Angles

Problem 1

Find $k$ for the planes $x-2y+4z=10$ and $18x+17y+kz=50$ to be perpendicular. Options: (a) -4 (b) 4 (c) 2 (d) -2

See Solution

Problem 2

Given a circle with center $O$ and diameter $MN$ , prove:
(i) $\angle MAN = 90^{\circ} + \angle PQR$ ,
(ii) $\angle QPR + 2 \times \angle MAN = 360^{\circ}$ .

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Problem 3

Find $m \angle 12$ if $m \angle 3=2x+14$ and $m \angle 16=4x-16$ , with angles 3 and 12 as alternate exterior angles.

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Problem 4

Find $m \angle 16$ if $m \angle 1=5x+8$ and $m \angle 16=7x-20$ .

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Problem 5

Find $m \angle 13$ given $m \angle 7 = 4x + 6$ and $m \angle 15 = 13x - 6$ for parallel lines.

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Problem 6

Find $m \angle 14$ given $m \angle 1 = 67^{\circ}$ and $m \angle 18 = 42^{\circ}$ with parallel lines $l$ and $m$ . Options: $113^{\circ}$ , $96^{\circ}$ , $105^{\circ}$ , $109^{\circ}$ .

See Solution

Problem 7

Prove that two lines are parallel if and only if alternate exterior angles are congruent: $\angle 1 \cong \angle 8$ .

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Problem 8

Find the sum of two alternate exterior angles: $6a - 33$ and $2x + 10$ .

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Problem 9

Prove that if $B P$ and $P Q$ are tangents, then (i) $A B \parallel P Q$ and (ii) $M P \times A M = B M \times M Q$ .

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Problem 10

Find the sum of two alternate exterior angles: $6x - 33$ and $2x + 10$ . What is the sum in degrees?

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Problem 11

If $\overline{F B} \| \overline{E C}$ and $m \angle E F B=103^{\circ}$ , find $m \angle C E F$ .

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Problem 12

Find the sum of the angle measures of two alternate exterior angles: $5x - 26$ and $2x + 10$ .

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Problem 13

Find the sum of two alternate exterior angles: $6x - 33$ and $3x + 13$ . What is their total measure in degrees?

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Problem 14

Three forces $F_{1}, F_{2}, F_{3}$ at point $O$ are in equilibrium. Given $F_{1} = 10.0 \mathrm{~N}$ , $F_{2} = 5.0 \mathrm{~N}$ , angle = 60°. Find $F_{3}$ .

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Problem 15

Anthony turned $90^{\circ}$ right, then $90^{\circ}$ , and $135^{\circ}$ more. Did he complete a circle? How much more to finish?

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Problem 16

Calcola due angoli sapendo che la loro somma è $90^{\circ} 30^{\prime}$ e la loro differenza è $10'30$ .

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Problem 17

Find the number of sides $n$ in a polygon if the interior angles sum to $1080^{\circ}$ . Use $=(n-2)180$ .

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Problem 18

Find $m \angle A B C$ if $m \angle A B D=76^{\circ}$ and $m \angle D B C=68^{\circ}$ .

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Problem 19

Find the number of sides of a regular polygon with an interior angle of $150^{\circ}$ . $n = [\text{?}]$

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Problem 20

Find the value of $x$ if $\mathrm{m} \angle \mathrm{PQR}=6x-7$ and $\mathrm{m} \angle \mathrm{RQT}=20x+5$ sum to 180°.

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Problem 21

Name the following: (a) an angle < $90^{\circ}$ , (b) a polygon with 5 sides, (c) a quadrilateral with 1 pair of parallel sides.

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Problem 22

Find the angle $x$ in a triangle with angles $x, 43^{\circ}, 79^{\circ}$ . Is it obtuse? Show your work.

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Problem 23

An n-sided polygon has 2 exterior angles of $75^{\circ}$ and the rest $21^{\circ}$ . Find $n$ .

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Problem 24

Find the angular magnification if an object has an angular diameter of $0.50^{\circ}$ and appears $8.0^{\circ}$ through a magnifier.

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Problem 25

Samantha adjusts her telescope from $32^{\circ}$ to $25^{\circ}$ . Answer these:
1. What is the angle change?
2. How far apart are the angles?

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Problem 26

You start at a bearing of 32 degrees, turn left 115 degrees, walk, then turn right 46 degrees. Find your final bearing.

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Problem 27

Сумма двух острых углов прямоугольного треугольника $126^{\circ}$ . Найдите их значения.

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Problem 28

Unghiul $M N O$ este măsurat. Ce valoare are dacă $A O B = 120^{\circ}$ , $O M$ e bisectoare și $M N \parallel O A$ ? a) $90^{\circ}$ ; b) $120^{\circ}$ ; c) $60^{\circ}$ ; d) $30^{\circ}$ .

See Solution

Problem 29

Сумма двух углов прямоугольного треугольника $126^{\circ}$ . Найдите острые углы.

See Solution

Problem 30

Постройте угол bisectrix с помощью циркуля и линейки.

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Problem 31

Find $m \angle Q O R$ if $m \angle P O Q=24$ and $m \angle P O R=59$ .

See Solution

Problem 32

Find $m \angle R O S$ if $m \angle Q O S=46$ , $m \angle P O R=61$ , and $m \angle P O Q=28$ .

See Solution

Problem 33

Find $m \angle P O S$ if $m \angle P O Q = 19$ , $m \angle Q O R = 31$ , and $m \angle R O S = 15$ .

See Solution

Problem 34

Find the missing angle in triangles ABE and DEC where $\angle ABE=25^{\circ}$ and $\angle EDC=100^{\circ}$ .

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Problem 35

1. Într-un triunghi dreptunghic $A B C$ cu $\angle C=60^{\circ}$ , găsiți măsura unghiului $A D C$ (bisectoarea $B A M$ ).
2. Calculați aria unui paralelogram cu laturile $A B=170 \mathrm{~m}, B C=80 \mathrm{~m}$ și diagonala $B D=150 \mathrm{~m}$ .

See Solution

Problem 36

Find angle $O Q P$ given points $P, Q, R$ on a circle with $m\angle Q P R=35^{\circ}$ and $m\angle O R P=30^{\circ}$ .

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Problem 37

Find the angles of a triangle with a ratio of $5: 3: 4$ and identify the type of triangle based on these angles.

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Problem 38

Convert the angle $45^{\circ} 12^{\prime} 23.2^{\prime \prime}$ to radians.

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Problem 39

A cyclist travels from A to B at 250 m/min, covering an angle of $1^{\circ} 5^{\prime}$ on Earth with radius $6440 \text{ km}$ . Find the time taken in hours.

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Problem 40

A cyclist travels from A to B at 250 m/min. The angle between A and B is $1^{\circ} 5^{\prime}$ , with Earth's radius $6440 \mathrm{~km}$ . Find the travel time in hours.

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Problem 41

Find the radius of the moon if it appears at an angle of $16^{\prime}$ from a distance of $237600 \mathrm{~km}$ .

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Problem 42

Find the number of sides in a regular polygon where the ratio of interior angle to exterior angle is $7: 1$ .

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Problem 43

Find the interior angle in terms of the exterior angle $x$ given the ratio of interior to exterior angles is $7:1$ .

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Problem 44

Solve for $x$ in the equations: $\frac{x}{2} + 8^{\circ} = \frac{6}{5} + 20^{\circ}$ and $\frac{x}{2} - \frac{x}{5} = 12^{\circ}$ . Find $a$ and $p$ if $a = p = \frac{x}{2} - 8^{\circ}$ .

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Problem 45

Encontre $a$ e $b$ onde as retas paralelas $r$ e $s$ são cortadas por uma transversal com a equação $\frac{x}{2} + \frac{x}{2} + \frac{x}{8} + 8 = \frac{x}{5} + 20$ .

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Problem 46

Find the equation of line $PQ$ , check if $PS$ is perpendicular to $PQ$ , find $QR$ parallel to $PS$ , calculate angle $\theta$ , and distance $QS$ .

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Problem 47

Find the angle $\theta$ at the center of sector $OAB$ with radius $9 \mathrm{~cm}$ and arc length \overparen{AB}=6 \pi \mathrm{~cm}.

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Problem 48

If the angle of a sector increases by $25\%$ and the radius decreases by $k\%$ , find $k$ when the arc length stays the same.

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Problem 49

Can nonadjacent angles share vertex $A$ and arm $A B$ ? If yes, provide an example; if no, explain why.

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Problem 50

You walk on a bearing of 32°, turn left 115°, then right 46°. What is your final bearing?

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Problem 51

You start at a bearing of 32 degrees, turn left 115 degrees, then right 46 degrees. Find your final bearing.

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Problem 52

You start at a bearing of 30 degrees and turn left to 290 degrees. How many degrees did you turn?

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Problem 53

Identify the true properties of parallelograms from these options: A. Adjacent sides congruent, B. Opposite angles congruent, C. Opposite angles parallel, D. Opposite sides parallel, E. Consecutive angles supplementary, F. Diagonals bisect each other.

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Problem 54

Which quadrilaterals have opposite angles that are always congruent? Check all that apply: A. Parallelogram B. Square C. Quadrilateral D. Rhombus

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Problem 55

Find (a) the complement and (b) the supplement of an angle measuring $17^{\circ} 15^{\prime}$ .

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Problem 56

Find the complement and supplement of an angle measuring $22^{\circ} 13^{\prime}$ .

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Problem 57

What is the angle in degrees that corresponds to $\frac{92}{360}$ of a full turn in a circle?

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Problem 58

Convert the angle $22^{\circ} 42^{\prime}$ to decimal degrees.

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Problem 59

Convert the angle $-80^{\circ} 36^{\prime}$ to decimal degrees.

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Problem 60

Is this true? The degree measure of a minor arc equals the measure of its central angle. A. Yes B. No C. Maybe D. Sometimes E. Not applicable

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Problem 61

Find the smaller angle between clock hands at 1:25.

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Problem 62

Convert the angle $\alpha=73^{\circ} 39^{\prime} 16^{\prime \prime}$ to decimal degrees.

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Problem 63

Convert the angle $\alpha=18.8211^{\circ}$ to degrees, minutes, and seconds.

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Problem 64

For a regular 20-sided polygon, what is the rotation angle in degrees? Use the formula $\frac{360}{n}$ where $n = 20$ .

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Problem 65

Find the angle $b$ at the center of a circle with a 295-degree arc.

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Problem 66

Find the value of $z$ given $x = z$ , $x = 6k + 13$ , and $y = 8k - 29$ with lines $m$ and $n$ parallel.

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Problem 67

Find $x$ , $m<SQR>$ , and $m<PQT>$ given $PQT = 4x + 43$ and $SQR = 7x - 20$ are congruent.

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Problem 68

Find $x$ if angles $<R$ and $<S$ are complementary with $m<R=(9x-7)$ and $m<S=(7x+1)$ .

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Problem 69

Find the value of $x$ if angles $<R$ and $<S$ are complementary, with $m<R=(9x-7)$ and $m<S=(7x+1)$ .

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Problem 70

Draw the angle $150^{\circ}$ , find its reference angle, and identify the quadrant of its terminal side.

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Problem 71

Gregor drives up a $15^{\circ}$ hill for $5 \mathrm{~km}$ . What is the vertical rise? Round to the nearest metre.

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Problem 72

Find the value of $x$ in a triangle where an exterior angle is 144 degrees and one interior angle is 93 degrees.

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Problem 73

The supplement of an angle is 20 more than three times the angle. Find the measures of the angles.

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Problem 74

A man on a 150-ft building sees a car move with angles of depression $25^{\circ}$ and $45^{\circ}$ . Find the distance traveled.

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Problem 75

A boat travels at 40 knots on a course of $65^{\circ}$ for 2 hours, then $155^{\circ}$ for 4 hours. Find the distance and bearing from Fort Lauderdale.

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Problem 76

Find the smaller angle between clock hands at 5:25.

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Problem 77

Convert the angle $10^{\circ} 48^{\prime}$ to decimal degrees.

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Problem 78

Calculate $90^{\circ} - 41^{\circ} 57^{\prime}$ .

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Problem 79

10. Identify pairs of vertically opposite, adjacent, linear pair, complementary, and supplementary angles. Given $\angle 4=110^{\circ}$ and $\angle 5=120^{\circ}$ , find the others.
11. What is the angle that equals its complement?
12. What is the angle that equals its supplement?

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Problem 80

Find the complementary angles where one is $(3x-9)^\circ$ and the other is $(6x)^\circ$ . Their sum is $90^\circ$ .

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Problem 81

Find the supplementary angle if the smaller angle is $(12x+1)^\circ$ . Supplementary angles sum to $180^\circ$ .

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Problem 82

Find the supplementary angle if one angle is $(17x+5)$ degrees. Their sum is $180^{\circ}$ .

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Problem 83

Find the measure of $\angle 2$ given that $\mathrm{m} \angle 1=(2 x+29)^{\circ}$ and $\mathrm{m} \angle 2=(3 x-17)^{\circ}$ , where they are vertical angles.

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Problem 84

Find the value of $x$ if $\angle 1$ and $\angle 2$ are vertical angles with $\mathrm{m} \angle 1=(2x-2)^{\circ}$ and $\mathrm{m} \angle 2=(3x-15)^{\circ}$ .

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Problem 85

Find $x$ if $m \angle PQR = x + 9$ , $m \angle SQR = x - 3$ , and $m \angle PQS = 100$ .

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Problem 86

If $\triangle \mathrm{JKL} \cong \triangle PQR$ and $m<P=52$ , $m<Q=48$ , $m<R=80$ , find $m<K$ . A. Cannot be determined B. 80 C. 52 D. 48

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Problem 87

Find the measure of angle $\mathrm{DOT}$ if $\overrightarrow{\mathrm{OG}}$ bisects $\angle D O T$ , with $m \angle 1 = 6x + 41$ and $m \angle 2 = 9x - 1$ .

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Problem 88

Find $m \angle ABC$ if $m \angle ABC = 6x - 4$ , $m \angle CBD = 3x + 2$ , and $m \angle ABD = 34$ .

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Problem 89

If $\angle 1 \cong \angle 2$ and $m \angle 1=2x+10$ , $m \angle 3=120^{\circ}$ , find $x$ . How many degrees in a line?

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Problem 90

If $\angle 1$ complements $\angle 2$ and $m \angle 1=23^{\circ}$ , what is $m \angle 2$ ?

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Problem 91

If $\angle 1 \cong \angle 2$ and $m \angle 1=2x+10$ , $m \angle 3=120^{\circ}$ , find $x$ .

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Problem 92

How many degrees must a gate arm move from $42^{\circ}$ to reach a horizontal position?

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Problem 93

Copy angle ABC onto ray DE to create angle FDE. Draw ray DF, then use compass to create points J and F.

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Problem 94

What is the best definition of an angle?

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Problem 95

Find $m$ if $\angle 1$ and $\angle 2$ are vertical angles with $m \angle 1=17x+1$ and $m \angle 2=20x-14$ .

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Problem 96

Convert the angle $\alpha=62^{\circ} 59^{\prime} 41^{\prime \prime}$ to decimal degrees.

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Problem 97

Convert the angle $38.32^{\circ}$ to degrees, minutes, and seconds.

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Problem 98

Convert the angle $\alpha=77.8211^{\circ}$ to degrees, minutes, and seconds.

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Problem 99

Find (a) the complement and (b) the supplement of the angle measuring $20^{\circ} 18^{\prime}$ .

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Problem 100

Angles $AXB$ and $BXC$ are supplementary. Find the measure of angle $AXB$ . Options: (A) $162^{\circ}$ (B) $146^{\circ}$ (C) $34^{\circ}$ (D) $18^{\circ}$

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Angles

Problem 1

Find k k k for the planes x−2y+4z=10 x-2y+4z=10 x−2y+4z=10 and 18x+17y+kz=50 18x+17y+kz=50 18x+17y+kz=50 to be perpendicular. Options: (a) -4 (b) 4 (c) 2 (d) -2

Problem 2

Given a circle with center OOO and diameter MNMNMN, prove:(i) ∠MAN=90∘+∠PQR\angle MAN = 90^{\circ} + \angle PQR∠MAN=90∘+∠PQR,(ii) ∠QPR+2×∠MAN=360∘\angle QPR + 2 \times \angle MAN = 360^{\circ}∠QPR+2×∠MAN=360∘.

Problem 3

Find m∠12 m \angle 12 m∠12 if m∠3=2x+14 m \angle 3=2x+14 m∠3=2x+14 and m∠16=4x−16 m \angle 16=4x-16 m∠16=4x−16, with angles 3 and 12 as alternate exterior angles.

Problem 4

Find m∠16 m \angle 16 m∠16 if m∠1=5x+8 m \angle 1=5x+8 m∠1=5x+8 and m∠16=7x−20 m \angle 16=7x-20 m∠16=7x−20.

Problem 5

Find m∠13 m \angle 13 m∠13 given m∠7=4x+6 m \angle 7 = 4x + 6 m∠7=4x+6 and m∠15=13x−6 m \angle 15 = 13x - 6 m∠15=13x−6 for parallel lines.

Problem 6

Find m∠14 m \angle 14 m∠14 given m∠1=67∘ m \angle 1 = 67^{\circ} m∠1=67∘ and m∠18=42∘ m \angle 18 = 42^{\circ} m∠18=42∘ with parallel lines l l l and m m m. Options: 113∘ 113^{\circ} 113∘, 96∘ 96^{\circ} 96∘, 105∘ 105^{\circ} 105∘, 109∘ 109^{\circ} 109∘.

Problem 7

Prove that two lines are parallel if and only if alternate exterior angles are congruent: ∠1≅∠8 \angle 1 \cong \angle 8 ∠1≅∠8.

Problem 8

Find the sum of two alternate exterior angles: 6a−33 6a - 33 6a−33 and 2x+10 2x + 10 2x+10.

Problem 9

Prove that if BP B P BP and PQ P Q PQ are tangents, then (i) AB∥PQ A B \parallel P Q AB∥PQ and (ii) MP×AM=BM×MQ M P \times A M = B M \times M Q MP×AM=BM×MQ.

Problem 10

Find the sum of two alternate exterior angles: 6x−33 6x - 33 6x−33 and 2x+10 2x + 10 2x+10. What is the sum in degrees?

Problem 11

If FB‾∥EC‾ \overline{F B} \| \overline{E C} FB∥EC and m∠EFB=103∘ m \angle E F B=103^{\circ} m∠EFB=103∘, find m∠CEF m \angle C E F m∠CEF.

Problem 12

Find the sum of the angle measures of two alternate exterior angles: 5x−26 5x - 26 5x−26 and 2x+10 2x + 10 2x+10.

Problem 13

Find the sum of two alternate exterior angles: 6x−33 6x - 33 6x−33 and 3x+13 3x + 13 3x+13. What is their total measure in degrees?

Problem 14

Three forces F1,F2,F3 F_{1}, F_{2}, F_{3} F1​,F2​,F3​ at point O O O are in equilibrium. Given F1=10.0 N F_{1} = 10.0 \mathrm{~N} F1​=10.0 N, F2=5.0 N F_{2} = 5.0 \mathrm{~N} F2​=5.0 N, angle = 60°. Find F3 F_{3} F3​.

Problem 15

Anthony turned 90∘90^{\circ}90∘ right, then 90∘90^{\circ}90∘, and 135∘135^{\circ}135∘ more. Did he complete a circle? How much more to finish?

Problem 16

Calcola due angoli sapendo che la loro somma è 90∘30′90^{\circ} 30^{\prime}90∘30′ e la loro differenza è 10′3010'3010′30.

Problem 17

Find the number of sides nnn in a polygon if the interior angles sum to 1080∘1080^{\circ}1080∘. Use =(n−2)180=(n-2)180=(n−2)180.

Problem 18

Find m∠ABCm \angle A B Cm∠ABC if m∠ABD=76∘m \angle A B D=76^{\circ}m∠ABD=76∘ and m∠DBC=68∘m \angle D B C=68^{\circ}m∠DBC=68∘.

Problem 19

Find the number of sides of a regular polygon with an interior angle of 150∘150^{\circ}150∘. n=[?] n = [\text{?}] n=[?]

Problem 20

Find the value of xxx if m∠PQR=6x−7\mathrm{m} \angle \mathrm{PQR}=6x-7m∠PQR=6x−7 and m∠RQT=20x+5\mathrm{m} \angle \mathrm{RQT}=20x+5m∠RQT=20x+5 sum to 180°.

Problem 21

Name the following: (a) an angle < 90∘90^{\circ}90∘, (b) a polygon with 5 sides, (c) a quadrilateral with 1 pair of parallel sides.

Problem 22

Find the angle xxx in a triangle with angles x,43∘,79∘x, 43^{\circ}, 79^{\circ}x,43∘,79∘. Is it obtuse? Show your work.

Problem 23

An n-sided polygon has 2 exterior angles of 75∘75^{\circ}75∘ and the rest 21∘21^{\circ}21∘. Find nnn.

Problem 24

Find the angular magnification if an object has an angular diameter of 0.50∘0.50^{\circ}0.50∘ and appears 8.0∘8.0^{\circ}8.0∘ through a magnifier.

Problem 25

Samantha adjusts her telescope from 32∘32^{\circ}32∘ to 25∘25^{\circ}25∘. Answer these: 1. What is the angle change?2. How far apart are the angles?

Problem 26

You start at a bearing of 32 degrees, turn left 115 degrees, walk, then turn right 46 degrees. Find your final bearing.

Problem 27

Сумма двух острых углов прямоугольного треугольника 126∘126^{\circ}126∘. Найдите их значения.

Problem 28

Unghiul MNOM N OMNO este măsurat. Ce valoare are dacă AOB=120∘A O B = 120^{\circ}AOB=120∘, OMO MOM e bisectoare și MN∥OAM N \parallel O AMN∥OA? a) 90∘90^{\circ}90∘; b) 120∘120^{\circ}120∘; c) 60∘60^{\circ}60∘; d) 30∘30^{\circ}30∘.

Problem 29

Сумма двух углов прямоугольного треугольника 126∘126^{\circ}126∘. Найдите острые углы.

Problem 30

Постройте угол bisectrix с помощью циркуля и линейки.

Problem 31

Find m∠QORm \angle Q O Rm∠QOR if m∠POQ=24m \angle P O Q=24m∠POQ=24 and m∠POR=59m \angle P O R=59m∠POR=59.

Problem 32

Find m∠ROSm \angle R O Sm∠ROS if m∠QOS=46m \angle Q O S=46m∠QOS=46, m∠POR=61m \angle P O R=61m∠POR=61, and m∠POQ=28m \angle P O Q=28m∠POQ=28.

Problem 33

Find m∠POSm \angle P O Sm∠POS if m∠POQ=19m \angle P O Q = 19m∠POQ=19, m∠QOR=31m \angle Q O R = 31m∠QOR=31, and m∠ROS=15m \angle R O S = 15m∠ROS=15.

Problem 34

Find the missing angle in triangles ABE and DEC where ∠ABE=25∘\angle ABE=25^{\circ}∠ABE=25∘ and ∠EDC=100∘\angle EDC=100^{\circ}∠EDC=100∘.

Problem 35

Problem 36

Find angle OQPO Q POQP given points P,Q,RP, Q, RP,Q,R on a circle with m∠QPR=35∘m\angle Q P R=35^{\circ}m∠QPR=35∘ and m∠ORP=30∘m\angle O R P=30^{\circ}m∠ORP=30∘.

Problem 37

Find the angles of a triangle with a ratio of 5:3:45: 3: 45:3:4 and identify the type of triangle based on these angles.

Problem 38

Convert the angle 45∘12′23.2′′45^{\circ} 12^{\prime} 23.2^{\prime \prime}45∘12′23.2′′ to radians.

Problem 39

A cyclist travels from A to B at 250 m/min, covering an angle of 1∘5′1^{\circ} 5^{\prime}1∘5′ on Earth with radius 6440 km6440 \text{ km}6440 km. Find the time taken in hours.

Problem 40

A cyclist travels from A to B at 250 m/min. The angle between A and B is 1∘5′1^{\circ} 5^{\prime}1∘5′, with Earth's radius 6440 km6440 \mathrm{~km}6440 km. Find the travel time in hours.

Find $k$ for the planes $x-2y+4z=10$ and $18x+17y+kz=50$ to be perpendicular. Options: (a) -4 (b) 4 (c) 2 (d) -2

Given a circle with center $O$ and diameter $MN$ , prove:
(i) $\angle MAN = 90^{\circ} + \angle PQR$ ,
(ii) $\angle QPR + 2 \times \angle MAN = 360^{\circ}$ .

Find $m \angle 12$ if $m \angle 3=2x+14$ and $m \angle 16=4x-16$ , with angles 3 and 12 as alternate exterior angles.

Find $m \angle 16$ if $m \angle 1=5x+8$ and $m \angle 16=7x-20$ .

Find $m \angle 13$ given $m \angle 7 = 4x + 6$ and $m \angle 15 = 13x - 6$ for parallel lines.

Find $m \angle 14$ given $m \angle 1 = 67^{\circ}$ and $m \angle 18 = 42^{\circ}$ with parallel lines $l$ and $m$ . Options: $113^{\circ}$ , $96^{\circ}$ , $105^{\circ}$ , $109^{\circ}$ .

Prove that two lines are parallel if and only if alternate exterior angles are congruent: $\angle 1 \cong \angle 8$ .

Find the sum of two alternate exterior angles: $6a - 33$ and $2x + 10$ .

Prove that if $B P$ and $P Q$ are tangents, then (i) $A B \parallel P Q$ and (ii) $M P \times A M = B M \times M Q$ .

Find the sum of two alternate exterior angles: $6x - 33$ and $2x + 10$ . What is the sum in degrees?

If $\overline{F B} \| \overline{E C}$ and $m \angle E F B=103^{\circ}$ , find $m \angle C E F$ .

Find the sum of the angle measures of two alternate exterior angles: $5x - 26$ and $2x + 10$ .

Find the sum of two alternate exterior angles: $6x - 33$ and $3x + 13$ . What is their total measure in degrees?

Three forces $F_{1}, F_{2}, F_{3}$ at point $O$ are in equilibrium. Given $F_{1} = 10.0 \mathrm{~N}$ , $F_{2} = 5.0 \mathrm{~N}$ , angle = 60°. Find $F_{3}$ .

Anthony turned $90^{\circ}$ right, then $90^{\circ}$ , and $135^{\circ}$ more. Did he complete a circle? How much more to finish?

Calcola due angoli sapendo che la loro somma è $90^{\circ} 30^{\prime}$ e la loro differenza è $10'30$ .

Find the number of sides $n$ in a polygon if the interior angles sum to $1080^{\circ}$ . Use $=(n-2)180$ .

Find $m \angle A B C$ if $m \angle A B D=76^{\circ}$ and $m \angle D B C=68^{\circ}$ .

Find the number of sides of a regular polygon with an interior angle of $150^{\circ}$ . $n = [\text{?}]$

Find the value of $x$ if $\mathrm{m} \angle \mathrm{PQR}=6x-7$ and $\mathrm{m} \angle \mathrm{RQT}=20x+5$ sum to 180°.

Name the following: (a) an angle < $90^{\circ}$ , (b) a polygon with 5 sides, (c) a quadrilateral with 1 pair of parallel sides.

Find the angle $x$ in a triangle with angles $x, 43^{\circ}, 79^{\circ}$ . Is it obtuse? Show your work.

An n-sided polygon has 2 exterior angles of $75^{\circ}$ and the rest $21^{\circ}$ . Find $n$ .

Find the angular magnification if an object has an angular diameter of $0.50^{\circ}$ and appears $8.0^{\circ}$ through a magnifier.

Samantha adjusts her telescope from $32^{\circ}$ to $25^{\circ}$ . Answer these:
1. What is the angle change?
2. How far apart are the angles?

Сумма двух острых углов прямоугольного треугольника $126^{\circ}$ . Найдите их значения.

Unghiul $M N O$ este măsurat. Ce valoare are dacă $A O B = 120^{\circ}$ , $O M$ e bisectoare și $M N \parallel O A$ ? a) $90^{\circ}$ ; b) $120^{\circ}$ ; c) $60^{\circ}$ ; d) $30^{\circ}$ .

Сумма двух углов прямоугольного треугольника $126^{\circ}$ . Найдите острые углы.

Find $m \angle Q O R$ if $m \angle P O Q=24$ and $m \angle P O R=59$ .

Find $m \angle R O S$ if $m \angle Q O S=46$ , $m \angle P O R=61$ , and $m \angle P O Q=28$ .

Find $m \angle P O S$ if $m \angle P O Q = 19$ , $m \angle Q O R = 31$ , and $m \angle R O S = 15$ .

Find the missing angle in triangles ABE and DEC where $\angle ABE=25^{\circ}$ and $\angle EDC=100^{\circ}$ .

Find angle $O Q P$ given points $P, Q, R$ on a circle with $m\angle Q P R=35^{\circ}$ and $m\angle O R P=30^{\circ}$ .

Find the angles of a triangle with a ratio of $5: 3: 4$ and identify the type of triangle based on these angles.

Convert the angle $45^{\circ} 12^{\prime} 23.2^{\prime \prime}$ to radians.

A cyclist travels from A to B at 250 m/min, covering an angle of $1^{\circ} 5^{\prime}$ on Earth with radius $6440 \text{ km}$ . Find the time taken in hours.

A cyclist travels from A to B at 250 m/min. The angle between A and B is $1^{\circ} 5^{\prime}$ , with Earth's radius $6440 \mathrm{~km}$ . Find the travel time in hours.

Find the radius of the moon if it appears at an angle of $16^{\prime}$ from a distance of $237600 \mathrm{~km}$ .

Find the number of sides in a regular polygon where the ratio of interior angle to exterior angle is $7: 1$ .

Find the interior angle in terms of the exterior angle $x$ given the ratio of interior to exterior angles is $7:1$ .

Solve for $x$ in the equations: $\frac{x}{2} + 8^{\circ} = \frac{6}{5} + 20^{\circ}$ and $\frac{x}{2} - \frac{x}{5} = 12^{\circ}$ . Find $a$ and $p$ if $a = p = \frac{x}{2} - 8^{\circ}$ .

Encontre $a$ e $b$ onde as retas paralelas $r$ e $s$ são cortadas por uma transversal com a equação $\frac{x}{2} + \frac{x}{2} + \frac{x}{8} + 8 = \frac{x}{5} + 20$ .

Find the equation of line $PQ$ , check if $PS$ is perpendicular to $PQ$ , find $QR$ parallel to $PS$ , calculate angle $\theta$ , and distance $QS$ .

Find the angle $\theta$ at the center of sector $OAB$ with radius $9 \mathrm{~cm}$ and arc length \overparen{AB}=6 \pi \mathrm{~cm}.

If the angle of a sector increases by $25\%$ and the radius decreases by $k\%$ , find $k$ when the arc length stays the same.

Can nonadjacent angles share vertex $A$ and arm $A B$ ? If yes, provide an example; if no, explain why.

Find (a) the complement and (b) the supplement of an angle measuring $17^{\circ} 15^{\prime}$ .

Find the complement and supplement of an angle measuring $22^{\circ} 13^{\prime}$ .

What is the angle in degrees that corresponds to $\frac{92}{360}$ of a full turn in a circle?

Convert the angle $22^{\circ} 42^{\prime}$ to decimal degrees.

Convert the angle $-80^{\circ} 36^{\prime}$ to decimal degrees.

Convert the angle $\alpha=73^{\circ} 39^{\prime} 16^{\prime \prime}$ to decimal degrees.

Convert the angle $\alpha=18.8211^{\circ}$ to degrees, minutes, and seconds.

For a regular 20-sided polygon, what is the rotation angle in degrees? Use the formula $\frac{360}{n}$ where $n = 20$ .

Find the angle $b$ at the center of a circle with a 295-degree arc.

Find the value of $z$ given $x = z$ , $x = 6k + 13$ , and $y = 8k - 29$ with lines $m$ and $n$ parallel.

Find $x$ , $m<SQR>$ , and $m<PQT>$ given $PQT = 4x + 43$ and $SQR = 7x - 20$ are congruent.

Find $x$ if angles $<R$ and $<S$ are complementary with $m<R=(9x-7)$ and $m<S=(7x+1)$ .

Find the value of $x$ if angles $<R$ and $<S$ are complementary, with $m<R=(9x-7)$ and $m<S=(7x+1)$ .

Draw the angle $150^{\circ}$ , find its reference angle, and identify the quadrant of its terminal side.

Gregor drives up a $15^{\circ}$ hill for $5 \mathrm{~km}$ . What is the vertical rise? Round to the nearest metre.

Find the value of $x$ in a triangle where an exterior angle is 144 degrees and one interior angle is 93 degrees.

A man on a 150-ft building sees a car move with angles of depression $25^{\circ}$ and $45^{\circ}$ . Find the distance traveled.