Secondary 2 Angles, Lines and Polygons

7 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. 7 of the 12 questions are printed below. The other 5 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

   An angle measures $x$ degrees, where $0<x\le 360$.

   1. For each class below, write the condition on $x$ as an inequality or an equation: acute, right, obtuse, straight, reflex, full angle.
   2. Bruno draws an angle whose measure is three times the smallest whole number of degrees that is obtuse. Give the measure of Bruno's angle and classify it.
3. Q3Classifying Polygons

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

4. Q4Congruent and Similar Figures

   A logo is drawn as a triangle $△ABC$ with $m\overline{AB}=6$ cm, $m\overline{BC}=10$ cm and $m\overline{CA}=14$ cm. It is redrawn larger for a banner as $△DEF$, with $△ABC~△DEF$ and $m\overline{DE}=9$ cm.

   1. Find the ratio of similarity from $△ABC$ to $△DEF$.
   2. Find $m\overline{EF}$ and $m\overline{FD}$.
   3. In the small logo, $m\angle B={47}^{\circ }$. Give $m\angle E$ and justify your answer.
5. Q5Geometry

   Each of the four statements below uses a geometric term or symbol incorrectly. Rewrite each one so that it is correct, and say in a few words what the mistake was.

   1. “$m\angle ABC=6$ cm.”
   2. “$△PQR~△STU$, so all six sides have the same length.”
   3. “These two lines are perpendicular, because they never meet.”
   4. “$\overline{AB}\parallel \angle C$.”
6. Q6Plane figures

   A student's list contains: a disc, a cube, a regular octagon, a cylinder, a right trapezoid, a circle, a triangular prism.

   1. Sort the seven items into plane figures and solids.
   2. Among the plane figures, which ones are polygons? Justify your sorting with the definition of a polygon.
   3. Explain the difference between a circle and a disc.
7. Q7Polygons

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

8. Q8Regular Polygons

   The floor of a gazebo in a municipal park is a regular octagon.

   1. Find the sum of the measures of its interior angles.
   2. Find the measure of one interior angle, and classify that angle.
   3. Eight identical triangular floor panels meet at the centre of the gazebo, one for each side. Find the angle each panel makes at the centre.
9. Q9Relationships Between Angles

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

10. Q10Similarity, Congruence, and Equivalence

    For each pair of plane figures, say which of the words congruent, similar and equivalent apply. More than one word may apply, or none.

    1. A rectangle $4$ cm by $9$ cm and a square of side $6$ cm.
    2. Two squares, each of side $7$ cm.
    3. A right triangle whose two sides forming the right angle measure $8$ cm and $5$ cm, and a right triangle whose two sides forming the right angle measure $10$ cm and $4$ cm.
    4. Two regular pentagons, one of side $3$ cm and one of side $7.5$ cm.

    Then explain why congruent figures are always both similar and equivalent, while equivalent figures need not be similar.

11. Q11Special Lines

    A triangular park $△ABC$ has $m\overline{BC}=30$ m, and the height drawn from $A$ to the side $\overline{BC}$ measures $12$ m. The median from $A$ meets $\overline{BC}$ at $M$.

    1. Find the area of the park, and find $m\overline{BM}$.
    2. The median $\overline{AM}$ splits the park into $△ABM$ and $△AMC$. Find the area of each, and justify why the median always splits a triangle into two equivalent parts.
    3. Is the height from $A$ always inside the triangle? Describe a triangle where it is not.
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 7 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.

Geometry at this level is a naming game, and the names are the marks

Nearly every question on this sheet can be answered in two lines of arithmetic once you have said which relation applies. Q5 makes that explicit by handing you four statements that use a geometric term or a symbol incorrectly and asking you to repair each one — a question with no computation in it at all, and one of the most useful on the sheet. If you can say why "m∠ABC = 6 cm" is nonsense, most of the rest follows.

Classifying angles, written as conditions (Q2)

Q2 asks for each class as an inequality or an equation rather than as a description, and that is a genuinely different demand: not "an obtuse angle is bigger than a right angle" but 90 < x < 180. Acute is 0 < x < 90, right is x = 90, obtuse is 90 < x < 180, straight is x = 180, reflex is 180 < x < 360, and a full angle is x = 360.

Two details decide whether the answer is right. The boundaries are excluded — 90° is not obtuse, it is right — so the inequalities are strict, and an equation, not an inequality, is what describes a class holding exactly one value. And the smallest whole number of degrees that is obtuse is 91, not 90.1 or 90: a question asking for a whole number of degrees has quietly told you which values are allowed.

Convex or not, and what a polygon actually is (Q3, Q6)

A polygon is a closed figure made of straight segments joined end to end. Straight is the operative word, which is why a disc and a circle are plane figures but not polygons, and why a cube and a cylinder are not plane figures at all — they are solids. Q6 sorts a mixed list on exactly those two questions, and asks for the distinction between a circle (the curved line) and a disc (the line together with everything inside it).

The angle sum, built rather than memorised (Q7, Q8)

Q7 has you count, from one chosen vertex, how many diagonals a pentagon, a heptagon and a decagon allow and how many triangles those diagonals cut the polygon into. The counts are n − 3 diagonals and n − 2 triangles, and the second of those is the whole reason the angle sum formula is what it is: each triangle contributes 180°, and every interior angle of the polygon is used exactly once.

For a regular polygon that sum divides evenly, so one interior angle is (n − 2) × 180° ÷ n. An octagon gives 1080° ÷ 8 = 135°, an obtuse angle, and Q8 asks you to classify it as well as compute it. The central angle is a separate and simpler idea: the full turn at the centre is shared equally between the sides, so it is 360° ÷ n, which is 45° for the octagon. Interior angle and central angle are different angles in different places — mixing them up is the most common error on this material.

Parallel lines, and the year's real novelty (Q9)

Q9 is where Secondary 2 shows its hand. Two angles formed by a transversal are given as expressions in x, and the geometry does not give you the answer — it gives you the equation. Corresponding angles are equal, so the two expressions are set equal to each other, and what follows is the first-degree solving from the algebra half of the course.

Then read the last step carefully. Solving gives x, which is not an angle and is not the answer; substitute it back into both expressions to get the two measures, and check that they agree, since that agreement is what the relation promised. To find a further angle at the same point, use the fact that angles on a straight line add to 180°, and name the relation you used — that naming is the part being assessed.

Similar figures: one ratio, used everywhere (Q4)

Two figures are similar when their corresponding angles are equal and their corresponding sides are all in the same ratio. Q4 gives one pair of corresponding sides, which fixes that ratio, and every other length follows by multiplication.

The angles are where marks are won cheaply. Under a similarity the angles do not change at all — the figure is scaled, not distorted — so a 47° angle stays 47° however much bigger the banner is. Two habits keep the work honest: match the vertices in the order the similarity statement gives them, and check that your ratio points the way you think it does. Going from the small figure to the large one, every length is multiplied by a number greater than 1; if your answers are coming out smaller, the ratio is upside down.

Congruent, similar, equivalent — three different words (Q10)

Congruent means same shape and same size: every corresponding side and angle equal. Similar means same shape, with all lengths in one fixed ratio. Equivalent means same area, and says nothing whatever about shape — which is how a 4 cm by 9 cm rectangle and a square of side 6 cm can be equivalent while being neither similar nor congruent.

The implications run one way only, and Q10 asks you to explain that direction. Congruent figures are always similar and always equivalent. Equivalent figures need not be similar, need not be congruent, and need not even have the same number of sides. When a question asks which words apply, check all three separately rather than assuming that one settles the others.

Median and height are not the same line (Q11)

A median runs from a vertex to the midpoint of the opposite side; a height runs from a vertex perpendicular to the opposite side, or to that side extended. In most triangles they are two different lines from the same vertex, and confusing them means using the wrong length in the area formula.

Two facts from Q11 are worth carrying. A median always splits a triangle into two equivalent triangles, because the two halves have equal bases and share the same height — an area argument, not a symmetry one. And a height is not always inside the triangle: in an obtuse triangle the foot of the height from one of the acute vertices lands outside the opposite side, which is why that side has to be extended before the perpendicular can meet it.

Getting the most out of it

Mark the relation on the drawing before you compute

Redraw the figure large, mark the pairs of angles you are treating as corresponding or alternate, and write the relation beside them in words. Most wrong answers on this material are the right method applied to the wrong pair, and that is a mistake visible in a drawing and invisible in the algebra.

When an angle is an expression, solve for x and then keep going

The value of x is never the answer to a geometry question. Substitute it back into each expression, state the angle measures with their degree symbols, and check that the two values are consistent with the relation you used. That check costs one line and catches an arithmetic slip on the spot.

Answer the naming questions in full sentences

The question that asks you to repair four incorrect statements, and the parts that ask which relation you used, are the most tempting to answer in your head. Write them out. The reasoning is what is being assessed on this topic, and a bare correct number does not show it.

Common questions