Chapter 4 Congruent Triangles Identify the corresponding parts

Chapter 4 Congruent Triangles • Identify the corresponding parts of congruent figures • Prove two triangles are congruent • Apply theorems and corollaries about isosceles triangles

4. 1 Congruent Figures Objectives • Identify the corresponding parts of congruent figures

What we already know… • Congruent Segments – Same length – AB CD – AB = 4 , CD = 4 • Congruent Angles – Same degree measure – ABC EFG – m ABC = 48 ◦ , m EFG = 48◦

Congruent Figures Exactly the same size and shape. Don’t ASSume ! C B A D E F

Definition of Congruency Two polygons are congruent if corresponding vertices can be matched up so that: 1. All corresponding sides are congruent 2. All corresponding angles are congruent.

What does corresponding mean again? • Matching • In the same position

Definition of Congruent Triangles ABC DEF You can slide and rotate the triangles around so that they MATCH up perfectly. A E C B The order in which you name the triangles matters ! F D

Based on the definition of congruency…. • Three pairs of corresponding angles • Three pairs of corresponding sides 1. A D 1. AB DE 2. B E 2. BC EF 3. C F 3. CA FD There are 6 pieces of information that we need to have in order to prove that two triangles are congruent!!

ABC XYZ • Based off this information with or without a diagram, we can conclude… • Letters X and A, appear first, naming corresponding vertices, which means… – X A. • The letters Y and B come next, so – Y B and –XY AB HINT: If there’s no drawing, create your own!!!

CAUTION !! • If the diagram doesn’t show the markings A B D C F • Use the information given to you… – Shared sides, shared angles, vertical angles, parallel lines F

White Boards • Suppose TIM BER IM ___

White Boards • Suppose TIM BER IM ER , Why ?

White Boards • Corresponding Parts of Congruent Triangles are Congruent (aka the definition of congruent triangles) • CPCTC - this is the abbreviated way to say the statement above

White Boards • Suppose TIM BER ___ R

White Boards • Suppose TIM BER M R, Why?

White Boards • Corresponding Parts of Congruent Triangles are Congruent

White Boards • Suppose TIM BER MTI ____

White Boards • Suppose TIM BER MTI RBE

A. B. C. D. Always Sometimes Never I don’t know • An acute triangle is _____ congruent to an obtuse triangle.

A. B. C. D. Always Sometimes Never I don’t know • A right triangle is ______ congruent to another right triangle.

A. B. C. D. Always Sometimes Never I don’t know • If ABC XYZ, B is ______ congruent to Y.

4. 2 Warm-up (on 4. 1 day) [Use pg. 122 to help fill in the blanks] FILL IN THE BLANKS…. • AB is opposite _____ • AB is included between L___ and L____ B • LA is opposite ____ • LA is included between ____ and ____ A C

4. 2 Some Ways to Prove Triangles Congruent Objectives • Learn about ways to prove triangles are congruent

Don’t ASSume • Triangles cannot be assumed to be congruent because they “look” congruent. and • It’s not practical to cut them out and match them up so,

We must show 6 congruent pairs • WHAT ARE THOSE 6 PAIRS? – 3 angle pairs and – 3 pairs of sides

WOW • That’s a lot of work

We are lucky…. . • There is a shortcut – We don’t have to show • ALL pairs of angles are congruent and • ALL pairs of sides are congruent • It’s like a lawyer not needing as much evidence to get a criminal convicted…

Experiment • Use the 3 connector pieces to create a triangle • Compare your groups triangle to your neighbors

SSS Postulate Each side matches congruent with the 3 sides of another triangle E B A C F D

**ORDER MATTERS!!! SAS Postulate • Two sides match up congruent and… • The angles between the 2 sides are congruent. E B A C F D

ASA Postulate • Two angles match up congruent and… • The side in between those angles E B A C F D

The order of the letters MEAN something • Is SAS the same as SSA or A$$ ? NO!!!! • SAS – TWO SIDES WITH THE ANGLE THAT IS IN BETWEEN (INCLUDED ) • ASA – TWO ANGLES WITH THE SIDE THAT IS BETWEEN THEM

CAUTION !! • If the diagram doesn’t show the markings or • You don’t have a reason – Shared sides, shared angles, vertical angles, parallel lines

A. B. C. D. SSS Postulate SAS Postulate ASA Postulate Cannot be proved congruent

A. B. C. D. SSS Postulate SAS Postulate ASA Postulate Cannot be proved congruent

A. B. C. D. SSS Postulate SAS Postulate ASA Postulate Cannot be proved congruent

A. B. C. D. SSS Postulate SAS Postulate ASA Postulate Cannot be proved congruent

A. B. C. D. SSS Postulate SAS Postulate ASA Postulate Cannot be proved congruent

A. B. C. D. SSS Postulate SAS Postulate ASA Postulate Cannot be proved congruent

A. B. C. D. SSS Postulate SAS Postulate ASA Postulate Cannot be proved congruent

A. B. C. D. SSS Postulate SAS Postulate ASA Postulate Cannot be proved congruent

White Board • Decide Whether you can deduce by the SSS, SAS, or ASA Postulate that the two triangles are congruent. If so, write the congruence ( ABC _ _ _ ). If not write not congruent.

ABC EDC SAS D A C B E

CDA ABC ASA A B C D

4. 3 Using Congruent Triangles Objectives • Use congruent triangles to prove other things

Pg. 124 • Copy down problem #10 • Talk through with your partner how you would prove the triangles are congruent • REMEMEBER… TO SAY THE TRIANGLES ARE CONGRUENT MEANS YOU HAVE THE EVIDENCE FOR 1 OF OUR SHORTCUTS (SSS, ASA, SAS) • Write an explanation in paragraph form

Pg. 125 • Copy down problem # 16 in your notes – Copy down everything! – Including the diagram – Complete the proof in your group

Once the triangles are congruent…. • If we can show two triangle are congruent, using the SSS, SAS, ASA postulates then. . – We can use the definition of Congruent Triangles to say other parts of the triangles are congruent. • Corresponding Parts of Congruent Triangles are Congruent. (CPCTC)

This is an abbreviated way to refer to the definition of congruency with respect to triangles. C orresponding P arts of C ongruent T riangles are C ongruent Used as a reason in a proof to say that the rest of the pieces of the triangles not mentioned are congruent too. (GETTING A CONVICTION) CPCTC cannot be used in a proof until after the triangle is proven congruent.

STEPS TO FOLLOW… 1. Identify two triangles in which the two segments or angles are corresponding parts. (criminals) 2. Prove that those two triangles are congruent (evidence) 3. State that the two parts are congruent using the reason CPCTC. ( You convict the other corresponding pieces as being congruent)

Given: m 1 = m 2 m 3=m 4 Prove: JM = MK J L 34 1 2 M • Mark the diagram with the given info • Check for free visual evidence. . • Remember that you must prove the triangles congruent first, before using CPCTC. • TALK IT OUT IN YOUR GROUPS WRITE AN EXPLANATION K

Given: m 1 = m 2 m 3=m 4 Prove: JM = MK J L 34 1 2 M K

Chapter 4 Quiz • Drawing two congruent triangles – What can you conclude? – Name the corresponding sides and angles • Given a diagram… – Are the two triangles congruent? – Name the two congruent triangles and the postulate that proves it (ORDER MATTERS) • Complete a proof – Using SSS, SAS, or ASA – Using CPCTC

4. 4 The Isosceles Triangle Theorem Objectives • Apply theorems and corollaries about isosceles triangles

Isosceles Triangle By definition, it is a triangle with two congruent sides called legs. X Legs Vertex Angle Does the base always have to be at the bottom? Base Angles Y Base Z

Experiment - Goal • Discover Properties of an Isosceles Triangle

Procedure 5. Label the triangle P Do we have an isosceles triangle? S R Q

Procedure 6. Since PRQ fits exactly over PSQ (because that’s the way we cut it), P PRQ PSQ S R Q

Procedure 7. What conclusions can you make? – – – Use the 2 smaller congruent triangles (cpctc) Every conclusion must be justified Be careful not to ASSume anything. P S R Q

Conclusions 1. PQ bisects RPS P 2. PQ bisects RS 3. PQ RS at Q S R Q

These conclusions are actually • Theorems and corollaries

Theorem If two on sides a triangle arecan congruent, then Based theof diagram, what we conclude opposite those sides? sides are ifthe weangles have two congruent. B A Always draw the arrows to show where the opposite angle is. C

Corollary equiangular. 1. An equilateral triangle is also _____ 2. equilateral triangle = equiangular triangle 60. 3. An equilateral triangle has angles that measure ____

Corollary If one of the following occurs, then they all do. . 1. Segment bisects vertex angle 2. Segment bisects base (segment) 3. Segment is perp. to base R P Q Segment coming from vertex to the base S

Theorem If two angles of a triangle arecan congruent, then the Based on the diagram, what we conclude? sides opposite those angles are congruent. THINK: Converse of previous theorem B A C

White Board Practice • Find the value of x 30º xº x = 75º

White Board Practice • Find the value of x 2 x - 4 2 x + 2 x+5 x=9

White Board Practice • Find the value of x X = 42 41 42 56 º 62 º x

4. 5 Other Methods of Proving Triangles Congruent Objectives • Learn two new ways to prove triangles are congruent

WARM –UP • CREATE A SEPARATE DRAWING REPRESENTING EACH OF THE FOLLOWING… 1. SSS congruency 2. SAS congruency 3. ASA congruency • Your drawings should use – – Shared sides, vertical angles, parallel lines Congruency markings and/or written in measurements

Proving Triangles We can already prove triangles are congruent by the ASA, SSS and SAS. There are two other ways to prove them congruent…

AAS Theorem • Two angles match up congruent and… • A side not between the angles is congruent. E B A C F How can we apply one of the previous postulates to help us prove this theorem? D

The Right Triangle Can you label the types of angles this triangle has? B What are the specific names for the sides of a right triangle? hyp leg A ote acute angles right angle leg nus e C

HL Theorem • hypotenuse is congruent in each triangle • Either leg is congruent B What is another way I could have illustrated this theorem? A C F E D

Five Ways to Prove ’s All Triangles: ASA ORDER MATTERS!!! SSS SAS AAS Right Triangles Only: HL

White Board Practice • State which of the congruence methods can be used to prove the triangles congruent. You may choose more than one answer. SSS Postulate SAS Postulate ASA Postulate AAS Theorem HL Theorem

SSS Postulate SAS Postulate ASA Postulate AAS Theorem HL Theorem

SSS Postulate SAS Postulate ASA Postulate AAS Theorem HL Theorem

SSS Postulate SAS Postulate ASA Postulate AAS Theorem HL Theorem

SSS Postulate SAS Postulate ASA Postulate AAS Theorem HL Theorem

SSS Postulate SAS Postulate ASA Postulate AAS Theorem HL Theorem

SSS Postulate SAS Postulate ASA Postulate AAS Theorem HL Theorem

Section 4 -7 Objectives • Define altitudes, medians and perpendicular bisectors.

Definition: Median of a Triangle A segment connecting a vertex to the midpoint of the opposite side. Determine the definition by looking at the diagram vertex midpoint What indicates that the segment is intersecting the side at it’s midpoint?

Median of a Triangle Each triangle has three Medians vertex midpoint

Altitude of a Triangle A segment drawn from a vertex perpendicular to the opposite side. vertex Determine the definition by looking at the diagram perpendicular Does the segment intersect at the midpoint?

Altitude of a Triangle Each triangle has three altitudes perpendicular vertex

Special Cases - Altitudes Obtuse Triangles: Two of the altitudes are drawn outside the triangle. Extend the sides of the triangle

Special Cases - Altitudes Right Triangles: Two of the altitudes are already drawn for you. **YOU WILL SEE THIS ON THE TEST!!

Altitudes • Acute – all 3 are inside the triangle • Obtuse – 1 inside – 2 outside • Right – 1 inside – other 2 are the legs of the triangle

Perpendicular Bisector • A segment (line or ray) that is… – perpendicular to a segment and. . – passes through the midpoint of segment. Must put the perpendicular and congruent markings !

Perpendicular Bisector in depth… • Draw a point on the red line…what can you say about that point in relation to A and B? • Any point on the perp. bisector is equidistant from the endpoints. A B

Angle Bisector A ray that cuts an angle into two congruent angles. A B C

Remember • When you measure distance from a point to a line, you have to make a perpendicular line. A

Theorem Pick a point on the angle bisector… what can you say about that point using the word equidistant? A B C

Group Work Name the following. . 1. Median 2. Altitude 3. Angle Bisector C D E B F A

Ch. 4 test · Take a written statement about 2 congruent triangles, and deduce information. o I. e XRT congruent BAG Important definitions: Understanding wording and based on a diagram o Median o o Altitude Angle bisector Perpendicular bisector

·Given a diagram… o Are the two triangles congruent? Name the two triangles and the postulate that shows they are congruent o ORDER MATTERS o § · · Free info Shared angles / sides Parallel lines vert angles

· Solving for x o Know rules for isosceles and equilateral triangles o Set up problem with general rule o Then plug in values and solve for x · Drawing the following… o Perp. Bisect o o Medians altitudes