Given: - Central Bucks School District



Day 1 Notes

Section 4.1 Corresponding Parts in a Congruence

Congruent Figures: _______________________________________________________________

Two polygons are __________________ if and only if their vertices can be matched up so that _____________________ _______________ (angles and sides) are congruent.

When describing congruent polygons vertices are listed in order of correspondence.

(ABC ( (DFE (BCA ( (______ ABCDE ( PQRST AEDCB ( ________

(CBA ( (______ (ACB ( (______ BCDEA ( ________ CDEAB ( ________

Definition of Congruent Triangle:

Two triangles are congruent if and only if their vertices can be matched up so the ______________________ _____________ of the triangles are congruent.

(When the definition of congruent triangles is used in a proof, the wording used is ____________________________________________________________ abbreviated as CPCTC.

Given: (CAT ( (DOG

Sides: CA ( _____ Angles: (C ( (_____

CT ( _____ (A ( (_____

AT ( _____ (T ( (_____

Given: (PIG ( (COW

Sides: PG ( _____ Angles: (P ( (_____

PI ( _____ (I ( (_____

IG ( _____ (G ( (_____

How else could you name the congruent triangles?

If (LMN ( (RST, then the following corresponding parts are congruent.

Angles: (L ( (_____ Sides: LM ( _____

(M ( (_____ MN ( _____

(N ( (_____ LN ( _____

Day 1 Practice!!

Since, the definition is an “if and only if” statement, it also means that if you know the above corresponding parts are congruent, then you can say the triangles are congruent.

Suppose ( WXY ( ( ABF. Complete the following.

1) ∠W ( _____ 2) m∠B =_____ 3) YX ( _____

4) AF = _____ 5) (YWX ( (_____ 6) (BFA ( (_____

The triangles shown are congruent. Complete the following.

7) (ABD ( (_____ 8) BC ( _____

9) DC = _____ 10) ∠ABD ( _____

11) BD is called a common side of the two (s.

What property allows you to conclude that BD ( BD?

In some special cases, more than one pairing of vertices is possible. For the pair of congruent figures, complete to show two possible pairings.

12) ∠A ( _____ & _____

∠B ( _____

∠C ( _____ & _____

(ABC ( _________ & _________

How many pairing are possible? _______

The quadrilaterals shown are congruent. Complete the following.

13) ∠E ( _____ ( _____ ( _____

14) EF ( _____ ( _____ ( _____

15) FG ( _____

16) EH ( _____

17) EFGH ( _______ & _______

Day 2 Notes

Section 4.2 Some Ways to Prove Triangles Congruent

If two triangles are congruent, then you know all six pairs of corresponding parts are congruent. There are five easier ways to prove that two triangles are congruent.

SSS SAS ASA AAS HL

SSS Postulate: If __________ _________ of one triangle are congruent to _________ __________ of another triangle, then the triangles are congruent.

( By the SSS Postulate, (ABC ( (_____ and (POE ( (_____.

SAS Postulate: If two sides and the ____________ angle of one triangle are congruent to two sides and the ____________ angle of another triangle, then the triangles are congruent.

( By the SAS Postulate, (ABC ( (_____ and (MEL ( (_____.

ASA Postulate: If two angles and the ____________ side of one triangle are congruent to two angles and the ____________ side of another triangle, then the triangles are congruent.

( By the ASA Postulate, (ABC ( (_____ and (MON ( (_____.

AAS Theorem: If two angles and a ________________ side of one triangle are congruent to the corresponding parts of another triangle, then the triangles are congruent.

HL Theorem: If the hypotenuse and a leg of one ___________ ___________ are congruent to the corresponding parts of another right triangle, then the triangles are congruent.

Can the two triangles be proved congruent? If so, write (a) the congruence and (b) the name of the postulate used. If not, write none.

1) a) (ABC ( (_______ 2) a) (EFG ( (_______ 3) a) (JKN ( (_______ 4) a) (PQS ( (_______

b) ___________ b) ___________ b) ___________ b) ____________

5) a) (TUV ( (_______ 6) a) (ABC ( (_______ 7) a) (LMN ( (_______ 8) a) (ABC ( (_______

b) ___________ b) ___________ b) ___________ b) ____________

Proof Practice Using SSS, SAS, ASA

1.) Given: AB = DE; C is the midpoint of AE and DB

Prove: ΔABC ” ΔEDC

Statements Reasons

2.) Given: C is the midpoint of AE and DB

Prove: ΔABC ” ΔEDC

Statements Reasons

3.) Given: C is the midpoint of DB; ∠B ” ∠D

Prove: ΔABC ” ΔEDC

Statements Reasons

Day 2 Practice Problems

1) Given: HI ll GJ; Statements Reasons

HG ll IJ

Prove: (GHJ ( (IJH 1) __________________ 1) __________________

2) __________________ 2) __________________

3) __________________ 3) __________________

4) (GHJ ( (_______ 4) __________________

2) Given: DC bisects AB; Statements Reasons

AC ( BC

Prove: (ADC ( (BDC 1) __________________ 1) __________________

2) __________________ 2) __________________

3) __________________ 3) __________________

4) __________________ 4) __________________

5) (ADC ( (______ 5) __________________

HW: pg 124-125 #1-15

Day 3 Practice

Proof Packet

Identify SAS

1. Which pair of triangles below illustrates the SAS postulate?

2. On the diagrams above that DO NOT show SAS, color the side of each triangle that would make it an example of SAS.

Guided Practice Proof

Given: C is the midpoint of BF; AC ≅ CE

Prove: ΔABC ≅ ΔEFC

|Statements |Reasons |

|1. C is the midpoint of BF; 1. |

|AC≅CE |

|2. (ACB ( ( ECF 2. |

|3. BC ≅ CF 3. |

|4. Δ ABC ≅ Δ EFC 4. |

Proof #1 Given: BD bisects (CDA ; CD ≅DA

Prove: ΔBCD ≅ Δ BAD

Statements Reasons

Identify ASA

1) Which pair of triangles on the right illustrates angle-side-angle relationship?

Proof #2 Given: OM bisects (LMN; (LOM ( (NOM

Prove: LM ≅ NM

Statements Reasons

Proof #3 Given: (BAC ( (DEC; BD bisects AE

Prove: (ABC ( (EDC

Statements Reasons

Proof #4 Given: HJ is a Perpendicular Bisector of KI

Prove: ΔHJK ≅Δ HJI

Statements Reasons

Activity: To the right is the proof that two

triangles are congruent by Side Angle Side.

Draw two triangles, whose diagram is

consistent with the proof.

Day 3 Homework

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Day 4 Notes

Section 4.4: The Isosceles Triangle Theorem Notes

Isosceles triangles: a triangle has ___________________________________________.

They have special names for their parts:

□ ________: the two congruent sides

□ ________: the third side

□ __________ ___________: the angles adjacent to the base.

□ __________ ___________: the angle opposite the base.

Isosceles Triangle Theorem:

If two sides of a triangle are congruent, then the __________ opposite those sides are ________________.

If AB ( AC, then (B ( (C

Converse of Isosceles Triangle Theorem:

If two angles of a triangle are congruent, then the __________ opposite those angles are congruent.

If (B ( (C, then AB ( AC

Example 1: Example 2:

Proof Example:

Given: XY ” XZ Prove: ∠3 ” ∠5

Corollary 1: An equilateral triangle is also _______________. (and vice versa)

Corollary 2: An equilateral triangle has three _____ angles.

Corollary 3: The bisector or the vertex angle of an isosceles triangle is perpendicular to the base at its _______________. Figure:

Practice Problems

Given two congruent parts,

a) Name the (

b) Use the Isos. ( Theorem or its converse to name the ( sides or angles.

c) State which if you used the theorem or the converse.

1) ∠V ( ∠Y 2) TZ ( UZ

a) (________ a) (________

b) _____ ( ______ b) _____ ( ______

c) _______________ c) ___________

Find the value of x.

3) x = _____ 4) x = _____ 5) x = _____

6) x = _____ 7) x = _____ 8) x = _____

9) Given: BC ( AC; m( 1 = 140, find:

m(2: _____

m(3: _____

m(4: _____

10) Given: BC ( DC; Statements Reasons

BF ( DE

Prove: (1 ( (2 1) __________________ 1) __________________

2) __________________ 2) __________________

3) (______ ( (______ 3) __________________

4) __________________ 4) __________________

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HW: pg 137 #1-8 all

Day 5 Notes

Section 4.7: Medians, Altitudes, and Perpendicular Bisectors Notes

Angle Bisector: A ray that divides the angle into two ___________, adjacent angles.

□ Every triangle has three ( bisectors because there are three angles.

← To identify an angle bisector, ___________________________________________

Median : In a (, a segment from a _____________ to the ________________ of the opposite side.

□ Every triangle has three medians because there are three vertices and three sides.

To identify a median, _______________________________________________

Algebra Connection

Altitude: In a (, a _______________ segment from a __________ to a line that contains the opposite side.

□ Every triangle has three altitudes because there are three vertices.

□ Altitudes can be found inside or outside the ( or can be a side of the (.

← To identify an altitude, ________________________________________________

1) Acute (s

□ All three altitudes are found inside the triangle.

2) Right (s

□ Two of the altitudes are part of the triangle (the ________ of the ()

□ The third is inside the triangle.

3) Obtuse (s

□ Two of the altitudes are outside the triangle.

□ The third is inside the triangle.

Perpendicular Bisector: A line (or ray or segment) that is __________________ to a segment at its __________________.

□ Cuts the segment into two equal parts (def. of _____________)

□ Every triangle has three ( bisectors because there are three sides.

← To identify a ( bisector, _______________________________________________

Chart

| |Goes Through Vertex |Forms a Right Angle |Goes Through Midpoint |Splits angle in two |

| | | | |congruent angles |

|Angle Bisector | | | | |

|Median | | | | |

|Altitude | | | | |

|Perpendicular Bisector | | | | |

Practice Problems

Complete the following.

1) If AB = BC, then ______ is a median of (APC.

2) If PC is a ( bisector of (______, then BC = DC.

3) If (APD is a rt (, then ______ and _____ are altitudes of (APD.

4) If PC is a median of (PBD, then ______ ( ______.

5) If BC = CD and PC ( BD, then ______ is a ( bisector of (______.

6) If PC and AC are both altitudes of (PCA, then (______ is a right angle.

For the triangle to the right draw:

7) the median from A to BC

8) the altitude from A to BC

9) the ( bisector of BC

10) the angle bisector of (A

11) AB ( AC. What kind of ( is it?

12) What conclusions can you draw?

13) Do you think your observations work only for this triangle or for all triangles?

HW: Pg 155 #1-9 all

Day 6 Notes

6.4 Inequality in One Triangle

THEOREM 6-2: If one side of a triangle is longer than a second side, then the ___________ opposite the first side is larger than the angle opposite the second side.

• The longest side is opposite the _____________________ angle.

• The shortest side is opposite the _____________________ angle.

Examples:

Name the (a) smallest and (b) largest angles in the given triangles.

1. (a) _____ 2. (a) _____ 3. (a) _____

(b) _____ (b) _____ (b) _____

THEOREM 6-3: If one angle of a triangle is larger than a second angle, then the __________ opposite the first angle is longer than the side opposite the second angle.

• The largest angle is opposite the _____________________ side.

• The smallest angle is opposite the _____________________ side.

Examples:

Name the (a) shortest and (b) longest side for each triangle.

4. (a) _____ 5. (a) _____ 6. (a) _____

(b) _____ (b) _____ (b) _____

THEOREM 6-4: The Triangles Inequality

The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

In (ABC, AB + BC _____ AC,

AB + AC _____ BC, and

AC + BC _____ AB.

Examples:

Is it possible for a triangle to have sides with the lengths indicated?

7. 3, 4, 8 8. 3, 4, 5 9. 1.5, 3.5, 5

10. 7, 7, 14.1 11. 0.6, 0.5, 1 12. 16, 11, 5

The lengths of the two sides of a triangle are given. Write the numbers that best complete the statement: The length of the third side must be (a) greater than ___, but (b) less than ___.

10. 5, 15 11. 10.5, 7.5

(a) _____ (a) _____

(b) _____ (b) _____

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Classwork: pg 119 #1-16

HW: pg 120 #1-11

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