and = . Since interior angles of a triangle add up to 180 , we - PowerPoint PPT Presentation

D AY 74 – AA CRITERION

I NTRODUCTION Having looked at postulates that can easily determine if two triangles are congruent, we are delighted to do the same with similarity. We do not have to go the long way to show that two figures are similar. For instance, if we have a triangular plan of a structure which has to be used establish a similar structure on the ground, we have to verify, only, a few things to show the two have the same shape as intended. In this lesson, we are going to discuss the AA criterion and show that it is valid when it comes to proving the existence of similar triangles.

V OCABULARY 1. Similarity This is a terms that is used to describe the existence of two figures with equal corresponding angles and proportional sides.

AA criterion It states that two pairs of corresponding angles are equal if and only if two triangles are similar. We verify that this is true. Let ABC and LMN be two triangles such that ∠𝐵 = ∠𝑀 and ∠𝐶 = ∠𝑁. Since interior angles of a triangle add up to 180° , we have ∠𝐷 = 180 − ∠𝐵 − ∠𝐶 and ∠𝑂 = 180 − ∠𝑀 − ∠𝑁 Since ∠𝐵 = ∠𝑀 and ∠𝐶 = ∠𝑁 , we have ∠𝐷 = 180 − ∠𝐵 − ∠𝐶 = 180 − ∠𝑀 − ∠𝑁 = ∠𝑂

Thus, ∠𝐷 = ∠𝑂. This shows that corresponding angles are equal. Consequently, the two figures have the same shape. This can only happen if one of the triangles is attained by proportionally reducing or increasing its sides to get the other. But proportional corresponding angles and equal corresponding angles implies the existence of dilation, hence similarity. Thus, the two triangles are similar.

 If two triangles are similar then corresponding angles which is three pairs are equal. This implies that at least two triangles are equal, thus, the AA criterion. Example Show that the two triangles, SVU and UVT are similar if UV is a bisector of angle SUT. U S T V

Solution SU = UT implying that triangle STU is an Isosceles triangle, thus angles USV and angle UTV are equal because they are the base angles. Also angle SUV and angle TUV are equal since UV is an angle bisector of angle SUT. Hence we have at least two pair of corresponding angles equal. By AA criterion, triangles SVU and TVU are similar.

HOMEWORK Use the AA criterion to find out if triangles KNL and KNM are similar. N K L M

A NSWERS TO HOMEWORK Angle NKL = Angle NKM (Common to both triangles) Angles KNL and KNM are not equal because they are at the same vertex but are not bound by the same lines. Angles KLN and KMN are not equal because lines NL and NM are not parallel since they share a common point. Hence AA criterion cannot apply. This implies that the two triangles are not similar.

THE END