SSAT Upper Level Math : How to find if two acute / obtuse triangles are similar

Study concepts, example questions & explanations for SSAT Upper Level Math

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Example Questions

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Example Question #581 : Geometry

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.

Evaluate .

Possible Answers:

Correct answer:

Explanation:

Corresponding sides of similar triangles are in proportion, so

Therefore,  as well. 

Again, by similarity,

Example Question #51 : Acute / Obtuse Triangles

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.

Which of the following scenarios is possible ?

I)  is an acute triangle.

II)  is an obtuse triangle with  the obtuse angle.

III)  is an obtuse triangle with  the obtuse angle.

IV)  is an obtuse triangle with  the obtuse angle.

Possible Answers:

I only

I and III only

II, III, and IV only

II and IV only

I, II, III, and IV

Correct answer:

I and III only

Explanation:

Corresponding angles of similar triangles are congruent, so  and . Since , it follows that . Since a triangle cannot have two angles that measure more than , both  and  are acute. No information is given about , so  can be acute, right, or obtuse. Therefore, scenarios (I) and (III) are possible, but not (II) or (IV).

Example Question #53 : Acute / Obtuse Triangles

.

Which of the following is true about  ?

Possible Answers:

 is isosceles and obtuse.

 is scalene and acute.

 is scalene and obtuse.

 is isosceles and acute.

None of the other responses is correct.

Correct answer:

 is isosceles and acute.

Explanation:

Corresponding sides of similar triangles are in proportion, so 

 and .

Substituting, we have from the first statement

Since  is isosceles. 

We can compare the sum of the squares of the lesser two sides to that of the greatest. 

The sum of the squares of the lesser two sides is greater than the square of the third, so  is acute.

Example Question #51 : Acute / Obtuse Triangles

 

Which of the following is true about ?

Possible Answers:

None of the other responses is correct.

 is isosceles and acute.

 is scalene and acute.

 is isosceles and obtuse.

 is scalene and obtuse.

Correct answer:

 is isosceles and acute.

Explanation:

Corresponding angles of similar triangles are congruent, so the measures of the angles of  are equal to those of .

, so . Also , so

.

All three angles have measure less than , so  is acute. Also, two of the angles are congruent, so by the Converse of the Isosceles Triangle Theorem,  is isosceles.

Example Question #51 : Acute / Obtuse Triangles

.

Which of the following is true about ?

Possible Answers:

 is scalene and acute.

 is isosceles and obtuse.

None of the other responses is correct.

 is isosceles and acute.

 is scalene and obtuse.

Correct answer:

 is isosceles and acute.

Explanation:

, so corresponding sides are in proportion; it follows that 

Therefore,  is isosceles.

Also, corresponding angles are congruent, so if  acute (or obtuse), so is . We can compare the sum of the squares of the lesser two sides to that of the greatest;

The sum of the squares of the lesser sides is greater than the square of the greatest side, so  is acute - and so is . The correct response is that  is isosceles and acute.

Example Question #11 : How To Find If Two Acute / Obtuse Triangles Are Similar

Which of the following statements would prove that the statement 

 

is false?

Possible Answers:

None of the other statements alone would prove the statement  to be false.

 and  have different perimeters

 and  have different areas

Correct answer:

Explanation:

Triangles that are similar need not have congruent sides, so it does not follow that , or that their perimeters are equal. Consequently, their areas need not be equal either.

However, if , then corresponding angles are congruent; specifically,   and . Therefore, . Contrapositively, if , then .

Example Question #53 : Acute / Obtuse Triangles

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.

What is the ratio of the area of  to that of  ?

Possible Answers:

Correct answer:

Explanation:

The similarity ratio of two triangles is the ratio of the lengths of their corresponding sides.

The similarity ratio of  to  is 

.

The similarity ratio of  to  is 

Multipliy these to get the similarity ratio of  to :

The ratio of the areas of two similar figures is the square of their similarity ratio, so the ratio of the areas of the triangles is 

.

The correct choice is .

Example Question #51 : Acute / Obtuse Triangles

Given:  and .

Which of the following statements would not be enough, along with what is given, to prove that ?

Possible Answers:

The given information is enough to prove the triangles similar.

Correct answer:

Explanation:

From both the given proportion statement and either  or , it follows that —all three pairs of corresponding sides are in proportion; by the Side-Side-Side Similarity Theorem,  . From the given proportion statement and , since these are the included angles of the sides that are in proportion, then by the Side-Angle-Side Similarity Theorem,  . From the given proportion statement and , since these are nonincluded angles of the sides that are in proportion, no similarity can be deduced.

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