Acute / Obtuse Triangles - ISEE Upper Level Quantitative Reasoning

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.

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.

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

The two angles at bottom are marked as congruent. One forms a linear pair with a angle, so it is supplementary to that angle, making its measure . Therefore, each marked angle measures .

The sum of the measures of the interior angles of a triangle is , so:

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

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

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

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

The measure of an exterior angle of a triangle, which here is , is equal to the sum of the measures of its remote interior angles, which here are and . Consequently,

and form a linear pair and, therefore,

.

The measure of an interior angle of a triangle is equal to 180 degrees minus that of its adjacent exterior angle, so

and

.

The sum of the degree measures of the three interior angles is 180, so

and form a linear pair, so their degree measures total ; consequently,

, so by the Isosceles Triangle Theorem,

The sum of the degree measures of a triangle is , so

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.

,

where and are the lengths of two sides and is the angle measure.

Plug in our given values:

,

where is the base and is the altitude.

The base can be seen as the (horizontal) line segment connecting and , the length of which is . The height is the pependicular distance from to the segment; since the segment is part of the -axis, this altitude is vertical and has a length equal to -coordinate .

The area of this triangle is therefore

.

The base can be seen as the (vertical) line segment connecting and , which has length . The height is the pependicular distance from to the segment; since the segment is part of the -axis, this altitude is horizontal and has length equal to -coordinate .

The area of this triangle is therefore

.

The altitude of a right triangle from the vertex of its right angle - which, here, is - divides the triangle into two triangles similar to each other as well as the large triangle.

The similarity ratio of to is the ratio of the lengths of their hypotenuses. The hypotenuse of the latter is 18; that of the former, from the Pythagorean Theorem, is

The similarity ratio is therefore . The ratio of their areas is the square of this, or

The area of is

of that of , so the choice closest to the correct percent is 25%.

A scalene triangle, by definition, has three sides of different lengths. We can identify the scalene triangle by converting measures to the same unit. We will convert to inches in this case.

5 feet = inches. This triangle can be eliminated, since two sides have length 60 inches.

7 feet = inches. This triangle can be eliminated, since two sides have length 84 inches.

These three measures are equal. This is an equilateral triangle, and it can be eliminated.

feet = inches. This triangle can be eliminated, since two sides have length 66 inches.

5 feet = 60 inches.

6 feet = inches.

The sides of this triangle measure 60, 72, and 84 inches, so the triangle is scalene.

,

where and are the lengths of two sides and is the angle measure.

Plug in our given values: