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Lines and Angles

3 questions

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Question 1Question →

Lines L1L_1 and L2L_2 are parallel. Points PP and QQ lie on line L1L_1, and points RR and SS lie on line L2L_2. Segment PSPS and segment QRQR intersect at point TT. If the measure of angle TPQTPQ is 38∘38^\circ and the measure of angle PTQPTQ is 105∘105^\circ, what is the measure, in degrees, of angle TRSTRS?

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Answer: 37

Answer

37
To find the measure of angle TRSTRS, we first find the measure of angle TQPTQP using the property that the sum of interior angles in triangle TPQTPQ is 180∘180^\circ. This gives ∠TQP=180∘−38∘−105∘=37∘\angle TQP = 180^\circ - 38^\circ - 105^\circ = 37^\circ. Since lines L1L_1 and L2L_2 are parallel, the transversal line QRQR creates alternate interior angles that are equal in measure. Therefore, angle TRSTRS is equal to angle TQPTQP, which is 37∘37^\circ.

Step-by-Step Solution

1
Calculate the measure of angle TQPTQP in triangle TPQTPQ.
∠TQP=37∘\angle TQP = 37^\circ
The sum of the angles in a triangle is 180∘180^\circ. Therefore, ∠TQP=180∘−∠TPQ−∠PTQ=180∘−38∘−105∘=37∘\angle TQP = 180^\circ - \angle TPQ - \angle PTQ = 180^\circ - 38^\circ - 105^\circ = 37^\circ.
2
Relate angle TQPTQP to angle TRSTRS using parallel line properties.
∠TRS=37∘\angle TRS = 37^\circ
Since line L1L_1 is parallel to line L2L_2, the transversal line QRQR creates equal alternate interior angles. Thus, ∠TRS=∠TQP=37∘\angle TRS = \angle TQP = 37^\circ.

Key Concept

Alternate interior angles and triangle angle sum theorem
Question 2Question →

Two straight support beams on a bridge intersect at a single point. One of the angles formed by their intersection measures 74∘74^\circ. An adjacent angle along the straight line of one of the beams has a measure of (2x+16)∘(2x + 16)^\circ. What is the value of xx?

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Answer: 45

Answer

45
Since the two angles form a linear pair along a straight support beam, their measures must sum to 180∘180^\circ. We can write the equation as 74+(2x+16)=18074 + (2x + 16) = 180. Combining like terms gives 2x+90=1802x + 90 = 180. Subtracting 9090 from both sides yields 2x=902x = 90. Finally, dividing by 22 gives x=45x = 45.

Step-by-Step Solution

1
Set up the equation using the property of supplementary angles.
74+(2x+16)=18074 + (2x + 16) = 180
Adjacent angles on a straight line form a linear pair, which means they are supplementary and sum to 180∘180^\circ.
2
Combine the constant terms on the left side of the equation.
2x+90=1802x + 90 = 180
Adding 7474 and 1616 simplifies the constant terms to 9090.
3
Isolate the variable term by subtracting 90 from both sides of the equation.
2x=902x = 90
Subtracting 9090 from 180180 leaves 9090 on the right side.
4
Solve for x by dividing both sides of the equation by 2.
x=45x = 45
Dividing 2x2x and 9090 by 22 isolates the variable xx.

Key Concept

Adjacent angles on a straight line are supplementary and add up to 180 degrees.
Estimated Time:45s
Question 3Question →

In the figure, lines ABAB and CDCD intersect at point OO, and ray OEOE is perpendicular to line ABAB. If the measure of ∠EOC\angle EOC is 27∘27^\circ, what is the measure, in degrees, of ∠BOD\angle BOD?

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Answer: 63

Answer

The measure of ∠BOD\angle BOD is 63 degrees.
The correct answer is 63. Since ray OEOE is perpendicular to line ABAB, the angle ∠EOA\angle EOA is a right angle measuring 90∘90^\circ. The adjacent angles ∠AOC\angle AOC and ∠EOC\angle EOC make up ∠EOA\angle EOA, which means ∠AOC=90∘−27∘=63∘\angle AOC = 90^\circ - 27^\circ = 63^\circ. Finally, because lines ABAB and CDCD intersect at point OO, the angle ∠BOD\angle BOD and the angle ∠AOC\angle AOC are vertical angles. Vertical angles are equal in measure, so the measure of ∠BOD\angle BOD is 63∘63^\circ.

Step-by-Step Solution

1
Identify the angle formed by perpendicular lines.
The measure of ∠EOA\angle EOA is 90∘90^\circ.
Since ray OEOE is perpendicular to line ABAB, the angle ∠EOA\angle EOA is a right angle.
2
Calculate the measure of ∠AOC\angle AOC.
The measure of ∠AOC\angle AOC is 63∘63^\circ.
Angles ∠AOC\angle AOC and ∠EOC\angle EOC are adjacent and form the right angle ∠EOA\angle EOA, meaning they are complementary: ∠AOC=90∘−27∘=63∘\angle AOC = 90^\circ - 27^\circ = 63^\circ.
3
Determine the measure of ∠BOD\angle BOD.
The measure of ∠BOD\angle BOD is 63∘63^\circ.
Lines ABAB and CDCD intersect at point OO, making ∠BOD\angle BOD and ∠AOC\angle AOC vertical angles. Since vertical angles are equal, the measure of ∠BOD\angle BOD is equal to the measure of ∠AOC\angle AOC.

Key Concept

Using properties of perpendicular lines and vertical angles to solve for unknown angle measures.
Estimated Time:1m 30s