DSQ: Calculating an angle of a line - GMAT Quantitative

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Question

$\angle 1$ and $\angle2$ are supplementary angles. Which one has the greater measure?

Statement 1: $m \angle 1 < 100$

Statement 2: $\angle2$ is an obtuse angle.

Answer

By definition, if $\angle 1$ and $\angle2$ are supplementary angles, then $m\angle 1 + m\angle 2 = 180$ .

If Statement 1 is assumed and $m \angle 1 < 100$ , then $m\angle 2 > 80$ . This does not answer our question, since, for example, it is possible that $m \angle 1 = 91$ and $m \angle 2 = 89$ , or vice versa.

If Statement 2 is assumed, then $m\angle 2 > 90$ , and subsequently, $m \angle 1 < 90$ ; by transitivity, $m\angle 2 > m\angle 1$ .

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Question

Note: Figure NOT drawn to scale.

Refer to the above diagram.

What is the measure of $\angle FCB$ ?

Statement 1: $m \angle FBC = 43 ^{\circ }$

Statement 2: $m \angle DCG = 43 ^{\circ }$

Answer

If we only know that $m \angle FBC = 43 ^{\circ }$ , then we cannot surmise anything from the diagram about the measure of $\angle FCB$ . But $\angle FCB$ and $\angle DCG$ are vertical angles, which must be congruent, so if we know $m \angle DCG = 43 ^{\circ }$ , then $\angle FCB= 43 ^{\circ }$ also.

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Question

What is the measure of $\angle 1$ ?

Statement 1: $\angle 1$ is complementary to an angle that measures $48 ^{\circ }$ .

Statement 2: $\angle 1$ is adjacent to an angle that measures $42 ^{\circ }$ .

Answer

Complementary angles have degree measures that total $90 ^{\circ }$ , so the measure of an angle complementary to a $48 ^{\circ }$ angle would have measure $(90-48)^{\circ } = 42 ^{\circ }$ . If Statement 1 is assumed, then $m \angle 1= 42 ^{\circ }$ .

Statement 2 gives no useful information. Adjacent angles do not have any numerical relationship; they simply share a ray and a vertex.

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Question

Find the angle made by and the -axis.

I) goes through the origin and the point .

II) makes a degree angle between itself and the -axis.

Answer

To find the angle of the line, recall that each quadrant has 90 degrees

I) Tells us that the line has a slope of one. This means that if we make a triangle using our line, the x-axis and a line coming up from the x-axis at 90 degrees we will have a 45/45/90 triangle. Therefore, I) tells us that our angle is 45 degrees.

II) Tells us that the line makes a 45 degree angle between itself and the y-axis. Therefore:

$\theta=90^{\circ}-45^{\circ}=45^{\circ}$

Therfore, we could use either statement.

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Question

Note: Figure NOT drawn to scale.

Refer to the above diagram. Evaluate $m \angle 3+ m \angle 4$ .

Statement 1: $m \angle1 + m \angle 5 = 120^{\circ }$

Statement 2: $\bigtriangleup OPQ$ is an equilateral triangle.

Answer

Assume Statement 1 alone. $\angle 1$ and $\angle 4$ are a pair of vertical angles, as are $\angle 3$ and $\angle 5$ . Therefore,

$m \angle 4 = m \angle 1$

$m \angle 3 = m \angle 5$

By substitution,

$m \angle 3+ m \angle 4 = m \angle 1 + m \angle 5 = 120 ^{\circ}$ .

Assume Statement 2 alone. The angles of an equilateral triangle all measure $60^{\circ }$ , so $m \angle POQ = 60 ^{\circ }$ .

$\angle 3$ , $\angle 4$ , and $\angle POQ$ together form a straight angle, so ,

$m \angle 3 + m \angle 4 + m \angle POQ= 180^{\circ }$

$m \angle 3 + m \angle 4 +60^{\circ } = 180^{\circ }$

$m \angle 3+ m \angle 4 = 120 ^{\circ}$

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Question

Note: Figure NOT drawn to scale.

Refer to the above diagram. What is the measure of $\angle 1$ ?

Statement 1: $\bigtriangleup OPQ$ is an equilateral triangle.

Statement 2: $m \angle 3 = 65^{\circ }$

Answer

$\angle 1$ , $\angle 2$ , and $\angle 3$ together form a straight angle, so their measures total $180^{\circ }$ ; therefore,

$m \angle 1+m \angle2+ m \angle 3 = 180^{\circ }$

Assume Statement 1 alone. The angles of an equilateral triangle all measure $60^{\circ }$ , so $m \angle POQ = 60 ^{\circ }$ ; $\angle POQ$ and $\angle 2$ form a pair of vertical angles, so they are congruent, and consequently, $m \angle 2 = 60^{\circ }$ . Therefore,

$m \angle 1+ 60^{\circ } + m \angle 3 = 180^{\circ }$

$m \angle 1 + m \angle 3 = 120^{\circ }$

But with no further information, $m \angle 1$ cannot be calculated.

Assume Statement 2 alone. It follows that

$m \angle 1+m \angle2+ 65^{\circ }= 180^{\circ }$

$m \angle 1+m \angle2 =115^{\circ }$

Again, with no further information, $m \angle 1$ cannot be calculated.

Assume both statements to be true. $m \angle 2 = 60^{\circ }$ as a result of Statement 1, and $m \angle 3 = 65^{\circ }$ from Statement 2, so

$m \angle 1+60^{\circ } + 65^{\circ }= 180^{\circ }$

$m \angle 1+125^{\circ }= 180^{\circ }$

$m \angle 1=55^{\circ }$

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Question

Note: Figure NOT drawn to scale.

Refer to the above diagram. Evaluate $m \angle 1+ m \angle 2$ .

Statement 1: $m \angle 2+ m \angle 3 = 122^{\circ }$

Statement 2: $m \angle 3+ m \angle 4 = 126^{\circ }$

Answer

Assume Statement 1 alone. $\angle 1$ , $\angle 2$ , and $\angle 3$ together form a straight angle, so their measures total $180^{\circ }$ ; therefore,

$m \angle 1+m \angle2+ m \angle 3 = 180^{\circ }$

$m \angle 1+122 ^{\circ } = 180^{\circ }$

$m \angle 1=58^{\circ }$

However, without any further information, we cannot determine the sum of the measures of $\angle 1$ and $\angle 2$ .

Assume Statement 2 alone. $\angle 2$ , $\angle 3$ , and $\angle 4$ together form a straight angle, so their measures total $180^{\circ }$ ; therefore,

$m \angle2+ m \angle 3 + m \angle 4= 180^{\circ }$

$m \angle 2+126 ^{\circ } = 180^{\circ }$

$m \angle 2 =54^{\circ }$

Again, without any further information, we cannot determine the sum of the measures of $\angle 1$ and $\angle 2$ .

Assume both statements are true. Since the measures of $\angle 1$ and $\angle 2$ can be calculated from Statements 1 and 2, respectively. We can add them:

$m \angle 1+ m \angle 2 = 58^{\circ }+ 54^{\circ }=112^{\circ }$

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Question

Note: Figure NOT drawn to scale.

Refer to the above diagram. What is the measure of $\angle 1$ ?

Statement 1: $m \angle 3+ m \angle 6 = 132^{\circ }$

Statement 2: $m \angle 1+ m \angle 4 = 140^{\circ }$

Answer

Assume Statement 1 alone. $\angle 3$ and $\angle 6$ are a pair of vertical angles and are therefore congruent, so the statement

$m \angle 3+ m \angle 6 = 132^{\circ }$

can be rewritten as

$m \angle 3+ m \angle3 = 132^{\circ }$

$2 \cdot m \angle3 = 132^{\circ }$

$m \angle3 = 66^{\circ }$

$\angle 1$ , $\angle 2$ , and $\angle 3$ together form a straight angle, so their measures total $180^{\circ }$ ; therefore,

$m \angle 1+m \angle2+ m \angle 3 = 180^{\circ }$

$m \angle 1+m \angle2+ 66 ^{\circ } = 180^{\circ }$

$m \angle 1+m \angle2 = 114^{\circ }$

But without further information, the measure of $\angle 1$ cannot be calculated.

Assume Statement 2 alone. $\angle 1$ and $\angle 4$ are a pair of vertical angles and are therefore congruent, so the statement

$m \angle 1+ m \angle 4 = 140^{\circ }$

can be rewritten as

$m \angle 1+ m \angle 1 = 140^{\circ }$

$2 \cdot m \angle 1 = 140^{\circ }$

$m \angle 1 = 70^{\circ }$

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Question

Note: Figure NOT drawn to scale.

Refer to the above diagram. What is the measure of $\angle 1$ ?

Statement 1: $\angle 6$ is a $54 ^{\circ }$ angle.

Statement 2: $\angle 2 \cong \angle 3$

Answer

Statement 1 alone gives insufficient information to find the measure of $\angle 1$ .

$\angle 6$ , $\angle 1$ , and $\angle 2$ together form a $180^{\circ }$ angle; therefore,

$m \angle 1+m \angle2+ m \angle 6 = 180^{\circ }$

$m \angle 6 = 54^{\circ }$ , so by substitution,

$m \angle 1+m \angle2+ 54^{\circ } = 180^{\circ }$

$m \angle 1+m \angle2= 126^{\circ }$

But with no further information, the measure of $\angle 1$ cannot be calculated.

Statement 2 alone gives insufficient information for a similar reason. $\angle 1$ , $\angle 2$ , and $\angle 3$ together form a $180^{\circ }$ angle; therefore,

$m \angle 1+m \angle2+ m \angle 3 = 180^{\circ }$

Since $\angle 2 \cong \angle 3$ , we can rewrite this statement as

$m \angle 1+m \angle2+ m \angle 2 = 180^{\circ }$

$m \angle 1+2 \cdot m \angle2 = 180^{\circ }$

Again, with no further information, the measure of $\angle 1$ cannot be calculated.

Assume both statements to be true. $\angle 2$ and $\angle 6$ are a pair of vertical angles, so $\angle 2 \cong \angle 6$ , and $m \angle 2 = m \angle 6 = 54^{\circ }$ . Since $\angle 2 \cong \angle 3$ , then $m \angle 3 = m \angle 2 = 54^{\circ }$ . Also,

$m \angle 1+m \angle2+ m \angle 3 = 180^{\circ }$

By substitution,

$m \angle 1+ 54^{\circ }+ 54^{\circ } = 180^{\circ }$

$m \angle 1+108^{\circ } = 180^{\circ }$

$m \angle 1=72^{\circ }$

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Question

Note: Figure NOT drawn to scale.

Refer to the above diagram. What is the measure of $\angle 1$ ?

Statement 1: $m \angle 1 = 30 ^{\circ } + 2 \cdot m \angle 2$

Statement 2: $\angle 3$ is a $130 ^{\circ }$ angle.

Answer

Assume Statement 1 alone. Since $\angle 1$ and $\angle 2$ form a linear pair, their measures total $180^{\circ }$ . Therefore, this fact, along with Statement 1, form a system of linear equations, which can be solved as follows:

$m \angle 1 = 30 ^{\circ } + 2 \cdot m \angle 2$

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

The second equation can be rewritten as

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

and a substitution can be made:

$m \angle 1 = 30 ^{\circ } + 2 \cdot ( 180^{\circ } - m \angle 1)$

$m \angle 1 = 30 ^{\circ } + 360^{\circ } - 2 \cdot m \angle 1$

$m \angle 1 = 390^{\circ } - 2 \cdot m \angle 1$

$m \angle 1 + 2 \cdot m \angle 1 = 390^{\circ } - 2 \cdot m \angle 1 + 2 \cdot m \angle 1$

$3 \cdot m \angle 1 = 390^{\circ }$

$3 \cdot m \angle 1 \div 3 = 390^{\circ } \div 3$

$m \angle 1 = 130^{\circ }$

Assume Statement 2 alone. $\angle 1$ and $\angle 3$ are a pair of vertical angles, which have the same measure, so $m \angle 1=m \angle 3 = 130^{\circ }$ .

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Question

Note: You may assume that and are not parallel lines, but you may not assume that and are parallel lines unless it is specifically stated.

Refer to the above diagram. Is the sum of the measures of $\angle ADC$ and $\angle DCB$ less than, equal to, or greater than $180^{\circ }$ ?

Statement 1: $m \angle 1 + m \angle 6 = 181^{\circ }$

Statement 2: $m \angle 7 + m \angle 12 = 179^{\circ }$

Answer

Assume Statement 1 alone. $\angle ADC$ and $\angle 1$ form a linear pair of angles, so their measures total $180^{\circ }$ ; the same holds for $\angle DCB$ and $\angle 6$ . Therefore,

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

$m \angle 6 + m \angle DBC = 180^{\circ }$

$m \angle 1 +m \angle ADC+ m \angle 6 + m \angle DBC = 180^{\circ } + 180^{\circ }$

$( m \angle ADC+ m \angle DBC) + (m \angle 1 +m \angle 6 )=360^{\circ }$

$( m \angle ADC+ m \angle DBC) +181^{\circ }=360^{\circ }$

$m \angle ADC+ m \angle DBC=179^{\circ } < 180^{\circ }$

Assume Statement 2 alone. $\angle 12$ and $\angle DAB$ form a linear pair of angles, so their measures total $180^{\circ }$ ; the same holds for $\angle 7$ and $\angle ABC$ . Therefore,

$m \angle 12 + m \angle DAB= 180^{\circ }$

$m \angle7 + m \angle ABC= 180^{\circ }$

$m \angle 12 +m \angle DAB + m \angle 7 + m \angle ABC= 180^{\circ } + 180^{\circ }$

$( m \angle DAB+ m \angle ABC) + (m \angle 7 +m \angle 12 )=360^{\circ }$

$( m \angle DAB+ m \angle ABC) + 179^{\circ } =360^{\circ }$

$m \angle DAB+ m \angle ABC =181^{\circ }$

$\angle DAB$ , $\angle ABC$ , $\angle DCB$ , and $\angle ADC$ are the four angles of Quadrilateral , so their degree measures total 360. Therefore,

$m \angle ADC+ m \angle DCB + \left (m \angle DAB+ m \angle ABC \right ) =360^{\circ }$

$m \angle ADC+ m \angle DCB + 181 ^{\circ } =360^{\circ }$

$m \angle ADC+ m \angle DCB =179^{\circ }< 180^{\circ }$

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Question

Note: You may assume that and are not parallel lines, but you may not assume that and are parallel lines unless it is specifically stated.

Refer to the above diagram. Is the sum of the measures of $\angle ADC$ and $\angle DCB$ less than, equal to, or greater than $180^{\circ }$ ?

Statement 1: There exists a point such that lies on $\overline{ZD}$ and lies on $\overline{ZC}$ .

Statement 2: Quadrilateral is not a trapezoid.

Answer

Assume Statement 1 alone. Since $\overline{ZD}$ exists and includes , $\overleftrightarrow{ZD}$ and $\overleftrightarrow{DA}$ are one and the same—and this is . Similarly, $\overleftrightarrow{ZC}$ is . This means that and have a point of intersection, which is . Since falls between and and falls between and , the lines intersect on the side of that includes points and . By Euclid's Fifth Postulate, the sum of the measures of $\angle ADC$ and $\angle DCB$ is less than $180^{\circ }$ .

Assume Statement 2 alone. Since it is given that , the other two sides, $\overline{AD}$ and $\overline{BC}$ are parallel if and only if Quadrilateral is a trapezoid, which it is not. Therefore, $\overline{AD}$ and $\overline{BC}$ are not parallel, and the sum of the degree measures of same-side interior angles $\angle ADC$ and $\angle DCB$ is not equal to $180^{\circ }$ . However, without further information, it is impossible to determine whether the sum of the measures is less than or greater than $180^{\circ }$ .

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Question

Note: Figure NOT drawn to scale.

Refer to the above figure. Evaluate $m \angle 1 + m \angle 2$ .

Statement 1: $\angle 4$ and $\angle 5$ are complementary.

Statement 2: $OD \cdot \sqrt{2} = OE$

Answer

Assume Statement 1 alone. $\angle 5$ and $\angle 6$ are vertical from $\angle 1$ and $\angle 2$ , respectively, so $m \angle 1 = m \angle 2$ and $m \angle 2 = m \angle 6$ . $\angle 5$ and $\angle 6$ form a complementary pair, so, by definition

$m \angle 5 + m \angle 6 = 90^{\circ}$

and by substitution,

$m \angle 1 + m \angle 2 = 90^{\circ}$ .

Assume Statement 2 alone. Since $\bigtriangleup ODE$ is a right triangle whose hypotenuse is $\sqrt{2}$ times as long as a leg, it follows that $\bigtriangleup ODE$ is a 45-45-90 triangle, so $m \angle DOE = 45^{\circ }$ .

$\angle 1$ , $\angle 2$ , $\angle 3$ , and $\angle DOE$ together form a straight angle, so their degree measures total $180^{\circ }$ .

$m \angle 1 + m \angle 2+ m \angle 3+m \angle DOE = 180^{\circ }$

$m \angle 1 + m \angle 2+ m \angle 3+ 45^{\circ } = 180^{\circ }$

$m \angle 1 + m \angle 2+ m \angle 3 = 135^{\circ }$

But without further information, the sum of the degree measures of only $\angle 1$ and $\angle 2$ cannot be calculated.

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Question

Note: Figure NOT drawn to scale.

Refer to the above figure. Evaluate $m \angle 3$ .

Statement 1: $\overline{BO} \cong \overline{BC}$

Statement 2: $m \angle 5 = 42^{\circ }$

Answer

Assume Statement 1 alone. $\overline{BO}$ and $\overline{BC}$ are congruent legs of right triangle $\bigtriangleup OBC$ , so their acute angles, one of which is $\angle BOC$ , measure $45^{\circ }$ . $\angle BOC$ and $\angle 3$ form a pair of vertical, and consequently, congruent, angles, so $m \angle 3 = 45^{\circ }$ .

Statement 2 alone gives insufficient information, as $\angle 3$ and $\angle 5$ has no particular relationship that would lead to an arithmetic relationship between their angle measures.

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Question

Note: Figure NOT drawn to scale.

Refer to the above figure. Evaluate $m \angle 1+ m \angle 2$ .

Statement 1: $m \angle 3 + m \angle 4 + m \angle 5 = 131^{\circ }$

Statement 2: $m \angle 6 + m \angle 7 + m \angle 8 = 131^{\circ }$

Answer

Assume Statement 1 alone. $\angle 2$ , $\angle 3$ , $\angle 4$ , and $\angle 5$ together form a straight angle, so their degree measures total $180^{\circ }$ .

$m \angle 2 + m \angle 3+ m \angle 4+m \angle 5 = 180^{\circ }$

$m \angle 2 + 131^{\circ } = 180^{\circ }$

$m \angle 2 = 49^{\circ }$

Without further information, no other angle measures, including that of $\angle 1$ , can be found.

Assume Statement 2 alone. $\angle 1$ , $\angle 6$ , $\angle 7$ , and $\angle 8$ together form a straight angle, so their degree measures total $180^{\circ }$ .

$m \angle 1 + m \angle 6+ m \angle 7+m \angle 8 = 180^{\circ }$

$m \angle 1 + 131^{\circ } = 180^{\circ }$

$m \angle 1 = 49^{\circ }$

Without further information, no other angle measures, including that of $\angle 2$ , can be found.

However, if both statements are assumed to be true, it follows from Statements 2 and 1 respectively, as seen before, that $m \angle 1 = 49^{\circ }$ and $m \angle 2 = 49^{\circ }$ , so

$m \angle 1 + m \angle 2 = 49^{\circ }+ 49^{\circ } = 98^{\circ }$ .

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Question

Note: Figure NOT drawn to scale.

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

Statement 1: $m \angle 2 + m \angle 3 = 91^{\circ }$

Statement 2: $m \angle 5 + m \angle 6 = 91^{\circ }$

Answer

Assume both statements to be true. We show that the two statements provide insufficient information by exploring two scenarios:

Case 1: $m \angle 1 = m \angle 3 = 45 ^{\circ }, m \angle 2 = 46^{\circ }$

$m \angle 2 + m \angle 3 = 46 ^{\circ }+ 45^{\circ } = 91^{\circ }$

$\angle 5$ and $\angle 6$ are vertical from $\angle 1$ and $\angle 2$ , respectively, so $m \angle 5 = m \angle 1$ and $m \angle 6 = m \angle 2$ , and

$m \angle 5 + m \angle 6 =m \angle 1+ m \angle 2 = 45 ^{\circ }+ 46^{\circ } = 91^{\circ }$

Case 2: $m \angle 1 = m \angle 3 = 46 ^{\circ }, m \angle 2 = 45^{\circ }$

$m \angle 2 + m \angle 3 = 45 ^{\circ }+ 46^{\circ } = 91^{\circ }$

$m \angle 5 + m \angle 6 =m \angle 1+ m \angle 2 = 46 ^{\circ }+ 45^{\circ } = 91^{\circ }$

The conditions of both statements are met, but $\angle 1$ assumes a different value in each scenario.

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Question

$\angle 1$ and $\angle2$ are supplementary angles. Which one has the greater measure?

Statement 1: $m \angle 1 < 100$

Statement 2: $\angle2$ is an obtuse angle.

Answer

By definition, if $\angle 1$ and $\angle2$ are supplementary angles, then $m\angle 1 + m\angle 2 = 180$ .

If Statement 1 is assumed and $m \angle 1 < 100$ , then $m\angle 2 > 80$ . This does not answer our question, since, for example, it is possible that $m \angle 1 = 91$ and $m \angle 2 = 89$ , or vice versa.

If Statement 2 is assumed, then $m\angle 2 > 90$ , and subsequently, $m \angle 1 < 90$ ; by transitivity, $m\angle 2 > m\angle 1$ .

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Question

Note: Figure NOT drawn to scale.

Refer to the above diagram.

What is the measure of $\angle FCB$ ?

Statement 1: $m \angle FBC = 43 ^{\circ }$

Statement 2: $m \angle DCG = 43 ^{\circ }$

Answer

If we only know that $m \angle FBC = 43 ^{\circ }$ , then we cannot surmise anything from the diagram about the measure of $\angle FCB$ . But $\angle FCB$ and $\angle DCG$ are vertical angles, which must be congruent, so if we know $m \angle DCG = 43 ^{\circ }$ , then $\angle FCB= 43 ^{\circ }$ also.

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Question

What is the measure of $\angle 1$ ?

Statement 1: $\angle 1$ is complementary to an angle that measures $48 ^{\circ }$ .

Statement 2: $\angle 1$ is adjacent to an angle that measures $42 ^{\circ }$ .

Answer

Complementary angles have degree measures that total $90 ^{\circ }$ , so the measure of an angle complementary to a $48 ^{\circ }$ angle would have measure $(90-48)^{\circ } = 42 ^{\circ }$ . If Statement 1 is assumed, then $m \angle 1= 42 ^{\circ }$ .

Statement 2 gives no useful information. Adjacent angles do not have any numerical relationship; they simply share a ray and a vertex.

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Question

Find the angle made by and the -axis.

I) goes through the origin and the point .

II) makes a degree angle between itself and the -axis.

Answer

To find the angle of the line, recall that each quadrant has 90 degrees

I) Tells us that the line has a slope of one. This means that if we make a triangle using our line, the x-axis and a line coming up from the x-axis at 90 degrees we will have a 45/45/90 triangle. Therefore, I) tells us that our angle is 45 degrees.

II) Tells us that the line makes a 45 degree angle between itself and the y-axis. Therefore:

$\theta=90^{\circ}-45^{\circ}=45^{\circ}$

Therfore, we could use either statement.

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