SAS theorem (Side-Angle-Side)īy applying the Side Angle Side Postulate (SAS), you can also be sure your two triangles are congruent. You will see that all the angles and all the sides are congruent in the two triangles, no matter which ones you pick to compare. So go ahead look at either ∠C and ∠T or ∠A and ∠T on △CAT.Ĭompare them to the corresponding angles on △BUG. The postulate says you can pick any two angles and their included side. You may think we rigged this, because we forced you to look at particular angles. You can only make one triangle (or its reflection) with given sides and angles. This is because interior angles of triangles add to 180°. This forces the remaining angle on our △CAT to be:ġ80 ° − ∠ C − ∠ A 180°-\angle C-\angle A 180° − ∠ C − ∠ A The two triangles have two angles congruent (equal) and the included side between those angles congruent. ![]() See the included side between ∠C and ∠A on △CAT? It is equal in length to the included side between ∠B and ∠U on △BUG. Notice that ∠C on △CAT is congruent to ∠B on △BUG, and ∠A on △CAT is congruent to ∠U on △BUG. In the sketch below, we have △CAT and △BUG. An included side is the side between two angles. The Angle Side Angle Postulate (ASA) says triangles are congruent if any two angles and their included side are equal in the triangles. Triangle Congruence Postulates and Theorems ASA theorem (Angle-Side-Angle) Let's take a look at the three postulates abbreviated ASA, SAS, and SSS. Testing to see if triangles are congruent involves three postulates. More important than those two words are the concepts about congruence. 8) 90 ∠ECB = 180 8) Since ∠B = 90 given 9) ∠ECB = 180 -90 = 90 9) By subtraction property 10) AB = EC 10) From (5) 11) BC = CB 11) Reflexive (Common side) 12) ∠ABC = ∠ECB 12) Each 90 0 13) ΔABC ≅ ΔECB 13) SAS postulate 14) AC = BE 14) CPCTC 15) 1/2AC = 1/2BE ⇒ 1/2AC = BD 15) Multiply by 1/2 but 1/2BE = BD by mid point definition.Do not worry if some texts call them postulates and some mathematicians call the theorems. Statements Reasons 1) AD = DC 1) Given 2) BD = DE 2) By construction 3) ∠ADB = ∠CDE 3) Vertically opposite angles 4) ΔADB ≅ ΔCDE 4) By SAS postulate 5) EC = AB and ∠CED = ∠ABD 5) CPCTC 6) CE || AB 6) If alternate interior angles are congruent then the lines are parallel 7) ∠ABC ∠ECB = 180 7) Angles formed on the same side of transveral are supplementary. ![]() Given : ΔABC in which ∠B = 90 0 and D is the mid point of AC.Ĭonstruction : Produce BD to E so that BD = DE. 5) ΔAOC ≅ ΔBOD 5) SAS postulate 6) AC = BD 6) CPCTC 7) ∠CAO = ∠DBO 7) CPCTC 8) AC || BD 8) If alternate interior angles are congruentĢ) If D is the mid point of the hypotenuse AC of a right triangle ABC, prove that BD = ½ AC. 3) ∠AOC = ∠BOD 3) Vertically opposite angles 4) CO = OD 4) By definition of mid point. 1) Given 2) AO = OB 2) By definition of mid point. Statements Reasons 1) O is the mid point. Prove that : i) ΔAOC ≅ ΔBOD ii) AC = BD and iii) AC || BD. ![]() Statements Reasons 1) AB = AC 1) Given 2) AD is a bisector 2) By construction 3) ∠BAD = ∠CAD 3) By definition of angle bisector 4) AD = AD 4) Reflexive (common side) 5) ΔABD ≅ ΔACD 5) SAS Postulate 6) ∠B = ∠C 6) CPCTCġ) O is the mid point of AB and CD. ![]() Theorem : Angles opposite to two equal sides of a triangle are equal.Ĭonstruction : Draw the bisector AD of ∠A which meets BC in D. Angle in standard position Side Angle Side Postulate Side angle side postulate ->If two sides and the included angle of a triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent by side angle side postulate.
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