Four straight lines intersect

2018 Handbook

2018 Test B Math · Question 5 of 57

Question

$Four straight lines cross at point P: one horizontal, one vertical and two slanted. Above the horizontal line, from right to left: a 60° angle between the horizontal line and the slanted line rising to the right, a 30° angle between that line and the vertical line, an unlabeled angle between the vertical line and the slanted line rising to the left, and the angle y° between that line and the horizontal line. Below the horizontal line, a 25° angle lies between the vertical line and the slanted line falling to the right.$

Four straight lines intersect at point P as shown above. What is the value of $y$ ?

Grid-in question: work out the answer and enter it as a number.

Show the answer

Correct answer: 65

Call the missing angle in the top half of the figure $x$ .

The sum of the four angles on the top of the figure is equal to 180°.

$x+y+30+60=180$

Since $x$ is a vertical angle with the 25° angle, then $x$ is also 25°. Use that to solve for $y$ .

$25+y+30+60=180$

$y+115=180$

$y=65$

Question analysis

This grid-in angle question shows four straight lines crossing at a single point P and asks for the size of an unknown angle y. Students combine vertical angles with the fact that angles along a straight line add to 180 degrees. One angle they need is unlabeled and must be found from its vertical partner.

NYS standards: 7.G.5 angle relationships and equations
Grade level: Grade 7 content, by NYS standard
Compared with grade 7: At grade 7 Using vertical angles and straight angles to write an equation for an unknown angle is a core grade 7 geometry task. Four lines create more angles than usual, but the reasoning stays at grade 7.
Estimated difficulty: Moderate: several steps, or a close reading with one strong distractor.
Thinking it demands: Depth of knowledge 2, skills and concepts · mainly apply
What makes it hard: Reading a figure, table or graph major
Combines ideas moderate
An answer in an unexpected form minor
Solution: 3 steps
Reading level: Question and choices: about grade 4, 16 words.
Students, last 30 days: Not enough answers yet to show (it takes 10).

Why it is hard, traps and wrong answers (shows the answer)

Why it is hard

The arithmetic is one subtraction; the work is in seeing the figure correctly. Students have to notice that a fourth angle sits between the 30° angle and y, recognize it as the vertical partner of the 25° angle, and then use the straight line. The drawing is not to scale: y is drawn close to the 60° angle across the vertical line, which rewards anyone who computes rather than eyeballs.

Reading a figure, table or graph major Understanding the question
Four lines make eight angles around P but only three are labeled; above the horizontal line sit 60°, 30°, an unlabeled angle and y°, and that unlabeled angle is easy to overlook. (Lesh, Post & Behr, 1987; Kirsch & Mosenthal, 1990)
Combines ideas moderate Working it out
The solution joins two facts: the unlabeled angle is vertical to the 25° angle below P, and the four angles above the horizontal line make a straight angle of 180°. (Embretson, 1998; Daniel & Embretson, 2010)
An answer in an unexpected form minor Choosing the answer
As a grid-in, nothing confirms the result, and the drawing shows y at about 60°, so a reading of the picture produces a number that looks right. (Enright, Morley & Sheehan, 2002)

Strategy: Pick a straight line through P and list the angles on one side of it; replace the unlabeled one with the labeled angle vertical to it, then subtract from 180.

Traps

Misreads the figure: Reads a value, label or tick spacing in a figure, table or graph wrongly.
Stops one step early: A choice is a number from partway through the solution.
Trusting the picture: The figure suggests something it does not state; work from the given facts.

The wrong answers, classified

90 Misreads the figure (Lesh, Post & Behr, 1987; Kirsch & Mosenthal, 1990)
Reveals partial knowledge Expected pull: strong
Misses the unlabeled angle between the 30° angle and y, so subtracts only 30 and 60 from 180.
115 Stops one step early (King et al., 2004; Gierl et al., 2017)
Reveals partial knowledge Expected pull: moderate
Adds the three known angles beside y, 25 + 30 + 60, and grids that sum without subtracting it from 180.
60 Trusting the picture (Lesh, Post & Behr, 1987)
Reveals a surface strategy Expected pull: moderate
Judges from the drawing, where y looks like a mirror image of the 60° angle across the vertical line, instead of computing it.

How the questions are analyzed, and the research behind it

Worked solution, step by step

Formulas used

Angles on a straight line: $a + b + c + d = 180$; a, b, c and d are angles that sit side by side along one side of a straight line, all meeting at one point. Together they make a half turn, which is 180 degrees. This works for any number of angles, not just four.
Vertical angles: $\angle 1 = \angle 2$; When two lines cross, they make an X. Angle 1 and angle 2 are opposite each other across the crossing point. Angles like these are called vertical angles. They are always equal.

Look closely at the figure
Four lines cross at point P. Look above the flat, or horizontal, line. From right to left there are four angles: 60°, 30°, an angle with no label, and y°.
Below the horizontal line there is a 25° angle. You need to find y.
Find the unlabeled angle
The unlabeled angle sits between the up-and-down, or vertical, line and the slanted line rising to the left.
That slanted line keeps going through P and down to the right. The 25° angle sits between that same line and the vertical line, below P.
So the two angles are opposite each other across P. They are vertical angles, so they are equal. The unlabeled angle is 25°.
Add the angles above the line
The four angles above the horizontal line fill one side of a straight line. So they add up to 180°.
$a + b + c + d = 180$
$y + 25 + 30 + 60 = 180$
$y + 115 = 180$
Solve for y
Take 115 away from 180 to get y alone.
$y = 180 - 115 = 65$
Watch out: Do not skip the unlabeled angle. Using only 30 and 60 gives 90. Do not stop at the sum, 115, either. And do not guess from the picture. Angle y looks like 60°, but the drawing is not exact.
So you would grid 65.

Check your answer

Add all four angles above the horizontal line: $65 + 25 + 30 + 60 = 180$ . They fill a straight line exactly, so 65 fits.

The original DOE question

Question 62 as printed on page 153 of the DOE handbook. — As printed in the DOE’s Specialized High Schools Student Handbook 2018–2019, page 153, question 62. Open page 153 of the handbook (PDF)