D AY 58 – C ONGRUENCE POSTULATE SAS
I NTRODUCTION Having looked at SSS and SAS postulates, we would like to proceed further to another postulate that can enable us to show if two given triangles are congruent or not. In this lesson, we are going to explain the SAS postulate and proof that it is actually true.
V OCABULARY Congruent triangles Refers to identical triangles
Consider the following triangles. K X W R Y D The postulates say that if two corresponding sides of the triangle and the corresponding included angle are congruent, SAS, then the two triangles are congruent. In triangles above, we have WY=RD, WX=RX and the angles between the lines, ∠𝑋 = ∠𝑆, thus, SAS.
We would like to prove that this is true. Consider the triangles below K X W R Y D Given SAS, we show that the triangles are congruent. We have 𝑋𝑌 = 𝑆𝐿 and 𝑋𝑍 = 𝑆𝐸 𝑋𝑌 𝑋𝑍 Thus 𝑆𝐿 = 1; 𝑆𝐸 = 1
Since the corresponding included angles are equal, we can fuse the two triangles such that X and K as well as Y and D are at the same point say M and N respectively. That would make WMRN be symmetrical at 𝑁𝑂 = 𝑌𝑍 = 𝐿𝐸. Thus, we have established that corresponding angles are equal making the ratio between the sides be 1 implying that the triangles are congruent.
Example Determine if the following pairs of triangles are congruent using SAS postulate. V 4 in Y L 109° 4 in 46° 25° E 9 in 9 in P
Solution In the figure above, YL and VR are corresponding and also equal, LP and RE are also corresponding and equal. We now focus on the corresponding included angles, ∠𝑀 and ∠𝑆. Angle VRE = 180° − 25° − 109° = 46° = ∠𝑀 . Thus, the SAS postulate is satisfied showing that the two triangles are congruent.
HOMEWORK Find out if the triangles are congruent based on SAS. F 5.5 in 106° 37° D E 10 in C 10 in A 27° 106° 5.5 in B
A NSWERS TO THE HOMEWORK Considering the two given sides, the corresponding included angles are not equal. Angle DEF = 37° ≠ angle 𝐵𝐷𝐶 = 180° − 106° − 27° = 47° . Thus, the figures are not congruent
END
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