Lines and Angles Practice Questions — GRE General Test

Question 1Question →

In the plane, line $L_1$ is parallel to line $L_2$ . A transversal line $T$ intersects $L_1$ and $L_2$ . One of the acute angles formed at the intersection of $L_1$ and $T$ measures $(4x - 10)^\circ$ , and an alternate interior angle on $L_2$ measures $(2x + 30)^\circ$ . What is the degree measure of one of the obtuse angles formed at the intersection of $L_1$ and $T$ ?

20^\circ

50^\circ

70^\circ

110^\circ

140^\circ

Show answer & explanation

Answer: $110^\circ$

Answer

The degree measure of the obtuse angle is

110^\circ

.

Since lines

L_1

and

L_2

are parallel, alternate interior angles are equal in measure. Setting

(4x - 10)^\circ = (2x + 30)^\circ

gives

2x = 40

, so

x = 20

. Substituting

x = 20

into

4x - 10

gives an acute angle of

70^\circ

. Because angles on a straight line are supplementary, the obtuse angle measures

180^\circ - 70^\circ = 110^\circ

, which corresponds to the correct choice.

Step-by-Step Solution

1

Set alternate interior angles equal to solve for

x

.

4x - 10 = 2x + 30 \implies 2x = 40 \implies x = 20

When two parallel lines are cut by a transversal, alternate interior angles are congruent.

2

Substitute

x = 20

into the expression for the acute angle.

Acute angle

= 4(20) - 10 = 70^\circ

This yields the degree measure of the acute angle formed by the intersection.

3

Calculate the measure of the supplementary obtuse angle.

Obtuse angle

= 180^\circ - 70^\circ = 110^\circ

Adjacent angles along a straight line are supplementary and sum to

180^\circ

.

Key Concept

Alternate Interior Angles and Supplementary Angles

Estimated Time:50s

Question 2Question →

In the figure, line $l_1$ is parallel to line $l_2$ . Point $A$ lies on line $l_1$ , while points $B$ and $C$ lie on line $l_2$ such that $B$ is to the left of $C$ . Line segments $AB$ and $AC$ extend from line $l_1$ to line $l_2$ to form $\triangle ABC$ . The measure of interior angle $\angle BAC$ is $(2x + 10)^\circ$ , the measure of interior angle $\angle ABC$ is $(3x - 15)^\circ$ , and the measure of the exterior angle at vertex $C$ along line $l_2$ is $(6x - 35)^\circ$ . What is the degree measure of angle $\angle BAC$ ?

30^\circ

35^\circ

50^\circ

70^\circ

75^\circ

Show answer & explanation

Answer: $70^\circ$

Answer

The degree measure of angle

\angle BAC

is

70^\circ

.

According to the Exterior Angle Theorem, the measure of an exterior angle of a triangle is equal to the sum of the measures of its two non-adjacent (remote) interior angles. Setting up the equation

(2x + 10) + (3x - 15) = 6x - 35

gives

5x - 5 = 6x - 35

, which simplifies to

x = 30

. Substituting

x = 30

into the expression for

\angle BAC

,

(2x + 10)^\circ

, yields

2(30) + 10 = 70^\circ

.

Step-by-Step Solution

1

Apply the Exterior Angle Theorem to set up an algebraic equation.

(2x + 10) + (3x - 15) = 6x - 35

The measure of an exterior angle of a triangle equals the sum of the measures of its two remote interior angles.

2

Simplify and solve the linear equation for

x

.

5x - 5 = 6x - 35 \implies x = 30

Combining like terms yields

5x - 5 = 6x - 35

. Subtracting

5x

from both sides gives

-5 = x - 35

, so

x = 30

.

3

Substitute

x = 30

into the expression for angle

\angle BAC

.

\text{m}\angle BAC = 2(30) + 10 = 70^\circ

The question asks for the degree measure of angle

\angle BAC

, which is given by

(2x + 10)^\circ

.

Key Concept

Exterior Angle Theorem and Angle Relationships in Parallel Lines

Question 3Question →

Two straight lines, $L_1$ and $L_2$ , intersect at point $O$ . One of the angles formed by their intersection measures $(3x - 15)^\circ$ , and the vertically opposite angle measures $(x + 25)^\circ$ . What is the degree measure of an angle adjacent to $(3x - 15)^\circ$ ?

20^\circ

45^\circ

135^\circ

67.5^\circ

112.5^\circ

Show answer & explanation

Answer: $135^\circ$

Answer

135^\circ

Since vertically opposite angles are equal, setting

3x - 15 = x + 25

yields

x = 20

. Substituting

x = 20

into

3x - 15

gives an angle of

45^\circ

. Because adjacent angles on intersecting lines form a straight line (a linear pair), they are supplementary. Therefore, the adjacent angle measures

180^\circ - 45^\circ = 135^\circ

.

Step-by-Step Solution

1

Set the vertically opposite angle expressions equal to each other.

3x - 15 = x + 25

Vertically opposite angles formed by two intersecting lines are equal in measure.

2

Solve for the variable

x

.

2x = 40 \implies x = 20

Subtract

x

and add

15

to both sides of the equation.

3

Calculate the degree measure of the angle

(3x - 15)^\circ

.

3(20) - 15 = 60 - 15 = 45^\circ

Substitute

x = 20

back into the angle expression.

4

Calculate the measure of an adjacent angle.

180^\circ - 45^\circ = 135^\circ

Adjacent angles along a straight line are supplementary and sum to

180^\circ

.

Key Concept

Vertically opposite angles are equal, and adjacent angles forming a linear pair are supplementary.

Question 4Question →

Lines $l_1$ and $l_2$ intersect at point $P$ to form an acute angle measuring $44^\circ$ . Line $b_1$ is the angle bisector of this acute angle. A third line, $l_3$ , is drawn perpendicular to $b_1$ and intersects line $l_1$ at point $Q$ (where $Q \neq P$ ). What is the measure, in degrees, of the acute angle formed by the intersection of line $l_2$ and line $l_3$ ?

22^\circ

46^\circ

68^\circ

78^\circ

112^\circ

Show answer & explanation

Answer: $68^\circ$

Answer

The measure of the acute angle formed by the intersection of line

l_2

and line

l_3

is

68^\circ

.

The correct answer is

68^\circ

. Line

b_1

divides the

44^\circ

angle between

l_1

and

l_2

into two

22^\circ

angles. Because line

l_3

is perpendicular to

b_1

, it forms a right triangle with

l_1

and

b_1

, making the angle between

l_1

and

l_3

equal to

90^\circ - 22^\circ = 68^\circ

. Considering the large triangle formed by lines

l_1

,

l_2

, and

l_3

, two of its angles are

44^\circ

and

68^\circ

. Thus, the third interior angle at the intersection of

l_2

and

l_3

is

180^\circ - (44^\circ + 68^\circ) = 68^\circ

, which is acute.

Step-by-Step Solution

1

Determine the angle formed by the angle bisector

b_1

with line

l_1

and line

l_2

.

Since line

b_1

bisects the

44^\circ

acute angle between

l_1

and

l_2

, the angle between

l_1

and

b_1

is

\frac{44^\circ}{2} = 22^\circ

, and the angle between

l_2

and

b_1

is also

22^\circ

.

An angle bisector divides an angle into two equal congruent parts.

2

Find the measure of the interior angle between line

l_1

and line

l_3

.

Line

l_3

is perpendicular to

b_1

, forming a right triangle with

l_1

and

b_1

. The interior angle between

l_1

and

l_3

is

180^\circ - 90^\circ - 22^\circ = 68^\circ

.

The sum of interior angles in any triangle is

180^\circ

.

3

Calculate the interior angle at the intersection of line

l_2

and line

l_3

in the main triangle formed by

l_1

,

l_2

, and

l_3

.

The interior angle at the intersection of

l_2

and

l_3

is

180^\circ - 44^\circ - 68^\circ = 68^\circ

.

The interior angles of the triangle formed by lines

l_1

,

l_2

, and

l_3

must sum to

180^\circ

.

4

Verify that the calculated angle is acute.

Since

68^\circ < 90^\circ

, the acute angle formed by lines

l_2

and

l_3

is

68^\circ

.

An angle measuring strictly less than

90^\circ

is defined as an acute angle.

Key Concept

Angle bisector properties, perpendicular lines, and interior angle sum theorem for triangles.

Question 5Question →

In the geometric plane, line $l_1$ is parallel to line $l_2$ . Point $A$ lies on line $l_1$ and point $B$ lies on line $l_2$ . Point $C$ is located between lines $l_1$ and $l_2$ such that $C$ lies to the right of both $A$ and $B$ .

The acute angle between segment $AC$ and the ray extending to the right from $A$ along line $l_1$ measures $x^\circ$ .

The acute angle between segment $BC$ and the ray extending to the right from $B$ along line $l_2$ measures $y^\circ$ .

The interior angle $\angle ACB = 118^\circ$ , and $y = 2x - 14$ .

Line $k$ passes through point $C$ and is perpendicular to segment $AC$ . Line $k$ intersects line $l_2$ at point $E$ , where point $E$ lies to the right of point $B$ .

What is the measure, in degrees, of the acute angle $\angle CEB$ formed by line $k$ and line $l_2$ ?

16^\circ

44^\circ

46^\circ

62^\circ

74^\circ

Show answer & explanation

Answer: $46^\circ$

Answer

The measure of the acute angle

\angle CEB

is

46^\circ

.

By the parallel lines angle property, drawing a parallel line through vertex

C

shows that

\angle ACB = x + y = 118^\circ

. Substituting

y = 2x - 14

yields

3x - 14 = 118

, giving

x = 44^\circ

. Since line

l_1

is parallel to line

l_2

, segment

AC

intersects line

l_2

at an acute angle of

44^\circ

. Line

k

is constructed perpendicular to segment

AC

, so the acute angle formed by line

k

and line

l_2

is complementary to

44^\circ

, which gives

90^\circ - 44^\circ = 46^\circ

.

Step-by-Step Solution

1

Set up the parallel lines zig-zag relationship to find

x

and

y

.

\angle ACB = x + y = 118^\circ

By drawing an auxiliary line through

C

parallel to

l_1

and

l_2

, the interior angle

\angle ACB

facing left equals the sum of the alternate interior angles

x

and

y

.

2

Substitute the given algebraic relation

y = 2x - 14

into the sum equation.

x + (2x - 14) = 118 \implies 3x - 14 = 118 \implies 3x = 132 \implies x = 44^\circ

Solving the linear system gives the exact value of angle

x

.

3

Determine the angle that line

AC

makes with line

l_2

.

Line

AC

intersects line

l_2

at an acute angle of

44^\circ

.

Since

l_1 \parallel l_2

, alternate interior angles formed by transversal line

AC

are equal (

x = 44^\circ

).

4

Calculate the acute angle

\angle CEB

between line

k

and line

l_2

.

\angle CEB = 90^\circ - 44^\circ = 46^\circ

Line

k

is perpendicular to segment

AC

, so the angle it forms with line

l_2

is the complementary angle to the angle line

AC

forms with line

l_2

.

Key Concept

Parallel Line Angle Relationships and Perpendicular Line Complements

Question 6Question →

Three straight lines, $R$ , $S$ , and $T$ , all intersect at a single point $P$ . Line $R$ is perpendicular to line $S$ . Line $T$ intersects line $R$ such that one of the acute angles formed between line $R$ and line $T$ measures $35^\circ$ . Which of the following degree measures represent angles formed between any two of the intersecting lines at point $P$ ? Select all that apply.

Select all that apply

55^\circ

90^\circ

145^\circ

115^\circ

70^\circ

Show answer & explanation

Answer: $55^\circ$ ; $90^\circ$ ; $145^\circ$

Answer

The degree measures

55^\circ

,

90^\circ

, and

145^\circ

are all valid angle measures formed by the intersecting lines at point

P

.

The intersecting lines form several distinct angle measures at point

P

. The right angle between lines

R

and

S

measures

90^\circ

. The acute angle between lines

T

and

S

is complementary to the

35^\circ

angle, giving

90^\circ - 35^\circ = 55^\circ

. The obtuse angle between lines

T

and

R

along straight line

R

is supplementary to

35^\circ

, giving

180^\circ - 35^\circ = 145^\circ

. Therefore, the options stating

55^\circ

,

90^\circ

, and

145^\circ

are all correct.

Step-by-Step Solution

1

Identify the angle measure between perpendicular lines R and S.

Since line

R \perp S

, the angle between them is

90^\circ

.

Perpendicular lines intersect at right angles (

90^\circ

).

2

Calculate the acute angle between line T and line S.

Angle between

T

and

S

=

90^\circ - 35^\circ = 55^\circ

.

The given

35^\circ

angle between

R

and

T

and the adjacent angle between

T

and

S

form the

90^\circ

right angle between

R

and

S

.

3

Determine the supplementary obtuse angles formed by line T with line R and line S.

Supplementary angle to

35^\circ

is

180^\circ - 35^\circ = 145^\circ

. Supplementary angle to

55^\circ

is

180^\circ - 55^\circ = 125^\circ

.

Adjacent angles along a straight line sum to

180^\circ

.

4

Compare calculated angle measures with the options.

The valid angle measures are

35^\circ

,

55^\circ

,

90^\circ

,

125^\circ

, and

145^\circ

. Thus

55^\circ

,

90^\circ

, and

145^\circ

are correct.

Matching calculated angle measures with the choices provided.

Key Concept

Perpendicular line relationships and supplementary angle properties of intersecting lines.

Estimated Time:1m 0s

Question 7Question →

In the geometric plane, line $k$ is parallel to line $m$ . A transversal line $t$ intersects line $k$ at point $P$ and line $m$ at point $Q$ . Ray $PR$ extends along line $k$ to the right of $P$ , and ray $QS$ extends along line $m$ to the right of $Q$ . The measure of interior angle $\angle RPQ$ is represented by $(3x + 20)^\circ$ and the measure of interior angle $\angle PQS$ is represented by $(2x + 10)^\circ$ . Which of the following statements must be true? Select all such statements.

Select all that apply

The value of

x

is

30

.

The measure of angle

\angle RPQ

is

110^\circ

.

The acute angle formed by line

t

and line

k

at point

P

measures

70^\circ

.

The value of

x

is

12

.

Angles

\angle RPQ

and

\angle PQS

are complementary angles.

Show answer & explanation

Answer: The value of $x$ is $30$ .; The measure of angle $\angle RPQ$ is $110^\circ$ .; The acute angle formed by line $t$ and line $k$ at point $P$ measures $70^\circ$ .

Answer

The true statements are that

x = 30

, the measure of angle

\angle RPQ

is

110^\circ

, and the acute angle formed by line

t

and line

k

at point

P

measures

70^\circ

.

Lines

k

and

m

are parallel, so interior angles on the same side of transversal

t

(angles

\angle RPQ

and

\angle PQS

) are supplementary. Solving

(3x + 20) + (2x + 10) = 180

gives

x = 30

. Substituting

x = 30

yields

\angle RPQ = 110^\circ

and

\angle PQS = 70^\circ

. The adjacent angle to

\angle RPQ

at point

P

is

180^\circ - 110^\circ = 70^\circ

, which is acute.

Step-by-Step Solution

1

Set up the geometric equation using the parallel line angle relationship.

(3x + 20) + (2x + 10) = 180

When two parallel lines are cut by a transversal, consecutive interior angles on the same side of the transversal are supplementary (their sum is

180^\circ

).

2

Solve the linear equation for

x

.

5x + 30 = 180 \implies 5x = 150 \implies x = 30

Combine like terms and isolate

x

using basic algebra.

3

Calculate the specific angle measures.

m

\angle RPQ = 3(30) + 20 = 110^\circ

and m

\angle PQS = 2(30) + 10 = 70^\circ

Substitute

x = 30

into the given algebraic angle expressions.

4

Determine the supplementary acute angle at point

P

.

180^\circ - 110^\circ = 70^\circ

Angles along a straight line form a linear pair and sum to

180^\circ

.

Key Concept

Parallel Lines and Consecutive Interior Angles

Question 8Question →

In a geometric plane, straight lines $L_1$ and $L_2$ intersect at point $P$ . The measure of the obtuse angle formed by the intersection of $L_1$ and $L_2$ is $(5x - 10)^\circ$ , and the measure of an adjacent acute angle is $(2x + 15)^\circ$ . Ray $PQ$ originates from point $P$ , is perpendicular to line $L_1$ , and lies entirely within the interior of the $(5x - 10)^\circ$ obtuse angle. What is the measure, in degrees, of the angle formed between ray $PQ$ and line $L_2$ ?

25^\circ

35^\circ

50^\circ

65^\circ

155^\circ

Show answer & explanation

Answer: $25^\circ$

Answer

25^\circ

Two intersecting lines form adjacent angles that sum to

180^\circ

. Solving

(5x - 10) + (2x + 15) = 180

gives

7x + 5 = 180

, so

x = 25

. The obtuse angle measure is

5(25) - 10 = 115^\circ

. Since ray

PQ

is perpendicular to line

L_1

, it forms a

90^\circ

angle with line

L_1

. The remaining angle between ray

PQ

and line

L_2

within the obtuse angle region is

115^\circ - 90^\circ = 25^\circ

.

Step-by-Step Solution

1

Set up an algebraic equation using the supplementary angle relationship.

(5x - 10) + (2x + 15) = 180

Adjacent angles formed by two intersecting straight lines lie on a straight line and are supplementary, summing to

180^\circ

.

2

Solve for the variable

x

.

7x + 5 = 180 \implies 7x = 175 \implies x = 25

Combining like terms simplifies the linear equation.

3

Calculate the degree measure of the obtuse angle.

Obtuse angle =

5(25) - 10 = 125 - 10 = 115^\circ

Substitute

x = 25

back into the expression

(5x - 10)^\circ

.

4

Calculate the angle between ray

PQ

and line

L_2

.

Angle =

115^\circ - 90^\circ = 25^\circ

Ray

PQ

is perpendicular to line

L_1

(

90^\circ

) and lies inside the

115^\circ

angle, dividing the obtuse angle into a

90^\circ

portion and the remaining angle adjacent to line

L_2

.

Key Concept

Supplementary angles on a straight line and angle subtraction with perpendicular rays

Estimated Time:1m 30s

Question 9Question →

Lines $L_1$ and $L_2$ intersect at point $P$ to form an acute angle of $54^\circ$ . Line $M$ passes through point $P$ and is perpendicular to line $L_1$ . Ray $R$ originates at point $P$ and lies in the interior of one of the obtuse angles formed by $L_1$ and $L_2$ . If Ray $R$ bisects the angle formed between line $M$ and line $L_2$ , what is the measure, in degrees, of the acute angle formed by Ray $R$ and line $L_1$ ?

Show answer & explanation

Answer: 72

Answer

The measure of the acute angle formed by Ray

R

and line

L_1

is

72^\circ

.

Lines

L_1

and

L_2

form an obtuse angle of

180^\circ - 54^\circ = 126^\circ

. Perpendicular line

M

consumes

90^\circ

of this angle relative to

L_1

, leaving

36^\circ

between line

M

and line

L_2

. Bisecting this remaining angle gives

18^\circ

. Ray

R

therefore lies

18^\circ

away from perpendicular line

M

, making an acute angle of

90^\circ - 18^\circ = 72^\circ

with line

L_1

.

Step-by-Step Solution

1

Determine the measure of the obtuse angle between lines

L_1

and

L_2

.

The obtuse angle measures

180^\circ - 54^\circ = 126^\circ

.

Adjacent angles along intersecting straight lines are supplementary and sum to

180^\circ

.

2

Find the angle between perpendicular line

M

and line

L_2

.

The angle between line

M

and line

L_2

within the obtuse region is

126^\circ - 90^\circ = 36^\circ

.

Line

M

is perpendicular to line

L_1

, taking up

90^\circ

of the

126^\circ

obtuse angle.

3

Determine the angle formed by Ray

R

after bisecting the

36^\circ

angle.

The angle between Ray

R

and line

M

is

36^\circ / 2 = 18^\circ

.

An angle bisector divides an angle into two equal parts.

4

Calculate the acute angle between Ray

R

and line

L_1

.

The acute angle formed between Ray

R

and line

L_1

is

90^\circ - 18^\circ = 72^\circ

.

Line

M

forms a

90^\circ

angle with line

L_1

. Subtracting the

18^\circ

offset created by Ray

R

yields the acute angle of

72^\circ

.

Key Concept

Supplementary angles, perpendicular lines, and angle bisectors

Estimated Time:2m 0s

Question 10Question →

In the geometric plane, line $k$ is parallel to line $m$ ( $k \parallel m$ ). Transversal line $t$ intersects line $k$ at point $A$ and line $m$ at point $B$ . Point $C$ lies on line $k$ to the left of $A$ , such that interior acute angle $\angle CAB = (3x - 10)^\circ$ . Ray $AD$ bisects $\angle CAB$ . Ray $BF$ is drawn into the region between lines $k$ and $m$ making an angle $\angle ABF = (x + 35)^\circ$ with transversal segment $AB$ . Ray $AD$ and ray $BF$ intersect at point $P$ inside the parallel region. Line $PB$ is extended past $P$ to intersect line $k$ at point $Q$ . If ray $AD$ is perpendicular to ray $BF$ , what is the measure of the obtuse angle formed at the intersection of line $QB$ and line $k$ ?

121^\circ

59^\circ

118^\circ

128^\circ

149^\circ

Show answer & explanation

Answer: $121^\circ$

Answer

The measure of the obtuse angle formed at the intersection of line

QB

and line

k

is

121^\circ

.

The measure

121^\circ

is correct because ray

AD

bisects

\angle CAB

, giving

\angle PAB = (1.5x - 5)^\circ

. Since

AD \perp BF

,

\triangle APB

is a right triangle where

(1.5x - 5) + (x + 35) = 90

, yielding

x = 24

. Consequently,

\angle QAB = 62^\circ

and

\angle ABQ = 59^\circ

. In

\triangle ABQ

, the acute angle

\angle AQB = 180^\circ - (62^\circ + 59^\circ) = 59^\circ

. The supplementary obtuse angle along line

k

is

180^\circ - 59^\circ = 121^\circ

.

Step-by-Step Solution

1

Express the angle

\angle PAB

in terms of

x

.

Since ray

AD

bisects

\angle CAB = (3x - 10)^\circ

,

\angle PAB = \frac{3x - 10}{2} = (1.5x - 5)^\circ

.

An angle bisector divides an angle into two equal halves.

2

Set up an equation using triangle

\triangle APB

.

In

\triangle APB

,

\angle APB = 90^\circ

because ray

AD \perp

ray

BF

. Therefore,

\angle PAB + \angle ABP = 90^\circ \implies (1.5x - 5) + (x + 35) = 90 \implies 2.5x + 30 = 90 \implies 2.5x = 60 \implies x = 24

.

The acute angles in a right triangle sum to

90^\circ

.

3

Calculate the measures of

\angle CAB

and

\angle ABQ

.

\angle CAB = 3(24) - 10 = 62^\circ

, and

\angle ABQ = \angle ABF = 24 + 35 = 59^\circ

.

Substitute

x = 24

back into the original angle expressions.

4

Determine the acute angle

\angle AQB

in triangle

\triangle ABQ

.

In

\triangle ABQ

, points

Q, C, A

lie on line

k

, so

\angle QAB = \angle CAB = 62^\circ

. Sum of angles in

\triangle ABQ

:

\angle AQB = 180^\circ - (62^\circ + 59^\circ) = 180^\circ - 121^\circ = 59^\circ

.

The interior angles of any triangle sum to

180^\circ

.

5

Find the supplementary obtuse angle at intersection point

Q

.

Obtuse angle

= 180^\circ - 59^\circ = 121^\circ

.

Angles forming a linear pair on a straight line are supplementary.

Key Concept

Parallel Lines, Transversals, Angle Bisectors, and Triangle Angle Sum Theorem

Question 11Question →

Four rays, $\vec{OA}$ , $\vec{OB}$ , $\vec{OC}$ , and $\vec{OD}$ , radiate from a common point $O$ in consecutive clockwise order such that $\vec{OA} \perp \vec{OC}$ and $\vec{OB} \perp \vec{OD}$ . If the measure of angle $\angle AOD$ is $3.5$ times the measure of angle $\angle BOC$ , what is the measure, in degrees, of angle $\angle AOB$ ?

36^\circ

40^\circ

50^\circ

65^\circ

140^\circ

Show answer & explanation

Answer: $50^\circ$

Answer

50^\circ

Because

\vec{OA} \perp \vec{OC}

and

\vec{OB} \perp \vec{OD}

, we know

\angle AOB + \angle BOC = 90^\circ

and

\angle BOC + \angle COD = 90^\circ

, which implies

\angle AOB = \angle COD

. The total angle

\angle AOD = \angle AOB + \angle BOD = \angle AOB + 90^\circ

. Using the given condition

\angle AOD = 3.5 \times \angle BOC

, we substitute

\angle BOC = 90^\circ - \angle AOB

to get

\angle AOB + 90^\circ = 3.5(90^\circ - \angle AOB)

, which solves to

\angle AOB = 50^\circ

.

Step-by-Step Solution

1

Set up angle variable definitions and perpendicular relationships.

Let

\angle AOB = x

,

\angle BOC = y

, and

\angle COD = z

. Since

\vec{OA} \perp \vec{OC}

, we have

x + y = 90^\circ

. Since

\vec{OB} \perp \vec{OD}

, we have

y + z = 90^\circ

.

Perpendicular rays form right angles measuring

90^\circ

.

2

Deduce the relationship between

x

,

y

, and

z

, and express

\angle AOD

in terms of

x

.

Subtracting

y

from both equations gives

x = 90^\circ - y

and

z = 90^\circ - y

, so

x = z

. Thus,

\angle AOD = x + y + z = x + 90^\circ

.

Adjacent angles sharing a vertex add up to form the overall combined angle.

3

Formulate and solve the equation based on the given ratio.

We are given

\angle AOD = 3.5 \times \angle BOC

, so

x + 90^\circ = 3.5y

. Substituting

y = 90^\circ - x

yields

x + 90^\circ = 3.5(90^\circ - x) \implies x + 90^\circ = 315^\circ - 3.5x \implies 4.5x = 225^\circ \implies x = 50^\circ

.

Substitution creates a single linear equation in terms of

x = \angle AOB

.

Key Concept

Perpendicular Ray Systems and Angle Addition

Estimated Time:1m 30s

Question 12Question →

In a plane, lines $l_1$ and $l_2$ are parallel ( $l_1 \parallel l_2$ ). A transversal line $t$ intersects $l_1$ at point $P$ and $l_2$ at point $Q$ . The measures of two consecutive interior angles on the same side of transversal $t$ are $(4x - 10)^\circ$ on line $l_1$ and $(3x + 50)^\circ$ on line $l_2$ . A third line $l_3$ passes through point $P$ and is perpendicular to line $l_2$ .

Which of the following statements MUST be true? Select all that apply.

Select all that apply

x = 20

The acute angle formed between line

t

and line

l_3

at point

P

measures

20^\circ

.

The obtuse angle formed at point

Q

between line

t

and line

l_2

measures

120^\circ

.

Line

l_3

is parallel to transversal line

t

.

The sum of all four interior angles located between lines

l_1

and

l_2

along transversal line

t

is

360^\circ

.

Show answer & explanation

Answer: $x = 20$ ; The acute angle formed between line $t$ and line $l_3$ at point $P$ measures $20^\circ$ .; The sum of all four interior angles located between lines $l_1$ and $l_2$ along transversal line $t$ is $360^\circ$ .

Answer

The correct statements are that

x = 20

, the acute angle formed between line

t

and line

l_3

at point

P

measures

20^\circ

, and the sum of all four interior angles located between lines

l_1

and

l_2

along transversal line

t

is

360^\circ

.

The statements confirming

x = 20

, the

20^\circ

acute angle between lines

t

and

l_3

, and the

360^\circ

interior angle sum are all correct because consecutive interior angles sum to

180^\circ

, perpendicular lines create

90^\circ

angles, and the four interior angles sum to

360^\circ

.

Step-by-Step Solution

1

Set up an equation for consecutive interior angles.

(4x - 10) + (3x + 50) = 180^\circ

Consecutive interior angles on the same side of a transversal intersecting parallel lines are supplementary.

2

Solve the algebraic equation for

x

.

7x + 40 = 180 \implies 7x = 140 \implies x = 20

Combine like terms and isolate

x

.

3

Calculate the interior angle measures at points

P

and

Q

.

Angle at

P = 4(20) - 10 = 70^\circ

; Angle at

Q = 3(20) + 50 = 110^\circ

Substitute

x = 20

back into the original expressions.

4

Determine perpendicular relationships and angle measure between line

t

and line

l_3

.

Angle between

t

and

l_3

at

P

is

90^\circ - 70^\circ = 20^\circ

Line

l_3 \perp l_2

implies

l_3 \perp l_1

because

l_1 \parallel l_2

. Thus

l_3

forms a

90^\circ

angle with

l_1

at

P

.

5

Sum the four interior angles between lines

l_1

and

l_2

.

70^\circ + 110^\circ + 70^\circ + 110^\circ = 360^\circ

The four interior angles comprise two pairs of interior angles measuring

70^\circ

and

110^\circ

each.

Key Concept

Parallel line angle theorems (consecutive interior angles, alternate interior angles) and perpendicular line relationships.

Question 13Question →

In a geometric plane, line $L_1$ is parallel to line $L_2$ . Points $A$ and $C$ lie on line $L_1$ , and points $B$ and $D$ lie on line $L_2$ . Line segments $AB$ and $CD$ intersect at point $X$ located between lines $L_1$ and $L_2$ . If measure of $\angle XAC = 42^\circ$ and measure of $\angle XDB = 35^\circ$ , what is the measure, in degrees, of $\angle AXC$ ?

Show answer & explanation

Answer: 103

Answer

The measure of

\angle AXC

is

103^\circ

.

Line

L_1

is parallel to line

L_2

, meaning segment

AC

is parallel to segment

BD

. Transversal line

CD

intersects both parallel lines, creating alternate interior angles

\angle XCA

and

\angle XDB

. Hence,

\angle XCA = \angle XDB = 35^\circ

. Inside triangle

ACX

, the three interior angles must sum to

180^\circ

. Substituting the values yields

\angle AXC = 180^\circ - (42^\circ + 35^\circ) = 103^\circ

.

Step-by-Step Solution

1

Identify parallel lines and the transversal line

Line segment

CD

acts as a transversal line intersecting parallel lines

L_1

and

L_2

.

Points

A

and

C

lie on line

L_1

while points

B

and

D

lie on line

L_2

with

L_1 \parallel L_2

.

2

Apply the alternate interior angles theorem

\angle XCA = \angle XDB = 35^\circ

When a transversal intersects two parallel lines, alternate interior angles are equal.

3

Calculate the target angle using the sum of interior angles in a triangle

\angle AXC = 180^\circ - (42^\circ + 35^\circ) = 103^\circ

The sum of interior angles in triangle

ACX

is

180^\circ

.

Key Concept

Properties of parallel lines intersected by a transversal and the triangle angle sum theorem.

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