ISEE Upper Level Math : Acute / Obtuse Triangles

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

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Example Questions

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Example Question #1 : Acute / Obtuse Triangles

Similar

NOTE: Figures NOT drawn to scale.

Refer to the above two triangles. 

What is ?

Possible Answers:

Insufficient information is given to answer the question.

Correct answer:

Explanation:

Corresponding sides of similar triangle are proportional, so if 

, then 

Substitute the known sidelengths, then solve for :

Example Question #1 : Acute / Obtuse Triangles

What is the perimeter of  ?

Possible Answers:

It cannot be determined from the information given.

Correct answer:

Explanation:

By definition, since, , side lengths are in proportion.

So,

 

The perimeter of  is

.

Example Question #3 : Acute / Obtuse Triangles

What is  ?

Possible Answers:

It is impossible to tell from the information given.

Correct answer:

Explanation:

By definition, since , all side lengths are in proportion.

 

Example Question #1 : Acute / Obtuse Triangles

Which of the following is true about a triangle with two angles that measure  and ?

Possible Answers:

This triangle cannot exist.

This triangle is scalene and obtuse.

This triangle is isosceles and obtuse.

This triangle is scalene and right.

This triangle is isosceles and right.

Correct answer:

This triangle cannot exist.

Explanation:

A triangle must have at least two acute angles; however, a triangle with angles that measure  and  could have at most one acute angle, an impossible situation. Therefore, this triangle is nonexistent.

Example Question #81 : Plane Geometry

Which of the following is true about a triangle with two angles that measure  each?

Possible Answers:

The triangle cannot exist.

The triangle is obtuse and scalene.

The triangle is obtuse and isosceles.

The triangle is acute and scalene.

The triangle is acute and isosceles.

Correct answer:

The triangle cannot exist.

Explanation:

A triangle must have at least two acute angles; however, a triangle with angles that measure  would have two obtuse angles and at most one acute angle. This is not possible, so this triangle cannot exist.

Example Question #1 : Solve Simple Equations For An Unknown Angle In A Figure: Ccss.Math.Content.7.G.B.5

One angle of an isosceles triangle has measure . What are the measures of the other two angles?

Possible Answers:

Not enough information is given to answer this question.

Correct answer:

Explanation:

An isosceles triangle not only has two sides of equal measure, it has two angles of equal measure. This means one of two things, which we examine separately:

Case 1: It has another  angle. This is impossible, since a triangle cannot have two obtuse angles.

Case 2: Its other two angles are the ones that are of equal measure. If we let  be their common measure, then, since the sum of the measures of a triangle is 

Both angles measure 

Example Question #7 : Acute / Obtuse Triangles

The angles of a triangle measure . Evaluate 

Possible Answers:

Correct answer:

Explanation:

The sum of the degree measures of the angles of a triangle is 180, so we solve for  in the following equation:

Example Question #8 : Acute / Obtuse Triangles

The acute angles of a right triangle measure  and 

Evaluate .

Possible Answers:

Correct answer:

Explanation:

The degree measures of the acute angles of a right triangle total 90, so we solve for  in the following equation:

Example Question #9 : Acute / Obtuse Triangles

Chords

Note: Figure NOT drawn to scale

Refer to the above figure. .

What is the measure of  ?

Possible Answers:

Correct answer:

Explanation:

Congruent chords of a circle have congruent minor arcs, so since , and their common measure is .

Since there are  in a circle, 

The inscribed angle  intercepts this arc and therefore has one-half its degree measure, which is 

Example Question #10 : Acute / Obtuse Triangles

Solve for :
Question11

Possible Answers:

Correct answer:

Explanation:

The sum of the internal angles of a triangle is equal to . Therefore:

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