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Download Triangle Sum Theorem The sum of the measures of the three angles
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Transcript
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. A Given: B ~ = Y C~ = Z AC ~ = XZ Prove: Proof ABC ~ = Statement XYZ X C B Reason Y Z AAS theorem for triangle congruence. A Given: B ~ = Y C~ = Z AC ~ = XZ Prove: Proof ABC ~ = Statement XYZ X C B Reason Y Z AAS theorem for triangle congruence. A Given: B ~ = Y C~ = Z AC ~ = XZ Prove: Proof ABC ~ = Statement XYZ X C B Reason Y Z AAS theorem for triangle congruence. A Given: B ~ = Y C~ = Z AC ~ = XZ Prove: Proof ABC ~ = Statement XYZ X C B Reason Y Z AAS theorem for triangle congruence. A Given: B ~ = Y C~ = Z AC ~ = XZ Prove: Proof ABC ~ = Statement XYZ X C B Reason Y Z AAS theorem for triangle congruence. A Given: B ~ = Y C~ = Z AC ~ = XZ Prove: Proof A 1. ABC ~ = XYZ Statement C~ = X C B Reason Z 1. Given Y Z AAS theorem for triangle congruence. A Given: B ~ = Y C~ = Z AC ~ = XZ Prove: Proof ABC ~ = Statement A 1. C ~ = Z S 2. AC ~ = XY XYZ X C B Reason 1. Given 2. Given Y Z AAS theorem for triangle congruence. A Given: B ~ = Y C~ = Z AC ~ = XZ Prove: Proof ABC ~ = Statement A 1. C ~ = Z S 2. AC ~ = XY 3. B ~ = Y XYZ X C B Reason 1. Given 2. Given 3. Given Y Z AAS theorem for triangle congruence. A Given: B ~ = Y C~ = Z AC ~ = XZ Prove: Proof ABC ~ = Statement A 1. C ~ = Z S 2. AC ~ = XY 3. B ~ = Y A 4. A ~ = X XYZ X C B Reason 1. Given 2. Given 3. Given 4. No-Choice Th. Y Z AAS theorem for triangle congruence. A Given: B ~ = Y C~ = Z AC ~ = XZ Prove: Proof ABC ~ = XYZ Statement A 1. C ~ = Z S 2. AC ~ = XY 3. B ~ = Y A 4. A ~ = X 5. ABC ~ = XYZ X C B Reason 1. Given 2. Given 3. Given 4. No-Choice Th. 5. ASA 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. a2 + b2 = c2 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. a2 + b2 = c2 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
