Data Loading...

Chapter Three: Triangle and Triangle Congruences Flipbook PDF

Chapter Three

120 Views
15 Downloads
FLIP PDF 1.04MB

A Learning Module in Plane and Solid Geometry

TRIANGLE & TRIANGLE CONGRUENCES Chapter Three

Mathematics –Physics – Statistics Department College of Arts and Sciences Benguet State University

Table of Contents LESSON 3.1. TRIANGLES ................................................................................................................................ 2 LESSON 3.2. KINDS OF TRIANGLES ................................................................................................................ 2 3.2.1. ACCORDING TO SIDES: .................................................................................................................... 2 3.2.2. ACCORDING TO ANGLES ................................................................................................................. 3 EXERCISE 3.1 ................................................................................................................................................. 6 LESSON 3.3. MEDIANS, ALTITUDE AND BISECTORS ...................................................................................... 7 PRACTICE EXERCISES ..................................................................................................................................... 8 LESSON 3.4 . CORRESPONDENCE AND CONGRUENCE ................................................................................ 10 PRACTICE EXERCISES ................................................................................................................................... 10 LESSON 3.5. PROVING TRIANGLES CONGRUENT ........................................................................................ 11 3.5.1. SSS POSTULATE: ............................................................................................................................ 11 3.5.2. SAS POSTULATE:............................................................................................................................ 11 3.5.3. ASA POSTULATE: ........................................................................................................................... 12 3.5.4. AAS/SAA POSTULATE: ................................................................................................................... 12 PRACTICE EXERCISES: .................................................................................................................................. 13 LESSON 3.6. CORRESPONDING PARTS OF A TRIANGLE............................................................................... 15 PRACTICE EXERCISES: .................................................................................................................................. 15 LESSON 3.7. PROVING RIGHT TRIANGLES CONGRUENT ............................................................................. 16 3.7.1. LA THEOREM ................................................................................................................................. 16 3.7.2. HA THEOREM ................................................................................................................................ 17 3.7.3. LL THEOREM:................................................................................................................................. 17 3.7.4. HL THEOREM: ................................................................................................................................ 17 EXERCISE 3.2 ............................................................................................................................................... 18 EXERCISE 3.3 ............................................................................................................................................... 19 EXERCISE 3.4 ............................................................................................................................................... 20 EXERCISE 3.5 ............................................................................................................................................... 22

|

CHAPTER THREE TRIANGLE AND TRIANGLE CONGRUENCES OBJECTIVES: At the end of the chapter, the students should be able to: 1. define and classify triangles; 2. identify the different parts of a triangle; 3. differentiate congruence from correspondence; and 4. prove triangles to be congruent.

Definition: A set of points is a triangle if and only if it consists of the figure formed by three segments (sides) connecting three non-collinear points called vertex. The three vertices are used to name a triangle. Capital letters are used to name the vertices of a triangle. Lowercase letters may be used to name the sides.

A triangle can be classified by its sides or by its angles. 3.2.1. ACCORDING TO SIDES: 1. Scalene – no sides are congruent 2. Isosceles – 2 sides are congruent • In an isosceles triangle: the two congruent sides are the legs. The third side is the base. The vertex angle is opposite the base. The base angles include the base. 3. Equilateral – all sides are congruent

Theorem 3-1: TRIANGLE ANGLE SUM THEOREM The sum of the measures of a triangle is 180°. Corollary 3-1-1: An equilateral triangle is also equiangular. Corollary 3-1-2: Each angle of an equilateral triangle has a measure of 60°. Corollary 3-1-3: An equiangular triangle is also equilateral. Example: Find x and the measure of each side of equilateral triangle RST. |

Theorem 3- 2: ISOSCELES TRIANGLE THEOREM If two sides of a triangle are congruent, then the angles opposite those sides are congruent. Corollary 3-2-1: The bisector of the vertex angle of an isosceles triangle is perpendicular to the base at its midpoint. In other words, in an isosceles triangle, the bisector of the vertex angle is also an altitude and a median of the triangle.

M Angle

Theorem 3-3: If two angles are congruent, then the sides opposite those angles are congruent.

Example: Find x, JM, MN, and JN if ∆JMN is an isosceles triangle with JM  MN .

3.2.2. ACCORDING TO ANGLES 1. Oblique – Triangles with no right angles a. Acute – three acute angles

|

Altit

b. Obtuse – with one right angle 2. Right - with one right angle 3. Equiangular – all angles are congruent

Example

D

Given: ∆ABC Prove: mA + mB + mC = 180

E

B 1

2

A

C

STATEMENTS

PROOF 1. Through a point not on a line, there is 1. Through B construct DE || AC . exactly one line parallel to the given line. 2. DBC and 2 form a linear pair 2. Definition of linear pair 3. DBC and 2 are supplementary angles 3. Linear Pair Postulate 4. mDBC + m2 = 180 4. Definition of Supplementary Angles 5. mDBC = m1 + mABC 5. Definition of betweenness of rays 6. m1 + mABC + m2 = 180 6. Substitution property 7. 1  A; 2  C 7. If || lines have a transversal, then alternate interior angles are congruent 8. m1 = mA; m2 = mC 8. Definition of congruent angles 9. mA + mB + mC = 180 9. Substitution property In this proof, DE is an added line. Such an addition is called an auxiliary line because it helps to prove a theorem. Corollary: In a triangle, there can be at most one right angle, or at most one obtuse angle. Corollary: The acute angles of a right triangle are complementary.

|

Corollary: It two angles of one triangle are congruent to two angles of a second triangle, then the third angles are congruent.

1

B

2

In this figure, each side of ∆ABC has been extended to from exterior angles: 1, 2, and 3 . Each exterior angle has an

A

3 C

adjacent interior angle and two remote interior angles. Exterior angle 2 is adjacent to interior angle ABC. Its two remote interior angles are BAC and ACB

Theorem 3 - 4: EXTERIOR ANGLE THEOREM The measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles. Example 1: Find the measure of FLW mLOW + mOWL = mFLW x + 32 = 2x – 48 32 = x – 48 80 = x mFLW = 2(80) – 48 = 112

Exterior Angle Theorem Substitution

Example 2: Determine the measure of the numbered angles

|

A. Use the figure at the right for problems 1-3.

5

2

1. Find m3 if m5 = 130 and m4 = 70. 2. Find m1 if m5 = 142 and m4 = 65. 3. Find m2 if m3 = 125 and m4 = 23. 11

3

4 1

8

B. Use the figure at the right for problems 4-7. 3. m6 + m7 + m8 = _______. 9

6

7

10

4. If m6 = x, m7 = x – 20, and m11 = 80, then x = _____. 5. If m8 = 4x, m7 = 30, and m9 = 6x -20, then x = _____. 6. m9 + m10 + m11 = _______. C. For 8 – 12, solve for x.

8.

9.

120

(5x)°

140°

x°

x°

35°

9.

(2x)º (4x)°

10.

11.

x°

(3x + 54)º

x°

(x - 20)°

12. Find m1, m2, m3, m4, and m5. 65°

82°

46°

1

|

2

5

4

3

142°

Medians, altitude and bisector are three types of segments associated with triangles. Every triangle has three medians and three angle bisectors. Definition: A segment is a median of a triangle if and only if it extends from a vertex of the triangle to the midpoint of the opposite side. The intersection of the three medians of a triangle is called the centroid.

B

D

Medians: AF, CD and BE Points D, E and F are midpoints of AB, AC and BC, respectively A

F C

E

Definition: A segment is an angle bisector if and only if it bisects an angle of the triangle and has one endpoint on the opposite side. The intersection of the three angle bisector of a triangle is called the incenter. A M

20°20°

30° 30° B

Every triangle has three altitudes. However, while medians and angle bisectors lie in the interior of the triangle, the position of the altitudes depends on the type of the triangle. The intersection of the three altitudes of a triangle is called the orthocenter. Definition: A segment is an altitude of a triangle if and only if it is perpendicular from a vertex to the triangle to the line containing the opposite side of the triangle. Theorem 3-5: If a point lies on the perpendicular bisector of a segment, then the point is equidistant from the endpoints of the segment.

|

G

40° 40°

S

O

H

C

S

N

D

N

Theorem 3-6: If a point is equidistant from the endpoints of a segment, then it lies on the perpendicular bisector of the segment. Corollary 3-6-1: If two points are each equidistant from the endpoints of a segment, then the line joining the points is the perpendicular bisector of the segment. Theorem 3-7: If a point lies on the bisector of an angle, then the point is equidistant from the sides of the angle. Theorem 3-8: If a point is equidistant from the sides of an angle, then the point lies on the bisector of the angle. B

1. Given: BD is a median of AB = x − 7

Find:

AD = x + 3 DC = 2 x −17 AB

A

C

D

2. Given: AD is an altitude of  ABC

A

BD = 2 x

Find:

DC = 3 x − 4  ADC = 4 x + 10 BC B

D

C

B

3.

Given: XB bisects  ABC A |

1

X

2

C

BX is an altitude Prove: BX is a median

Statements

Reasons

B

4.

Given: BD is a median BD ⊥ AC Prove: BD bisects  ABC A

Statements

|

D

Reasons

C

Congruence is a basic geometric relationship. Congruent figures have the same size and shape. Definition: Two or more triangles that have all angles and sides respectively equal (same measure) are called congruent (  ) triangles. Correspondence is association of members of a set with members of another set. Such association or pairing is one-to-one such that exactly one member is paired with exactly another one member and vice versa. The order in which objects are paired is important. Correspondences ABC ↔123 and ABC↔231 are different. The correspondence BIG↔SML identifies three pairs of corresponding angles and three pairs of corresponding sides.

•

B  S BI  SM I  M BG  SL G  L IG  ML Some correspondences appear different, yet represent the same pairing of vertices. The correspondence HAL ↔ TOM is the same as, or is equivalent to AHL ↔ OTM, because in both correspondences A ↔ O, H ↔ T and L ↔ M. T

A

L

H

•

M

O

Sometimes a correspondence between triangles is also a congruence. Two triangles are congruent if and only if there is correspondence between the vertices of the triangles such that the corresponding angles are congruent and the corresponding sides are congruent.

A. Determine which correspondences are equivalent to ∆XYZ↔∆MNQ? Explain. 1. ∆YZX↔∆NQM 2. ∆ZXY↔∆QMN 3. ∆XZY↔∆MNQ 4. ∆YZX↔∆NMQ B. Given ∆AMY  ∆LIN. Complete the following congruence statements.

|

1. A  ____ 2. ____  LI 3. _____  N 4. MY  _____

It is usually not necessary to use the definition of congruent triangles to prove two triangles congruent. There are more concise methods. The three postulates and the theorem that follow are instrumental in proving triangles congruent. 3.5.1. SSS POSTULATE: If three sides of one triangle are congruent to three sides of another triangle, then the two triangles are congruent. Example: Statements Reasons 1. AB  CB

1.

2. AD  CD

2.

3. BD  BD 4. ΔABD  ΔCBD

3. 4.

3.5.2. SAS POSTULATE: If two sides and the included angle of a triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent. Example: Given: RE  TE and DE  CE Prove: ΔRED  ΔTEC Statements 1. RE  TE 2. DE  CE 3. RED  TEC 4. ΔRED  ΔTEC

Reasons 1. 2. 3. 4.

|

3.5.3. ASA POSTULATE: If two angles and the include side of a triangle are congruent to two angle and the included side of another triangle, then the two triangles are congruent. 3.5.4. AAS/SAA POSTULATE: If two angles and the non-included side of one triangle are congruent, respectively, to the corresponding angles and the non-included side of another triangle, then the two triangles are congruent.

Example 3: Given: MNP  OPN, MPN  ONP Prove: MNP  OPN Statements 1. MNP  OPN 2. MPN  ONP 3. NP  NP

Reasons 1. 2. 3. 4.

4. MNP  OPN

Example 4: Given: XQ || TR, XR

bisects

QT

Prove: XMQ  RMT

Statements 1. XQ || TR

Reasons 1.

2. Q  T

2.

3. X  R

3.

4. XR bisects QT

4.

5. TM  QM

5.

6. XMQ  RMT

6.

|

1. Prove Triangles are Congruent Given: CD  EA , AD is the perpendicular bisector of CE Prove: CBD  EBA

Statements

Reasons

1. AD is the perpendicular bisector of CE 1. 2. CBD and EBA are right angles 2. 3. CBD and EBA are right triangles 3. 4. B is midpoint of CE 4. 5. CB  EB 5. 6. CD  EA 7. CBD  EBA

6. 7.

2. Prove Triangles are Congruent Given: WJ  KZ , JWZ and ZKJ are right angles Prove: WJZ  KZJ

Statements 1. JWZ and ZKJ are right angles 2. WJZ and KZJ are right triangles 3. JZ  JZ

1. 2.

4. WJ  KZ 5. WJZ  KZJ

4. 5.

|

Reasons

3.

3. Use triangle congruence theorems to find the value of the variable. a. WXY  ZYX Find p.

X

Y 2

2p V 20

(7p+13)

W

Z

b. ADC  CBA . Find x.

D

C

(2x + 7)°

1

2

(x – 8x)°

A

B

|

When two triangles are congruent, six parts of one triangle are equal respectively to six parts of the other. The corresponding parts of congruent triangles are often used to prove statements about overlapping triangles and sequences of congruence. CPCTC is the abbreviation of corresponding parts of congruent triangles are congruent. O

T

Example 1:

R

Given: RT  RY ; RS  RO Prove: TS  YO

Y

S

PROOF: STATEMENTS

REASONS

1. RT  RY ; RS  RO

1. Given

2. TRS  YRO 3. RTS  RYO

2. Vertical angles are congruent 3. SAS Postulate 4. CPCTC

4. TS  YO

1. Given: AC  EC , BC  DC Prove: A  E

|

2. Given: AB  CD, AE  FD, A  D Prove: EC  FB

c. Given: EOF  HOG, OFE  OGH,EG  FH

Prove: EOH is isosceles

There are four special theorems that can be used to prove that pairs of right triangles are congruent. In a right triangle, the non-right angles must be acute. The side of a right triangle that is opposite the right angle is hypotenuse. The sides that are opposite the acute angles are legs. Since the right angles of right triangles are always congruent, the next four theorems each require finding only two other congruent corresponding parts. 3.7.1. LA THEOREM: If a leg and an acute angle of one right triangle are congruent to the corresponding parts of another right triangle, then the triangles are congruent.

|

3.7.2. HA THEOREM: If the hypotenuse and an acute angle of one right triangle are congruent to the corresponding parts of another right triangle, then the triangles are congruent. 3.7.3. LL THEOREM: If the two legs of one right triangle are congruent to the two legs of another right triangle, then the triangles are congruent. 3.7.4. HL THEOREM: If the hypotenuse and a leg of one right triangle are congruent to the corresponding parts of another right triangle, then the triangles are congruent.

Determine whether the two triangles congruent? If so, write the triangle congruency statement and the theorem that makes them congruent. If not, tell why.

B

5. Prove that ADB  CDB Given:

BD ⊥ AC

AB  CB A

|

D

C

A. For each pair of triangles, tell which postulates, if any, make the triangles congruent. 1. ABC  EFD

______________2. ABC  CDA

C

______________ C

B

D

A

F

A

B

D

E

3. ABC  EFD

______________4. ADC  BDC

______________ C

F

C

B

A

5. MAD  MBC

D

A

E

______________6. ABE  CDE

D

B

D

______________ D

C

C

E

A A

B

M

7. ACB  ADB

B

______________ 8. MNP  MQP

C

B

A

D

|

______________

A. For each pair of triangles, tell which postulate, if any, can be used to prove the triangles congruent. 1. AEB  DEC

2. CDE  ABF ______________

______________

A D E

C

C

F

B

E A

B

D

3. DEA  BEC

4. AGE  CDF ______________

______________

A

B E

D

C

5. RTS  CBA T

6. ABC  ADC ______________

______________

B

S C A

C

R A

B D

7. BAP  BCP ______________ Given: BD bisects ABC

8. SAT  SAR ______________ R

A

B

D

P

S

A

T C

|

A. For each pair of triangles, tell: (a) Are they congruent (b) Write the triangle congruency statement. (c) Give the postulate that makes them congruent.

1.

2.

D

3. Given: T is the midpoint of WR

O

C

E

A E E

L A

V W

B

a. ______________ b. _____   _____ c. ______________

a. ______________ b. _____   _____ c. ______________

4.

a. ______________ b. _____   _____ c. ______________

5. Given: IH Bisects

WIS

6.

I

W

R

T

S

H

a. ______________ b. _____   _____ c. ______________

a. ______________ b. _____   _____ c. ______________

7.

8.

L

U

G

E

a. ______________ b. _____   _____ c. ______________ 9.

H

P

P A

A T

T M

a. ______________ a. ______________ b. _____   _____ b. _____   _____ c. ______________ c. ______________

|

a. ______________ b. _____   _____ c. ______________

M

10. Given: I is the midpoint of ME and SL M

11.

12. C

F

L I

S

E A

a. ______________ b. _____   _____ c. ______________

a. ______________ b. _____   _____ c. ______________

B

D

E

a. ______________ b. _____   _____ c. ______________

B. Using the given postulate, tell which parts of the pair of triangles should be shown congruent.

1.

SAS

2. ASA

3. SSS

C

F E

A

D

B

F A

B

B

A

C

C

E

D

_______  ________

________  _______

4. AAS

5. HL

D

_____  ____ 6. ASA

P

D

P

C

S T A

R

R

Q

_______  ________

S

________  ________

|

Q

______  _____

B

A. For which value(s) of x are the triangles congruent? 1. x = _______________ A

2. x = _______________ A

B

m 3 = x2 m4 = 7x - 10

1

3

E

7x - 4

4x + 8

2

4

C

C

D

3. x = _______________

B

R

4. x = _______________ W

A

x2 + 3x

x2 + 2x

B

D C

Z

x2 + 24

9x - 8 R

|

S

T