Triangle Sum Theorem The sum of the measures of the three angles of a triangle is 180º.

Triangle Sum Theorem The sum of the measures of the three angles of a triangle is 180º.

Triangle Sum Theorem The sum of the measures of the three angles of a triangle is 180º. No-Choice Theorem If two angles of one triangle are congruent to two angles of another triangle, then the third angles are congruent.

Theorem: If there exists a correspondence between the vertices of two triangles such that two angles and a nonincluded side of one are congruent to the corresponding parts of the other, then the triangles are congruent. (AAS)

AAS theorem for triangle congruence. Given: B C AC Y Z XZ A X Prove: ABC XYZ Proof Statement B Reason C Y Z

AAS theorem for triangle congruence. Given: B C AC Y Z XZ A X Prove: ABC XYZ Proof Statement B Reason C Y Z

AAS theorem for triangle congruence. Given: B C AC Y Z XZ A X Prove: ABC XYZ Proof Statement B Reason C Y Z

AAS theorem for triangle congruence. Given: B C AC Y Z XZ A X Prove: ABC XYZ Proof Statement B Reason C Y Z

AAS theorem for triangle congruence. Given: B C AC Y Z XZ A X Prove: ABC XYZ Proof Statement B Reason C Y Z

AAS theorem for triangle congruence. Given: B C AC Y Z XZ A X Prove: ABC XYZ Proof Statement A B 1. C Z 1. Given Reason C Y Z

AAS theorem for triangle congruence. Given: B C AC Y Z XZ A X Prove: ABC XYZ Proof Statement A S B Reason 1. C Z 1. Given 2. AC XY 2. Given C Y Z

AAS theorem for triangle congruence. Given: B C AC Y Z XZ A X Prove: ABC XYZ Proof Statement A S B Reason 1. C Z 1. Given 2. AC XY 2. Given 3. B Y 3. Given C Y Z

AAS theorem for triangle congruence. Given: B C AC Y Z XZ Prove: ABC XYZ Proof Statement A B Reason A 1. C Z 1. Given S 2. AC XY 2. Given 3. B Y 3. Given A 4. A X 4. No-Choice Th. C X Y Z

AAS theorem for triangle congruence. Given: B C AC Y Z XZ Prove: ABC XYZ Proof Statement A B Reason A 1. C Z 1. Given S 2. AC XY 2. Given 3. B Y 3. Given A 4. A X 4. No-Choice Th. 5. ABC XYZ 5. ASA C X Y Z

SSS: yes

SSS: yes SAS: yes

SSS: yes SAS: yes ASA: yes

SSS: yes SAS: yes ASA: yes AAS (SAA): yes

SSS: yes SAS: yes ASA: yes AAS (SAA): yes AAA: no

SSS: yes SAS: yes ASA: yes AAS (SAA): yes AAA: no Does SSA work for congruence?

In this case, two different triangles can be formed given two sides and a nonincluded angle.

Postulate: If there exists a correspondence between the vertices of two right triangles such that the hypotenuse and a leg of one triangle are congruent to the corresponding parts of the other triangle, the two right triangles are congruent. (HL)

Pythagorean Theorem: The sum of the squares of the lengths of the legs of a right triangle is equal to the square of the length of the hypotenuse. a 2 + b 2 = c 2

Triangle Sum Theorem The sum of the measures of the three angles of a triangle is 180º.

Triangle Sum Theorem The sum of the measures of the three angles of a triangle is 180º. No-Choice Theorem If two angles of one triangle are congruent to two angles of another triangle, then the third angles are congruent.

Pythagorean Theorem: The sum of the squares of the lengths of the legs of a right triangle is equal to the square of the length of the hypotenuse. a 2 + b 2 = c 2

Theorem: If there exists a correspondence between the vertices of two triangles such that two angles and a nonincluded side of one are congruent to the corresponding parts of the other, then the triangles are congruent. (AAS)

Postulate: If there exists a correspondence between the vertices of two right triangles such that the hypotenuse and a leg of one triangle are congruent to the corresponding parts of the other triangle, the two right triangles are congruent. (HL)

SSS: Three congruent sides

SSS: Three congruent sides SAS: Two sides and the included angle (Can be called LL if a right angle is included angle.)

SSS: Three congruent sides SAS: Two sides and the included angle (Can be called LL if a right angle is included angle.) ASA: Two angles and the included side

SSS: Three congruent sides SAS: Two sides and the included angle (Can be called LL if a right angle is included angle.) ASA: Two angles and the included side AAS: Two angles and a nonincluded side

SSS: Three congruent sides SAS: Two sides and the included angle (Can be called LL if a right angle is included angle.) ASA: Two angles and the included side AAS: Two angles and a nonincluded side HL: The hypotenuse and one leg of right triangles