice? base angles leg base

Theorem 4.6 Base Angles Theorem If 2 sides of a triangle are congruent, then … the angles opposite them are congruent.

Theorem 4.7 Base Angles Converse If 2 angles of a triangle are congruent, then the sides opposite them are congruent.

B C A D  A   C by the Base Angles Theorem 1. What angles are congruent?

B C A D AB  BC 2. What sides are congruent? by the Base Angles Converse

3. Find m  B. AB C 75˚ m  B = 75˚

4. Find m  B. A B C 68˚ m  B = 68˚ 44˚ ?

5. Find x. AB C 2x + 4 2x + 4 = x = 7 2x = 14

6. Find x. A B C 6x x – 10 = 5 5 6x = 15 x = 2.5 4

x˚x˚ 7. Find x and y. 50˚ y˚y˚ x =y = 115˚x˚x˚65˚ y˚y˚ ?? ?

Corollaries to Theorem 4.6/4.7 (hint: don’t write these yet) If a triangle is equilateral, then it is equiangular. AND If a triangle is equiangular, then it is equilateral. A triangle is equilateral if and only if it is equiangular. Corollaries to Theorem 4.6/4.7

8. Find x. AB C 24 7x + 3 = 24 7x + 3 x = 3 7x = 21 10x – 6 10x – 6 = 7x + 3 3x = 9 10x - 6 = 24 10x = 30

9. Find x. A B C What is the measure of each angle? 2x = x = 30 2x˚ 60˚

x˚x˚ 10. Find x and y. 50˚ y˚y˚ x = y = 60˚ 70˚ 50˚ ?? ? ? 60˚

Experiment Using a right angle, a hypotenuse, and a leg, can you ONLY create 2 triangles that are congruent?

Hypotenuse-Leg Congruence Theorem X Y Z A C B The triangles MUST be right triangles. If Hyp BC  YZ Leg AB  XY then ΔABC  ΔXYZ by HL If the hypotenuse and a leg of two right triangles are congruent, then the two triangles are congruent.

11. Does the diagram give enough info to use HL Congruence? W X Z Y NO Reflexive Property

12. X is a midpoint. Does the diagram give enough info to use HL? V W X ZY YZ X Def. of midpoint

13. Does the diagram give enough info to use HL Congruence? W X Z Y YZ X Base Angles Converse Reflexive Property

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