ISEE Upper Level Quantitative Reasoning - How to find an angle in an acute / obtuse triangle

Triangle

Figure NOT drawn to scale.

Refer to the above figure. Evaluate $m \angle ACB$ .

$102 ^{\circ }$

$97^{\circ }$

$107^{\circ }$

$112^{\circ }$

Explanation

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

$m \angle A + m \angle B = m \angle ACD$

$6 \cdot 2 = 0.5x \cdot 2$

$m \angle ACD =( 6.5 x) ^{\circ }= (6.5 \cdot 12 )^{\circ }= 78^{\circ }$

$\angle ACB$ and $\angle ACD$ form a linear pair and, therefore,

$m \angle ACB=180 ^{\circ } - m \angle ACD = 180 ^{\circ } -78^{\circ } = 102 ^{\circ }$ .

The acute angles of a right triangle measure $\left ( 2x - 8 \right )^{\circ }$ and $\left ( 3x - 4 \right )^{\circ }$ .

Evaluate .

Explanation

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

$\left ( 2x - 8 \right ) + \left ( 3x - 4 \right ) = 90$

$5x \div 5 = 102 \div 5$

The angles of a triangle measure $x^{\circ }, x^{\circ },\left ( x -42 \right ) ^{\circ }$ . Evaluate .

Explanation

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

$3x \div 3 = 222 \div 3$

Which of the following is true about a triangle with two angles that measure $120^{\circ }$ and $90^{\circ }$ ?

This triangle cannot exist.

This triangle is scalene and obtuse.

This triangle is scalene and right.

This triangle is isosceles and obtuse.

This triangle is isosceles and right.

Explanation

A triangle must have at least two acute angles; however, a triangle with angles that measure $120^{\circ }$ and $90^{\circ }$ could have at most one acute angle, an impossible situation. Therefore, this triangle is nonexistent.

Triangle 3

In the above figure, $\overline{AC} \cong \overline{AD}$ .

Give the measure of $\angle CAD$ .

$36 ^{\circ }$

$42^{\circ }$

$32 ^{\circ }$

$40 ^{\circ }$

Explanation

$\angle BCA$ and $\angle ACD$ form a linear pair, so their degree measures total $180 ^{\circ }$ ; consequently,

$m \angle ACD = 180 ^{\circ } - m \angle CAD$

$= 180 ^{\circ } - 108 ^{\circ }$

$= 72 ^{\circ }$

$\overline{AC} \cong \overline{AD}$ , so by the Isosceles Triangle Theorem,

$m \angle ADC = m \angle ACD = 72 ^{\circ }$

The sum of the degree measures of a triangle is $180^{\circ }$ , so

$\angle CAD + m \angle ADC + m \angle ACD = 180 ^{\circ }$

$\angle CAD + 72^{\circ } +72^{\circ } = 180 ^{\circ }$

$\angle CAD +144^{\circ } = 180 ^{\circ }$

$\angle CAD = 36 ^{\circ }$

Which of the following is true about a triangle with two angles that measure $91^{\circ }$ each?

The triangle cannot exist.

The triangle is acute and scalene.

The triangle is obtuse and scalene.

The triangle is acute and isosceles.

The triangle is obtuse and isosceles.

Explanation

A triangle must have at least two acute angles; however, a triangle with angles that measure $91 ^{\circ }$ would have two obtuse angles and at most one acute angle. This is not possible, so this triangle cannot exist.

Solve for :
Question11

Explanation

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

Chords

Note: Figure NOT drawn to scale

Refer to the above figure. $\overline{AB } \cong \overline{AC}$ ; $m \widehat{AC} = 140 ^{\circ }$ .

What is the measure of $\angle A$ ?

$40 ^{\circ }$

$60 ^{\circ }$

$70 ^{\circ }$

$50 ^{\circ }$

Explanation

Congruent chords of a circle have congruent minor arcs, so since $\overline{AB } \cong \overline{AC}$ , $\widehat{AB } \cong \widehat{AC}$ , and their common measure is $m\widehat{AB } =m \widehat{AC} = 140^{\circ}$ .

Since there are $360^{\circ}$ in a circle,

$m\widehat{BC } + m\widehat{AB } + m \widehat{AC} = 360^{\circ}$

$m\widehat{BC } + 140^{\circ} + 140^{\circ}= 360^{\circ}$

$m\widehat{BC } + 280^{\circ}= 360^{\circ}$

$m\widehat{BC } = 80^{\circ}$

The inscribed angle $\angle A$ intercepts this arc and therefore has one-half its degree measure, which is $40 ^{\circ }$

Triangle 2

Refer to the above figure. Express in terms of .

$y =\frac{1}{3} x + 8$

$y =\frac{1}{3} x - 8$

Explanation

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

$m \angle ABC = (180 - 82)^{\circ } = 98^{\circ }$

and

$m \angle ACB = (180 - x)^{\circ }$ .

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

$m \angle BAC + m \angle ABC +m \angle ACB = 180 ^{\circ }$

One angle of an isosceles triangle has measure $110^{\circ }$ . What are the measures of the other two angles?

$35^{\circ },35^{\circ }$

Not enough information is given to answer this question.

$110 ^{\circ }, 40 ^{\circ }$

$45^{\circ }, 45^{\circ }$

$70^{\circ }, 70^{\circ }$

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 $110^{\circ }$ 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 $180^{\circ }$ ,