AA Similarity
In short: AA similarity states that two triangles are similar as soon as two angles of one triangle are equal to two angles of the other, because the third angles then match automatically.
GeometryGeometryG-SRT.A.3
What is AA Similarity?
AA similarity, short for angle-angle, is the quickest test for similar triangles. Check two pairs of angles: if both pairs are equal, the triangles have the same shape, and no side lengths need to be known at all.
The reason is the angle sum. Every triangle holds \(180^{\circ}\), so once two angles agree, the third is forced: \(180 – a – b\) in both triangles. Equal angles then give proportional sides, with a single scale factor linking the two.
How to spot it
- A figure marks two angles with arcs in each of two triangles, with no tick marks on the sides; AA similarity is the natural tool.
- Two triangles share an angle, or contain a pair of vertical angles, or sit between parallel lines.
- A right angle in each triangle plus one more equal acute angle is given.
- A problem wants a side length but only angle measures are known for the first part.
The method
- Find two angle pairs that are equal: given marks, a shared corner, vertical angles or parallel lines.
- State that the triangles are similar by AA, listing the vertices in matching order.
- Set up the proportion of corresponding sides and cross-multiply.
- Solve for the unknown side, and check that its size agrees with the scale factor.
AA Similarity: solved example
Problem
Pennant \(PQR\) has \(\angle P = 48^{\circ}\) and \(\angle Q = 67^{\circ}\). Pennant \(XYZ\) has \(\angle X = 48^{\circ}\) and \(\angle Z = 65^{\circ}\). If \(PQ = 9\), \(XY = 15\) and \(QR = 12\), find \(YZ\).
Solution
In \(\triangle PQR\): \(\angle R = 180 – 48 – 67 = 65^{\circ}\). In \(\triangle XYZ\): \(\angle Y = 180 – 48 – 65 = 67^{\circ}\). Two angle pairs match (\(P = X\), \(Q = Y\)), so the triangles are similar by AA, with \(R\) matching \(Z\).
The scale factor is \(\frac{XY}{PQ} = \frac{15}{9} = \frac{5}{3}\), so \(YZ = 12 \times \frac{5}{3} = \) 20.
Try it
Question 1
Triangle one has angles of \(40^{\circ}\) and \(75^{\circ}\); triangle two has angles of \(65^{\circ}\) and \(40^{\circ}\). Are they similar?
Show the answer
The third angles are \(65^{\circ}\) and \(75^{\circ}\), so both triangles have angles 40, 65 and 75. They are similar by AA.
Question 2
Triangle A has angles of \(50^{\circ}\) and \(60^{\circ}\); triangle B has angles of \(50^{\circ}\) and \(80^{\circ}\). Are they similar?
Show the answer
A has angles 50, 60, 70 and B has 50, 80, 50. Only one angle value is shared, so they are not similar.
AA Similarity vs. Triangle Congruence
Two equal angles settle shape only. The congruence shortcut with two angles, AAS, also needs one pair of equal sides, and that extra side fixes size. Without it, AA similarity proves triangles of possibly different sizes, never exact copies.
Keep going
Lessons
Worksheets and quizzes
- Similar Triangle Theorems – Worksheet
- Similar Triangle Theorems – Quiz
- Dilations and Similarity – Worksheet
- Dilations and Similarity – Quiz
Related words
- Similar Triangles
- Triangle Angle Sum Theorem
- Scale Factor
- Vertical Angles
- Triangle Congruence (SSS, SAS, ASA, AAS)
Whole dictionary: A–Z · Geometry · Geometry