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Views and congruence

Ten questions of mixed difficulty, covering Views and congruence. Print it, or work through it on screen — the answer key starts on its own page.

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Views and congruence

Math 8 · maddyhelps.com

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  1. Two polygons are congruent when they have

    1. a) the same shape, but maybe different sizes
    2. b) the same area
    3. c) the same number of sides
    4. d) the same size and the same shape

  2. Three cubes are stacked one on top of another. What does the top view show?

    1. a) 6 squares
    2. b) 3 squares in a row
    3. c) 3 squares in a column
    4. d) 1 square

  3. Triangle A has sides of 5 cm, 7 cm and 9 cm. Which triangle must be congruent to it?

    1. a) a triangle with sides of 10 cm, 14 cm and 18 cm
    2. b) a triangle with sides of 9 cm, 5 cm and 7 cm
    3. c) any triangle with two sides of 5 cm and 7 cm
    4. d) any triangle with a perimeter of 21 cm

  4. A square and a rhombus both have sides of 5 cm. Are they congruent?

    1. a) Only if the rhombus's angles are all 90°; equal sides are not enough.
    2. b) No, because a square and a rhombus can never be congruent.
    3. c) Yes, because all four sides of one match all four sides of the other.
    4. d) Yes, because both have two pairs of parallel sides.

  5. Triangle JKL is congruent to triangle MNO, with the vertices matching in that order. Angle J = 50° and angle K = 70°. What is angle O?

    1. a) 70°
    2. b) 50°
    3. c) 60°
    4. d) 120°

  6. A top view of an object made of cubes has a number written in each square. What does each number tell you?

    1. a) how many faces of that cube can be seen
    2. b) the order the cubes were put down in, first to last
    3. c) how many cubes are in that row
    4. d) how many cubes are stacked on that square

  7. A top view shows how many cubes are in each stack. The back row reads 3, 2 and the front row reads 1, 1, from left to right. What does the front view show?

    1. a) a column of 1 square on the left and 1 on the right
    2. b) 4 squares in a 2 by 2 block
    3. c) a column of 3 squares on the left and 2 on the right
    4. d) a column of 4 squares on the left and 3 on the right

  8. Triangle ABC is congruent to triangle DEF, with A matching D, B matching E and C matching F. Side AB is 7 cm. Which side of triangle DEF is 7 cm?

    1. a) It could be any of the three.
    2. b) DE
    3. c) DF
    4. d) EF

  9. Two rectangles each have an area of 24 cm². Must they be congruent?

    1. a) Yes: two rectangles with the same area always match.
    2. b) No: they would also need the same angles.
    3. c) No: 3 cm by 8 cm and 4 cm by 6 cm have that area.
    4. d) Yes: all their angles are 90°, so they match.

  10. Three cubes sit in a row on a table, and a fourth cube sits on top of the left-hand one. What does the view from the right side show?

    1. a) 4 squares in an L shape
    2. b) 1 square
    3. c) 3 squares in a row
    4. d) 2 squares in a column

Answer key · Views and congruence

Math 8 · maddyhelps.com

  1. d) the same size and the same shape — Congruent means identical in size and shape: one would fit exactly on top of the other, perhaps after being turned or flipped. The same shape at a different size is similar, not congruent, and a 2 by 8 rectangle and a 4 by 4 square have the same area without matching at all.
  2. d) 1 square — Looking straight down, you see only the top face of the top cube; the other two are hidden directly underneath it. 3 squares in a column is the front view, which shows the height — a top view never does.
  3. b) a triangle with sides of 9 cm, 5 cm and 7 cm — Three side lengths fix a triangle's size and shape, and the order they are listed in does not matter — 9, 5 and 7 are the same three sides. 10, 14 and 18 is the same shape twice as big, which is similar, not congruent, and a matching perimeter or two matching sides still leaves room for a different triangle.
  4. a) Only if the rhombus's angles are all 90°; equal sides are not enough. — For shapes with more than three sides, matching sides do not fix the shape: a rhombus with 5 cm sides can lean over, changing its angles while its sides stay 5 cm. It matches the square only when its angles are all 90° — and then it is a square, since a square is a special kind of rhombus.
  5. c) 60° — Angle O matches angle L, and the angles of triangle JKL add to 180°: L = 180° − 50° − 70° = 60°, so angle O = 60°. 70° is angle N, which matches K, and 120° adds the two given angles instead of taking them away from 180°.
  6. d) how many cubes are stacked on that square — A top view shows only where the stacks stand, not how tall they are, so the number in each square records the height: how many cubes are in that stack. With the numbers, the top view describes the whole object, and the front and side views can be drawn from it.
  7. c) a column of 3 squares on the left and 2 on the right — From the front, each column shows its tallest stack, and taller stacks at the back rise above the short ones in front. The left column has stacks of 3 and 1, so it shows 3 squares; the right has 2 and 1, so it shows 2. Showing only the front row forgets the stacks behind it, and 4 and 3 add the stacks, counting cubes that are hidden.
  8. b) DE — The letters give the matching: A goes with D and B goes with E, so side AB matches side DE, and DE = 7 cm. Sides do not match by where they sit on the page — a congruent copy can be turned or flipped — which is why the order of the letters matters.
  9. c) No: 3 cm by 8 cm and 4 cm by 6 cm have that area. — Congruent polygons match in every side and every angle. All rectangles have four 90° angles, so the angles always match — it is the sides that can differ, and 3 by 8 and 4 by 6 both give 24 cm² without being the same shape. Equal area, or equal angles, is not enough.
  10. d) 2 squares in a column — From the right, the three cubes on the table line up one behind another, so they look like a single square, and the extra cube, on top of the one furthest away, shows above it: 2 squares in a column. 1 square forgets that a side view shows the tallest stack in each line, even at the back, and the L shape is the front view.