# ISEE Upper Level Math : Acute / Obtuse Triangles

## Example Questions

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### Example Question #1 : Acute / Obtuse Triangles

NOTE: Figures NOT drawn to scale.

Refer to the above two triangles.

What is ?

Insufficient information is given to answer the question.

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  ?

It cannot be determined from the information given.

Explanation:

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

So,

The perimeter of  is

.

### Example Question #3 : Acute / Obtuse Triangles

What is  ?

It is impossible to tell from the information given.

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 ?

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.

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?

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.

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?

Not enough information is given to answer this question.

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

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 .

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

Note: Figure NOT drawn to scale

Refer to the above figure. .

What is the measure of  ?

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

Solve for :