Acute / Obtuse Triangles - ISEE Upper Level Quantitative Reasoning

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Question

Examine the above diagram. If $l \parallel m$ , give in terms of .

Answer

The two marked angles are same-side exterior angles of parallel lines, which are supplementary - that is, their measures have sum 180. We can solve for in this equation:

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Question

Examine the above diagram. If $l \parallel m$ , give in terms of .

Answer

The two marked angles are same-side interior angles of parallel lines, which are supplementary - that is, their measures have sum 180. We can solve for in this equation:

$2y \div 2 = \left ( 184 -x \right )\div 2$

$y= 92 - \frac{1}{2}x$

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Question

Examine the above diagram. Which of the following statements must be true whether or not and are parallel?

Answer

Four statements can be eliminated by the various parallel theorems and postulates. Congruence of alternate interior angles or corresponding angles forces the lines to be parallel, so

$\angle 5 \cong \angle 4 \Rightarrow l \parallel m$ and

$\angle 2 \cong \angle 6 \Rightarrow l \parallel m$ .

Also, if same-side interior angles or same-side exterior angles are supplementary, the lines are parallel, so

$m \angle 4 + m \angle 7 = 180 ^{\circ } \Rightarrow l \parallel m$ and

$m \angle 1 + m \angle 6 = 180 ^{\circ } \Rightarrow l \parallel m$ .

However, $\angle 5 \cong \angle 8$ whether or not $l \parallel m$ since they are vertical angles, which are always congruent.

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Question

Examine the above diagram. What is ?

Answer

By angle addition,

$\left ( x - 13\right )+ 100 + (y + 25) = 180$

$x - 13\right+ 100 + y + 25 = 180$

$x + y - 13\right+ 100+ 25 = 180$

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Question

$\angle 1$ and $\angle 2$ are supplementary; $\angle 1$ and $\angle 3$ are complementary.

$m \angle 2 = 100 ^{\circ }$ .

What is $m \angle 3$ ?

Answer

Supplementary angles and complementary angles have measures totaling $180^{\circ }$ and $90^{\circ }$ , respectively.

$m \angle 2 = 100 ^{\circ }$ , so its supplement $\angle 1$ has measure

$m \angle 1 = 180^{\circ } - m \angle 2 = 180^{\circ } - 100 ^{\circ } = 80^{\circ }$

$\angle 3$ , the complement of $\angle 1$ , has measure

$m \angle 3 = 90^{\circ } - m \angle 1 = 90^{\circ } - 80 ^{\circ } = 10^{\circ }$

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Question

Note: Figure NOT drawn to scale.

In the above figure, $m \angle 1=\left ( x+17 \right )^{\circ}$ and $m \angle 2= \left (7x+11 \right )^{\circ}$ . Which of the following is equal to $m \angle 1$ ?

Answer

$\angle 1$ and $\angle 2$ form a linear pair, so their angle measures total $180^{\circ}$ . Set up and solve the following equation:

$m \angle 1+ m \angle 2 = 180^{\circ}$

$\left ( x+17 \right ) + \left (7x+11 \right ) = 180$

$m \angle 1=\left ( x+17 \right )^{\circ} = \left ( 19+17 \right )^{\circ} = 36^{\circ}$

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Question

Two angles which form a linear pair have measures $(2x+72)^{\circ}$ and $(5x-125)^{\circ}$ . Which is the lesser of the measures (or the common measure) of the two angles?

Answer

Two angles that form a linear pair are supplementary - that is, they have measures that total $180^{\circ}$ . Therefore, we set and solve for in this equation:

$x = \frac{233}{7}= 33 \frac{2}{7}$

The two angles have measure

$2 \cdot 33 \frac{2}{7} +72= 138 \frac{4}{7} ^{\circ}$

and

$5 \cdot 33 \frac{2}{7} -125 = 41\frac{3}{7}^{\circ}$

$41\frac{3}{7}^{\circ}$ is the lesser of the two measures and is the correct choice.

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Question

Two vertical angles have measures $(3x+14)^{\circ }$ and $\left ( 5x-17\right )^{\circ }$ . Which is the lesser of the measures (or the common measure) of the two angles?

Answer

Two vertical angles - angles which share a vertex and whose union is a pair of lines - have the same measure. Therefore, we set up and solve the equation

$x= 15\frac{1}{2}$

$3x+14 = 3 \cdot 15\frac{1}{2} + 14 = 46\frac{1}{2} + 14= 60\frac{1}{2} ^{\circ }$

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Question

A line intersects parallel lines and . $\angle 1$ and $\angle 2$ are corresponding angles; $\angle 1$ and $\angle 3$ are same side interior angles.

$m \angle 1 = \left (3x+2y \right )^{\circ }$

$m \angle 2 = \left ( 4x+21\right )^{\circ }$

$m \angle 3 = \left ( 2y-27\right )^{\circ}$

Evaluate .

Answer

When a transversal such as crosses two parallel lines, two corresponding angles - angles in the same relative position to their respective lines - are congruent. Therefore,

Two same-side interior angles are supplementary - that is, their angle measures total 180 - so

We can solve this system by the substitution method as follows:

Backsolve:

, which is the correct response.

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Question

Note: Figure NOT drawn to scale.

Refer to the above diagram. Give the measure of $\angle 1$ .

Answer

The top and bottom angles, being vertical angles - angles which share a vertex and whose union is a pair of lines - have the same measure, so

,

or, simplified,

The right and bottom angles form a linear pair, so their degree measures total 180. That is,

Substitute for :

The left and right angles, being vertical angles, have the same measure, so, since the right angle measures $\left (8x \right )^{\circ } =\left (8 \cdot 15 \right )^{\circ } = 120 ^{\circ }$ , this is also the measure of the left angle, $\angle 1$ .

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Question

Figure NOT drawn to scale

The above figure shows Trapezoid , with $\overline{RT}$ and $\overline{RA }$ tangent to the circle. $m \angle T = 106 ^{\circ }$ ; evaluate $m \angle A$ .

Answer

By the Same-Side Interior Angle Theorem, since $\overline{TR} || \overline{AP}$ , $\angle T$ and $\angle P$ are supplementary - that is, their degree measures total $180 ^{\circ }$ . Therefore,

$m \angle P + m \angle T = 180 ^{\circ }$

$m \angle P = 180 ^{\circ } - m \angle T$

$m \angle P = 180 ^{\circ } - 106 ^{\circ } = 74^{\circ }$

$\angle P$ is an inscribed angle, so the arc it intercepts, $\overarc{TA}$ , has twice its degree measure;

$m \overarc{TA} = 2 \cdot m \angle P =2 \cdot 74^{\circ } = 148 ^{\circ }$ .

The corresponding major arc, $\overarc{TPA}$ , has as its measure

$m \overarc{TPA} = 360 ^{\circ }- m \overarc{TA} = 360 ^{\circ }- 148 ^{\circ }= 212 ^{\circ }$

The measure of an angle formed by two tangents to a circle is equal to half the difference of those of its intercepted arcs:

$m \angle R = \frac{1}{2} \left (m \overarc{TPA} - m \overarc{TA} \right )$

$m \angle R = \frac{1}{2} \left (212 ^{\circ } -148 ^{\circ } \right )= \frac{1}{2} \cdot 64 ^{\circ } = 32^{\circ }$

Again, by the Same-Side Interior Angles Theorem, $\angle R$ and $\angle A$ are supplementary, so

$m \angle A + m \angle R= 180 ^{\circ }$

$m \angle A = 180 ^{\circ } - m \angle R$

$m \angle A = 180 ^{\circ } - 32 ^{\circ } = 148^{\circ }$

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Question

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

Answer

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.

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Question

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

Answer

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.

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Question

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

Answer

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 $180^{\circ }$ ,

$2x \div 2= 70 \div 2$

Both angles measure $35^{\circ}$

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Question

Note: Figure NOT drawn to scale.

What is the measure of angle

Answer

The two angles at bottom are marked as congruent. One forms a linear pair with a $152 ^{\circ }$ angle, so it is supplementary to that angle, making its measure $(180-152)^{\circ } = 28^{\circ }$ . Therefore, each marked angle measures $28^{\circ }$ .

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

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Question

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

Answer

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

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

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Question

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

Evaluate .

Answer

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

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

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

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Question

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 $\angle A$ ?

Answer

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 $40 ^{\circ }$

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Question

Solve for :

Answer

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

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Question

Figure NOT drawn to scale.

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

Answer

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,

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

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