Angle – Angle Similarity, Day 2

Warm Up

In an isosceles triangle, the two equal sides are called legs and the third side is called the

base. The angle formed by the two congruent sides is called the vertex angle.

The other two angles are called base angles. The base angles of an isosceles triangle are

congruent.

If the vertex angles of two isosceles triangles are congruent, are the triangles necessarily

similar? Explain your thinking.

How can you determine when two triangles are similar?

Two triangles are similar if it can be shown that two angles of one triangle are congruent to two angles of the other triangle.

In nonmathematical situations, similar can be used to mean like or

resembling in a general way. In mathematical terms, similar figures

are similar in a specific way: the corresponding angles are

congruent. The lengths of their corresponding sides are not

necessarily congruent but are proportional.

Are all equilateral triangles similar?

All equilateral triangles have three 60° angles. So, the AA Similarity Postulate holds for all equilateral

triangles.

The AA Similarity Postulate is one way of proving that triangles are similar. The SSS

Similarity Postulate is another way. It states that if the ratios of the measures of the corresponding sides are equal, then

the triangles are similar. The SAS Similarity Postulate is yet another way. It states that if the ratios of the measures of two pairs of

corresponding sides are equal, and the angles formed by those two sides in each triangle are congruent, then the triangles

are similar.

What do you need to show in order to prove that two triangles are similar?

To prove two triangles are similar, you need to prove that at least two angles in one triangle are congruent to two angles in the other triangle. Or, you

can show that the lengths of all three of their corresponding sides are

proportional.

Triangle AGD, shown here, is an isosceles triangle with sides

AG and DG congruent. Two line segments, segments EB

and FC, have been drawn perpendicular to side AD. Use what you have learned about

the AA Similarity Postulate and finding missing measures in similar triangles to describe

how you would find the length of line segment CD.

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