Lesson 7.1 Right Triangles pp. 262-266

Objectives: 1. To prove special congruence theorems for right triangles. 2. To apply right triangle congruence theorems in other proofs.

Review ABC is a rt.  B is the rt.  The side opposite B is AC, called the hypotenuse. AB and BC are called the legs.

SAS ASA AAS SSS

Theorem 7.1 HL 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 Theorem I H J L N M

HL Congruence Theorem I H J L N M

Theorem 7.2 LL 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.

LL Congruence Theorem I H J L N M

Theorem 7.3 HA 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 Theorem I H J L N M

Theorem 7.4 LA 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 Theorem I H 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.

Practice: Is ∆ADC  ∆ABC? 1. HL 2. LL 3. HA 4. LA 5. Not enough information B C D A

Practice: Is ∆EFG  ∆EHG? 1. HL 2. LL 3. HA 4. LA 5. Not enough information F G H E

Practice: Is ∆LMN  ∆PQR? 1. HL 2. LL 3. HA 4. LA 5. Not enough information L M N Q P R

Practice: Is ∆XYZ  ∆YXW? 1. HL 2. LL 3. HA 4. LA 5. Not enough information W X Y Z

Practice: Is ∆LMO  ∆PNO? 1. HL 2. LL 3. HA 4. LA 5. Not enough information L M N P O

Homework pp. 264-266

►A. Exercises Identify 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. Exercises Identify 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. Exercises Identify any triangle congruence theorem or postulate corresponding to each right triangle congruence theorem below. (Remember that the right angles are congruent.) 5. HL

►A. Exercises 9. Use the diagram to state a triangle congruence. Which right triangle theorem justifies the statement? M N H P Q

►A. Exercises 10. Prove HA. R S T U V W

10. Statements Reasons 1. RST & UVW are rt. ’s; RT  UW; R U 1. Given 2. S & V are rt. ’s 2. Def. of rt. ’s 3. S  V 3. All rt. ’s are  4. RST  UVW 4. SAA

►B. Exercises Use 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. Exercises 11. LA (opposite case) Given: ∆RST and ∆UVW are right triangles; RS  UV; T  W Prove: ∆RST  ∆UVW R S U V T W

►B. Exercises 12. LA (adjacent case) Given: ∆RST and ∆UVW are right triangles; RS  UV; R  U Prove: ∆RST  ∆UVW R S U V T W

12. Statements Reasons 1. RST & UVW are rt. ’s; RS  UV; R U 1. Given 2. S & V are rt. ’s 2. Def. of rt. ’s 3. S  V 3. All rt. ’s are  4. RST  UVW 4. ASA

►B. Exercises Use the following diagram to prove exercise 13. 13. Given: P and Q are right angles; PR  QR Prove: PT  QT Q P R T

►B. Exercises Use the following diagram to prove exercises 15-19. 15. Given: WY  XZ; X  Z Prove: ∆XYW  ∆ZYW W X Y Z

■ Cumulative Review 22. two opposite rays. Give the measure of the angle(s) formed by 22. two opposite rays.

■ Cumulative Review 23. perpendicular lines. Give the measure of the angle(s) formed by 23. perpendicular lines.

■ Cumulative Review 24. an equiangular triangle. Give the measure of the angle(s) formed by 24. an equiangular triangle.

■ Cumulative Review 25. the bisector of a right angle. Give the measure of the angle(s) formed by 25. the bisector of a right angle.

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