POSTULATE ASA I NTRODUCTION When triangles are identical, that is, - PowerPoint PPT Presentation

D AY 57 – C ONGRUENCE POSTULATE ASA

I NTRODUCTION When triangles are identical, that is, equal and overlapping each other, we say that they are congruent. We wish to develop easier methods of determining if triangles are congruent since this would be a good method to apply in an industry to confirm if the triangularly shaped items manufactured are actually equal. In this lesson, we are going to explain the ASA postulate and proof that it is actually true.

V OCABULARY  Congruent triangles Refers to identical triangles

Consider the following triangles. M B C L A K The postulates say that if lines CA = ML, ∠𝐷𝐵𝐶 = 𝑀𝑁𝐿 and ∠𝐵𝐷𝐶 = ∠𝑁𝑀𝐿 , then the two triangles ABC and KLM are congruent. The angles referred to as A and A in the postulate must be at the end points of the side referred to as S. The side S (between the angles) is referred to as the included angle.

We would like to prove that this is true. Consider the set up below T G E S U H EHG and GHF are triangles such that ∠𝑉 = ∠𝐼, ∠𝐹 = ∠𝑇 and line 𝐹𝐼 = 𝑉𝑇. Since the sum of interiors angles add up to 180°, we have ∠𝑈 = 180 − ∠𝑉 − ∠𝑇 also ∠𝐻 = 180 − ∠𝐹 − ∠𝐼. Since ∠𝑉 = ∠𝐼, ∠𝐹 = ∠𝑇 , have ∠𝐻 = 180 − ∠𝑇 − ∠𝑉 = ∠𝑈. Thus, ∠𝑈 = ∠𝐻.

Thus, corresponding angles are equal implying that the two triangles are either similar. T G E S U H 𝐹𝐺 Since EF=US, 𝑉𝑇 = 1 The scale factor is 1 implying that the trianglesare identical hence congruent. Thus, the postulate ASA is true.

Example Determine if the following pairs of triangles are congruent. C (i). F 2 in 4 in S 4 in R B 2 in A (ii). J Y L 8 in 9 in G P H

Solution (i). In the diagram above, line AB = line RF Second, corresponding angles at the endpoints of AB and FR are congruent. That is, ∠𝐵 = ∠𝑆, ∠𝐶 = ∠𝐺. Thus, ASA postulate is satisfied implying that triangles ABC and RFS are congruent. (ii).In the figure below, corresponding angles at the endpoints of PL and HJ are congruent but the sides, PL and HJ are not equal. Thus, ASA postulate is not satisfied implying that triangles PLY and HJG are not congruent.

HOMEWORK  Find out if the triangles are congruent based on ASA. K C 13 in 15 in T 13 in B A 15 in

A NSWERS TO THE HOMEWORK  Based on the given information, ∠𝐿 = ∠𝐵, ∠𝑍 = ∠𝐷 but 𝐵𝐷 ≠ 𝐿𝑍 implying that ASA postulate is not satisfied. This shows that the two triangles are congruent.

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