Secondary 1 Angles and Lines Worksheet

8 of the 12 questions

Each question targets one named concept from the sheet. Read them here, or print the PDF — it has working space under each one. 8 of the 12 questions are printed below. The other 4 are built on a diagram or a table of values that does not translate to the page, so they are in the free PDF — marked below where they would have come.

1. Q1Angles

   This question is built around a diagram or a table of values. Open it in the PDF.

2. Q2Classifying Angles
   1. Classify each of these angles as acute, right, obtuse, straight, reflex or a full angle: ${89}^{\circ }$, ${90}^{\circ }$, ${151}^{\circ }$, ${180}^{\circ }$, ${237}^{\circ }$, ${360}^{\circ }$.
   2. At exactly $4$ o'clock, the two hands of a clock form an angle. Give its measure and classify it.
3. Q3Constructing Perpendicular and Parallel lines

   Antoine has a straight line $d$ drawn on his page and a point $P$ that is not on $d$. He wants to construct the line through $P$ that is parallel to $d$, using only a ruler and a set square. His four steps are listed here in the wrong order:

   1. Slide the set square along the ruler, keeping it pressed against the ruler, until the edge that was lying on $d$ passes through $P$.
   2. Hold the ruler firmly against a second edge of the set square, and do not move the ruler again.
   3. Trace the line along the edge that now passes through $P$.
   4. Place one edge of the set square exactly along the line $d$.

   Give the correct order of the four steps, then explain in one or two sentences why the line obtained really is parallel to $d$.

4. Q4Constructing a Bisector

   In a stained-glass panel, the ray $BD$ is the bisector of $\angle ABC$.

   1. If $m\angle ABD={38}^{\circ }$, find $m\angle ABC$.
   2. In a second panel Rosalie measures $m\angle ABD={38}^{\circ }$ and $m\angle DBC={39}^{\circ }$, and says that $BD$ is the bisector there as well. Is she right? Explain.
   3. In the compass-and-ruler construction of a bisector, the first two arcs are drawn from the vertex and must have the same radius. Say what would go wrong if the two radii were different.
5. Q5Constructing a Median

   In $△KLM$, the median drawn from the vertex $K$ meets the side $LM$ at the point $N$, and $LM$ measures $18$ cm.

   1. Find the length of $\overline{LN}$.
   2. Charlotte instead draws the segment from $K$ that meets $LM$ at a right angle, and calls it the median. Name the segment she actually drew, and say what would have to be true about the triangle for her segment and the median to be the same one.
6. Q6Constructing a Perpendicular Bisector

   Two goal posts $P$ and $Q$ are planted $14$ m apart on a level field, and a coach constructs the perpendicular bisector of the segment $\overline{PQ}$.

   1. At what distance from $P$ does the perpendicular bisector cross $\overline{PQ}$?
   2. What angle does it make with $\overline{PQ}$, and how is that angle classified?
   3. A cone is placed at a point $T$ on the perpendicular bisector, $9$ m away from $P$. How far is the cone from $Q$? Justify your answer.
7. Q7Geometry

   Recognizing which relationship applies is most of the work in geometry. For each situation below, name the object or relationship you would use, and state the one thing that must be checked before you are allowed to use it.

   1. Two angles sit side by side and their outer sides together form one straight line.
   2. Two straight lines cross, and you are looking at the two angles that face each other across the crossing point.
   3. Two lines are cut by a third line, and you want to conclude that a certain pair of angles is congruent.
   4. A segment is drawn from a vertex of a triangle to a point on the opposite side, and you want to conclude that it cuts that side into two equal parts.
8. Q8Relationships Between Angles

   This question is built around a diagram or a table of values. Open it in the PDF.

9. Q9Relationships Between Two Lines

   A street plan shows several straight streets. For each pair below, name the relationship between the two lines (distinct parallel, coincident, intersecting, or perpendicular).

   1. Rue des Ormes and Rue Bellevue never meet, however far each one is extended.
   2. Boulevard Central crosses Rue des Ormes, and the four angles at the crossing are all right angles.
   3. Rue du Parc and Avenue Sud cross at a single point, forming angles of ${55}^{\circ }$ and ${125}^{\circ }$.
   4. On an older map the same road is drawn twice, once as Rue Principale and once as Route 138, one exactly on top of the other over its whole length.

   Then say which of these pairs are also intersecting lines, and explain the one that surprises people.

10. Q10Special Lines

    Four special lines are described inside $△ABC$. Name each one — median, height, angle bisector, or perpendicular bisector.

    1. It starts at $A$, ends on $\overline{BC}$, and meets $\overline{BC}$ at a right angle.
    2. It starts at $A$ and ends at the midpoint of $\overline{BC}$.
    3. It starts at $A$ and cuts $\angle BAC$ into two congruent angles.
    4. It crosses $\overline{BC}$ at its midpoint and is perpendicular to $\overline{BC}$, but it is not required to pass through any vertex.

    Then say which of the four is the only one that does not have to pass through a vertex of the triangle, and why.

11. Q11Types of Lines

    This question is built around a diagram or a table of values. Open it in the PDF.

12. Q12Synthesis — drawing on several sheets in this topic

    This question is built around a diagram or a table of values. Open it in the PDF.

The 6 challenge problems for this topic are a separate, paid sheet and are not reproduced here.

How to do every concept on this sheet

This is the part worksheet sites usually leave out. Below is the actual reasoning behind each group of questions — not a full solution set, the decisions that get you to one. Read it before you start, or after you get stuck.

An angle has a name before it has a measure (Q1)

Q1 opens with three light beams leaving one point and asks for the angle between two of them by name. The three-letter form has one rule and it never changes: the middle letter is the vertex. So ∠AOC and ∠COA are the same angle, and both say that the corner is at O and that the two sides are the rays OA and OC. Writing ∠AOC when you mean ∠AOB is not a small slip — it names a different angle, and everything after it is about something else.

The measure then comes from a fact so ordinary it is easy to miss: when a ray lies inside an angle it cuts that angle into two parts, and the two parts add to the whole. That single sentence answers both of the remaining parts of Q1 — once forwards, adding the two pieces to get the total, and once backwards, subtracting a known piece from a total that has not changed. Whenever an answer comes out by subtraction, add your two pieces back together as a check; it costs one line.

The six classes, and the two angles a clock makes (Q2)

Classification is decided by boundaries, not by appearance: acute below 90°, right at exactly 90°, obtuse between 90° and 180°, straight at exactly 180°, reflex between 180° and 360°, and a full angle at 360°. The values that get misplaced are the ones sitting right on a boundary — 90° and 180° each have their own name and belong to no other class.

Parallel lines cut by a transversal (Q8)

Eight angles appear, and between them they take only two different values. Q8 gives one value and asks for three others, each with its relationship named. Locate a pair by asking two questions about it, in this order:

The four special lines, and the one that ignores the vertex (Q4, Q5, Q6, Q10)

Four names, four definitions, and the whole difficulty is that they overlap in the figure students draw most often. Q10 asks you to name each one from its description, so learn them as descriptions rather than as pictures:

Q5 is that confusion made concrete: a segment drawn from a vertex at a right angle to the opposite side is the height, not the median, and the two coincide only in a triangle with two equal sides. Q6 adds the property that makes the perpendicular bisector genuinely useful — every point on it is the same distance from the two endpoints — which is why a cone placed anywhere along it is automatically equidistant from both posts, with nothing to compute. Q4 pins the bisector down the same way: two congruent angles, so a 38° half means a 76° whole, and 38° beside 39° is simply not a bisector, however close it looks.

A construction is only finished when you can say why it works (Q3)

Q3 gives the four steps for drawing a parallel with a ruler and a set square, out of order, and asks for the order and the reason. That second demand is the point of every construction question at this level. The reason here: once the ruler is held still, sliding the set square along it moves the set square without ever turning it, so the tracing edge keeps the exact direction it had while lying on the original line. The ruler is then a transversal cutting both lines at equal corresponding angles — which is the condition for parallel.

The same habit answers the compass question in Q4. The two opening arcs must share a radius so that the two points they mark are equidistant from the vertex; that equal distance is what makes the rest of the figure symmetric about the ray being built, and the symmetry is what makes the ray a genuine bisector. Change one radius and the symmetry goes, and the guarantee goes with it. A construction step you cannot justify is a step you will misremember.

Lines, rays, segments and what counts as a polygon (Q9, Q11)

Q11 sorts four drawings by two independent questions — open or closed? and broken, curved, or a mixture? — and only then asks which of them is a polygon. A polygon is a closed broken line that does not cross itself: every part a straight segment, the last endpoint back at the start, and no crossings anywhere. A closed figure with one curved part fails the first condition; a closed path whose segments cross one another partway along fails the last one.

The final part of Q11 is the one worth remembering. Only a segment has a measurable length, because only a segment has two endpoints. A straight line runs forever in both directions and a ray runs forever in one, so asking for the length of either is asking for a number that does not exist.

Q12: the synthesis question

The last free question puts three of the sheet's ideas in a row — halve an angle with a bisector, carry a measure across two parallel lines with corresponding angles, then use both results to decide whether a new pair of lines is parallel. The decision in that final part is the interesting one, because it runs the parallel test backwards: two lines cut by a transversal are parallel only when their corresponding angles are congruent, and here the two numbers you produced are different. Different corresponding angles is a proof of not parallel, not merely a failure to prove parallel.

Getting the most out of it

Mark the figure before you compute

Write every measure you are given onto the drawing, and write each one you deduce onto it as soon as you have it. Half the questions here become obvious once the diagram is complete, and a figure carrying your own numbers is the only way to notice that two of them contradict each other.

Name the relationship in words, every time

“118°” is half an answer. “118°, because those two angles are adjacent supplementary” is the whole one, and the naming questions on this sheet exist to make that a habit. If you cannot name the relationship you used, you have not finished checking that you were allowed to use it.

Check by a second route

Angles round a point add to 360°, angles along a straight line add to 180°, and the two parts of a divided angle add back to the whole. Every answer on this sheet can be confirmed by one of those three sums, and confirming it takes a single line.

Common questions