Triangle Sum Theorem The sum of the measures of the three angles of a triangle is 180º.
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1 Triangle Sum Theorem The sum of the measures of the three angles of a triangle is 180º.
2 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.
3 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)
4 AAS theorem for triangle congruence. Given: B C AC Y Z XZ A X Prove: ABC XYZ Proof Statement B Reason C Y Z
5 AAS theorem for triangle congruence. Given: B C AC Y Z XZ A X Prove: ABC XYZ Proof Statement B Reason C Y Z
6 AAS theorem for triangle congruence. Given: B C AC Y Z XZ A X Prove: ABC XYZ Proof Statement B Reason C Y Z
7 AAS theorem for triangle congruence. Given: B C AC Y Z XZ A X Prove: ABC XYZ Proof Statement B Reason C Y Z
8 AAS theorem for triangle congruence. Given: B C AC Y Z XZ A X Prove: ABC XYZ Proof Statement B Reason C Y Z
9 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
10 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
11 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
12 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
13 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
14 SSS: yes
15 SSS: yes SAS: yes
16 SSS: yes SAS: yes ASA: yes
17 SSS: yes SAS: yes ASA: yes AAS (SAA): yes
18 SSS: yes SAS: yes ASA: yes AAS (SAA): yes AAA: no
19 SSS: yes SAS: yes ASA: yes AAS (SAA): yes AAA: no Does SSA work for congruence?
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27 In this case, two different triangles can be formed given two sides and a nonincluded angle.
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36 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)
37 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
38 Triangle Sum Theorem The sum of the measures of the three angles of a triangle is 180º.
39 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.
40 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
41 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)
42 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)
43 SSS: Three congruent sides
44 SSS: Three congruent sides SAS: Two sides and the included angle (Can be called LL if a right angle is included angle.)
45 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
46 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
47 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
Exploring Congruent Triangles
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