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
►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
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}