In a plane, lines l_1 and l_2 are parallel (l_1 \parallel l_2). A transversal line t inter… — GRE General Test

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.

$x = 20$ Cevap
The acute angle formed between line $t$ and line $l_3$ at point $P$ measures $20^\circ$ .Cevap
C
The obtuse angle formed at point $Q$ between line $t$ and line $l_2$ measures $120^\circ$ .
D
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 $360^\circ$ .Cevap

Cevap

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

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

Adım Adım Çözüm

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.

Solve the algebraic equation for

x

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

Combine like terms and isolate

x

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.

Determine perpendicular relationships and angle measure between line

t

and line

l_3

Angle between

t

and

l_3

P

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

P

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.

Anahtar Kavram

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

Soru

Cevap

Adım Adım Çözüm

Anahtar Kavram

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