Lesson 7.1 Right Triangles pp. 262-266

Lesson 7.1Right Triangles

pp. 262-266

Lesson 7.1Right Triangles

pp. 262-266

Objectives:1. To prove special congruence

theorems for right triangles.2. To apply right triangle congruence

theorems in other proofs.

Objectives:1. To prove special congruence

theorems for right triangles.2. To apply right triangle congruence

theorems in other proofs.

ReviewReview

AA

CC

BB

ABC is a rt. ABC is a rt.

B is the rt. B is the rt.

The side opposite B is AC, called the hypotenuse. The side opposite B is AC, called the hypotenuse.

AB and BC are called the legs.AB and BC are called the legs.

SASSAS

ASAASA

SSSSSS

AASAAS

Theorem 7.1HL Congruence Theorem. If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and corresponding leg of another right triangle, then the two triangles are congruent.

Theorem 7.1HL Congruence Theorem. If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and corresponding leg of another right triangle, then the two triangles are congruent.

HL Congruence TheoremHL Congruence Theorem

IH

J

L

N

M

HL Congruence TheoremHL Congruence Theorem

IH

J

L

N

M

Theorem 7.2LL Congruence Theorem. If the two legs of one right triangle are congruent to the two legs of another right triangle, then the two triangles are congruent.

Theorem 7.2LL Congruence Theorem. If the two legs of one right triangle are congruent to the two legs of another right triangle, then the two triangles are congruent.

IH

J

L

N

M

LL Congruence TheoremLL Congruence Theorem

Theorem 7.3HA Congruence Theorem. If the hypotenuse and an acute angle of one right triangle are congruent to the hypotenuse and corresponding acute angle of another right triangle, then the two triangles are congruent.

Theorem 7.3HA Congruence Theorem. If the hypotenuse and an acute angle of one right triangle are congruent to the hypotenuse and corresponding acute angle of another right triangle, then the two triangles are congruent.

HA Congruence TheoremHA Congruence Theorem

IH

J

L

N

M

Theorem 7.4LA Congruence Theorem. If a leg and one of the acute angles of a right triangle are congruent to the corresponding leg and acute angle of another right triangle, then the two triangles are congruent.

Theorem 7.4LA Congruence Theorem. If a leg and one of the acute angles of a right triangle are congruent to the corresponding leg and acute angle of another right triangle, then the two triangles are congruent.

LA Congruence TheoremLA Congruence Theorem

IH

J

L

N

M

For the next 5 questions decide whether the right triangles are congruent. If they are, identify the theorem that justifies it. Be prepared to give the congruence statement.

For the next 5 questions decide whether the right triangles are congruent. If they are, identify the theorem that justifies it. Be prepared to give the congruence statement.

Practice: Is ∆ADC ∆ABC?1. HL2. LL3. HA4. LA5. Not

enoughinformation

Practice: Is ∆ADC ∆ABC?1. HL2. LL3. HA4. LA5. Not

enoughinformation

B

C

DA

Practice: Is ∆EFG ∆EHG?1. HL2. LL3. HA4. LA5. Not

enoughinformation

Practice: Is ∆EFG ∆EHG?1. HL2. LL3. HA4. LA5. Not

enoughinformation

F

G

HE

Practice: Is ∆LMN ∆PQR?1. HL2. LL3. HA4. LA5. Not

enoughinformation

Practice: Is ∆LMN ∆PQR?1. HL2. LL3. HA4. LA5. Not

enoughinformation

L M

N

Q P

R

Practice: Is ∆XYZ ∆YXW?1. HL2. LL3. HA4. LA5. Not

enoughinformation

Practice: Is ∆XYZ ∆YXW?1. HL2. LL3. HA4. LA5. Not

enoughinformation

W

X Y

Z

Practice: Is ∆LMO ∆PNO?1. HL2. LL3. HA4. LA5. Not

enoughinformation

Practice: Is ∆LMO ∆PNO?1. HL2. LL3. HA4. LA5. Not

enoughinformation

L

M

N

P

O

Homeworkpp. 264-266Homeworkpp. 264-266

►A. ExercisesIdentify any triangle congruence theorem or postulate corresponding to each right triangle congruence theorem below. (Remember that the right angles are congruent.)

3. LA, adjacent case

►A. ExercisesIdentify any triangle congruence theorem or postulate corresponding to each right triangle congruence theorem below. (Remember that the right angles are congruent.)

3. LA, adjacent case

►A. ExercisesIdentify any triangle congruence theorem or postulate corresponding to each right triangle congruence theorem below. (Remember that the right angles are congruent.)

4. LA, opposite case

►A. ExercisesIdentify any triangle congruence theorem or postulate corresponding to each right triangle congruence theorem below. (Remember that the right angles are congruent.)

4. LA, opposite case

►A. ExercisesIdentify any triangle congruence theorem or postulate corresponding to each right triangle congruence theorem below. (Remember that the right angles are congruent.)

5. HL

►A. ExercisesIdentify any triangle congruence theorem or postulate corresponding to each right triangle congruence theorem below. (Remember that the right angles are congruent.)

5. HL

►A. Exercises9. Use the diagram to state a triangle

congruence. Which right triangle theorem justifies the statement?

►A. Exercises9. Use the diagram to state a triangle

congruence. Which right triangle theorem justifies the statement?

MM

NN

HH PP

QQ

►A. Exercises10. Prove HA.►A. Exercises10. Prove HA.

RR SS

TT

UU VV

WW

10.10.

1. Given1. Given

2. S & V are rt. ’s 2. S & V are rt. ’s

2. Def. of rt. ’s2. Def. of rt. ’s

3. S V3. S V 3. All rt. ’s are 3. All rt. ’s are 4. RST UVW 4. RST UVW

Statements ReasonsStatements Reasons

4. SAA 4. SAA

1. RST & UVW are rt. ’s; RT

UW; R U

1. RST & UVW are rt. ’s; RT

UW; R U

►B. ExercisesUse the same diagram as in exercise 10 for the proofs in exercises 11-12 (the two cases of the LA Congruence Theorem).11. LA (opposite case)Given: ∆RST and ∆UVW are right

triangles; RS UV;T W

Prove: ∆RST ∆UVW

►B. ExercisesUse the same diagram as in exercise 10 for the proofs in exercises 11-12 (the two cases of the LA Congruence Theorem).11. LA (opposite case)Given: ∆RST and ∆UVW are right

triangles; RS UV;T W

Prove: ∆RST ∆UVW

►B. Exercises11. LA (opposite case)Given: ∆RST and ∆UVW are right

triangles; RS UV;T W

Prove: ∆RST ∆UVW

►B. Exercises11. LA (opposite case)Given: ∆RST and ∆UVW are right

triangles; RS UV;T W

Prove: ∆RST ∆UVW

RR SS UU VV

TT WW

►B. Exercises12. LA (adjacent case)Given: ∆RST and ∆UVW are right

triangles; RS UV;R U

Prove: ∆RST ∆UVW

►B. Exercises12. LA (adjacent case)Given: ∆RST and ∆UVW are right

triangles; RS UV;R U

Prove: ∆RST ∆UVW

RR SS UU VV

TT WW

12.12.

1. Given1. Given

2. S & V are rt. ’s 2. S & V are rt. ’s

2. Def. of rt. ’s2. Def. of rt. ’s

3. S V3. S V 3. All rt. ’s are 3. All rt. ’s are 4. RST UVW 4. RST UVW

Statements ReasonsStatements Reasons

4. ASA 4. ASA

1. RST & UVW are rt. ’s; RS

UV; R U

1. RST & UVW are rt. ’s; RS

UV; R U

►B. ExercisesUse the following diagram to prove exercise 13.13. Given:P and Q are right angles;

PR QRProve: PT QT

►B. ExercisesUse the following diagram to prove exercise 13.13. Given:P and Q are right angles;

PR QRProve: PT QT

QQ

PP

RR TT

►B. ExercisesUse the following diagram to prove exercises 15-19.15. Given:WY XZ; X ZProve: ∆XYW ∆ZYW

►B. ExercisesUse the following diagram to prove exercises 15-19.15. Given:WY XZ; X ZProve: ∆XYW ∆ZYW

WW

XX YY ZZ

■ Cumulative ReviewGive the measure of the angle(s) formed by22. two opposite rays.

■ Cumulative ReviewGive the measure of the angle(s) formed by22. two opposite rays.

■ Cumulative ReviewGive the measure of the angle(s) formed by23. perpendicular lines.

■ Cumulative ReviewGive the measure of the angle(s) formed by23. perpendicular lines.

■ Cumulative ReviewGive the measure of the angle(s) formed by24. an equiangular triangle.

■ Cumulative ReviewGive the measure of the angle(s) formed by24. an equiangular triangle.

■ Cumulative ReviewGive the measure of the angle(s) formed by25. the bisector of a right angle.

■ Cumulative ReviewGive the measure of the angle(s) formed by25. the bisector of a right angle.

■ Cumulative Review

26. Which symbol does not represent a set?ABC, ∆ABC, A-B-C, {A, B, C}

■ Cumulative Review

26. Which symbol does not represent a set?ABC, ∆ABC, A-B-C, {A, B, C}