AA Similarity Postulate

Key Concepts

• Understanding Similarity of Polygons
• Understanding Similarity of Triangles using AA Similarity Criteria
• Understanding Similarity of Triangles using SSS similarity Criteria
• Understanding Similarity of Triangles using SAS similarity Criteria

Understanding AA similarity criteria 

Angle-Angle (AA) Similarity Postulate 

If two angles AA Similarity Postulate  of one triangle are congruent to two angles of another triangle, then the two triangles are similar. 

Consider the triangle CFD and triangle DFE of the figure. Prove that they are similar. 

In triangle CFD and DEF 

∠CFD = ∠DFE = 90° 

Angle sum property  

∠CDF = 180 – (90 + 32) = 58° 

∠CDF = ∠DEF = 58° 

Therefore, by AA similarity criteria, triangle CFD and triangle DFE are similar. 

Understanding SSS similarity criteria 

If the corresponding side lengths of two triangles are proportional, then the triangles are similar. 

PQ / DE = QR / EF = PR / DF

2.5 / 5 = 2 / 4 = 3 / 6 = 1 / 2

Consider the two triangles LMN and XYZ. Prove that the two triangles are similar. 

In triangle LMN and triangle XYZ  

LM / YZ = NL /XY = NM / XZ

20 /30 = 26 /39 = 324 / 36 = 2 / 3

The corresponding sides of the triangles are proportional. Therefore, triangles LMN and  

XYZ is similar to the SSS similarity criteria. 

Understanding SAS similarity criteria 

If an angle of one triangle is congruent to an angle of a second triangle and the lengths of the sides, including these angles, are proportional, then the triangles are similar.    

∠A = ∠D, AC / DC = AB / DE

Consider the two triangles and prove that they are similar. 

In triangles SRT and PNQ 

Corresponding sides are proportional, and the angle between them is congruent. Therefore, by SAS similarity criteria, the triangles SRT and PNQ are similar. 

Exercise

• Find the scale factor of similar triangles in the figure.

• If the triangles are similar, then find side UT.

• Which is the similarity criteria used to prove that the two triangles are similar?

• Find the value of x if the following pair of triangles are similar.

• Use similarity criteria to prove that Δ ABC ~ Δ ADE.

• If the triangles are similar. Find the missing value x.

• If the two triangles are similar. Find the length of OD.

• Solve for x.

• Given MN ÷ PR = ON ÷ RQ and ∠N = ∠R. Prove that two triangles are similar using similarity criteria.

• Given ∠S = ∠W. Prove that the given triangles are similar using similarity criteria.

Concept Map

What have we learned

• What is a similarity in polygons?

Two polygons are similar if corresponding angles are congruent and corresponding sides lengths are proportional.

• What is the AA similarity criteria for triangles to be similar?

If two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar.

• What is the SSS similarity criteria for triangles to be similar?

If the corresponding side lengths of two triangles are proportional, then the triangles are similar.

• What is the SAS similarity criteria for triangles to be similar?

If an angle of one triangle is congruent to an angle of a second triangle and the lengths of the sides including these angles are proportional, then the triangles are similar.