ACT Math : How to find if of acute / obtuse isosceles triangle are similar

Study concepts, example questions & explanations for ACT Math

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Example Questions

Example Question #21 : Isosceles Triangles

Triangle A and Triangle B are similar isosceles triangles. Triangle A's sides measure , and . Two of the angles in Triangle A each measure . Triangle B's sides measure , and . What is the measure of the smallest angle in Triangle B?

Possible Answers:

Correct answer:

Explanation:

Because the interior angles of a triangle add up to , and two of Triangle A's interior angles measure , we must simply add the two given angles and subtract from  to find the missing angle:

Therefore, the missing angle (and the smallest) from Triangle A measures . If the two triangles are similar, their interior angles must be congruent, meaning that the smallest angle is Triangle B is also .

The side measurements presented in the question are not needed to find the answer!

Example Question #221 : Geometry

Triangle A and Triangle B are similar isosceles triangles. Triangle A has a base of  and a height of . Triangle B has a base of . What is the length of Triangle B's two congruent sides?

Possible Answers:

Correct answer:

Explanation:

We must first find the length of the congruent sides in Triangle A. We do this by setting up a right triangle with the base and the height, and using the Pythagorean Theorem to solve for the missing side (). Because the height line cuts the base in half, however, we must use  for the length of the base's side in the equation instead of . This is illustrated in the figure below:

Triangle a

Using the base of  and the height of , we use the Pythagorean Theorem to solve for :

Therefore, the two congruent sides of Triangle A measure ; however, the question asks for the two congruent sides of Triangle B. In similar triangles, the ratio of the corresponding sides must be equal. We know that the base of Triangle A is  and the base of Triangle B is . We then set up a cross-multiplication using the ratio of the two bases and the ratio of  to the side we're trying to find (), as follows:

Therefore, the length of the congruent sides of Triangle B is .

Example Question #222 : Geometry

Isosceles triangles  and  share common side  is an obtuse triangle with sides  is also an obtuse isosceles triangle, where . What is the measure of ?

Possible Answers:

Correct answer:

Explanation:

In order to prove triangle congruence, the triangles must have SAS, SSS, AAS, or ASA congruence. Here, we have one common side (S), and no other demonstrated congruence. Hence, we cannot guarantee that side  is not one of the two congruent sides of , so we cannot state congruence with .

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