Geometry - ISEE Upper Level Mathematics Achievement

Card 0 of 20

Question

Find the distance between and .

Answer

To find the distance, first remember the distance formula: $\sqrt{(x_{2}-x_{1})^2+(y_{2}-y_{1})^2}$ . Plug in so that you have: $\sqrt{(1+2)^2+(4-3)^2}$ . Simplify so that you get $\sqrt{3^2+1^2}$ . This yields $\sqrt{10}$ .

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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

If the sum of angles in a hexagon is equal to $650^\circ$ , what is the value of the remaining sixth angle?

Answer

The sum of the angles in a polygon can be found using the equation below, in which t is equal to the total sum of the angles, and n is equal to the number of sides.

Given that a hexagon has 6 angles, the total number of angles will be:

$t=4\cdot 180$

Given that 5 of the angles have a sum of 650 degrees, we would subtract 650 from 720, resulting in 70 degrees.

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Question

Solve for :

Answer

Find the sum of the interior angles of the polygon using the following equation where n is equal to the number of sides.

The sum of the angles must equal 360.

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Question

The measures of the angles of a pentagon are $x^{\circ }, x^{\circ }, x^{\circ }, y^{\circ }, y^{\circ }$ . If , what is ?

Answer

The angles of a pentagon measure a total of $180 (5-2) = 540 ^{\circ }$ . From the information, we know that:

If , then

$3x + 2 \cdot 100 = 540$

$x = 340 \div 3$

$x = 113 \frac{1}{3}^{\circ}$

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Question

A convex pentagon has four angles that measure $150^{\circ }$ each. What is the measure of the fifth angle?

Answer

The angles of a pentagon measure a total of $180 (5-2) = 540 ^{\circ }$ . If we let the unknown angle measure be $x^{\circ }$ , then from this information:

Since an angle measure cannot be negative, this pentagon cannot exist.

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Question

The measures of the angles of a pentagon are $x ^{\circ }, x ^{\circ }, x ^{\circ }, y ^{\circ }, y ^{\circ }$ . If $x = 100 ^{\circ }$ , what is ?

Answer

The angles of a pentagon measure a total of $180 (5-2) = 540 ^{\circ }$ . From the information, we know:

If $x = 100 ^{\circ }$ , then the above becomes:

$3 \cdot 100 + 2y = 540$

$2y \div 2 = 240 \div 2$

$y = 120^{\circ}$

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Question

Solve for :

Answer

The sum of the interior angles of a pentagon can be determined by the following equation, where is the number of sides:

Therefore:

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Question

Consider the rhombus below. Solve for .

Answer

The total sum of the interior angles of a quadrilateral is degrees. In this problem, we are only considering half of the interior angles:

$\frac{360}{2}=180$

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Question

Note: Figure NOT drawn to scale.

The above depicts a rhombus and one of its diagonals. What is ?

Answer

The diagonals of a rhombus bisect the angles.

The angle bisected must be supplementary to the $122^{\circ }$ angle since they are consecutive angles of a parallelogram; therefore, that angle has measure $\left (180 - 122 \right ) ^{\circ } = 58^{\circ }$ , and is half that, or $29^{\circ }$ .

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