Math 8 · Views and congruence
Flat views of a solid, and shapes that match exactly
A set of house plans shows the building from above, from the front and from the side, and a builder can put it together from those flat drawings. This unit draws and reads views of objects built from cubes, then turns to congruence: when two polygons are exact copies, and how to show it by matching every side and every angle.
The words, first
The idea: Corresponding is the word the second half depends on: the parts that land on each other when one shape is placed on the other.
| Word | What it means |
|---|---|
| Top, front and side views | What you see looking straight down, straight at the front, and straight at one side of a 3-D object, drawn flat. Every visible cube face shows as a square. |
| Polygon | A closed flat shape with straight sides, such as a triangle or a quadrilateral (4 sides). |
| Vertex | A corner of a polygon. A polygon is named by its vertices in order around it: ABCD. |
| Congruent, ≅ | Exactly the same size and shape. ABCD ≅ PQRS is read “ABCD is congruent to PQRS”. |
| Corresponding sides and angles | The sides and angles that match up when one polygon is placed exactly on top of the other. |
| ∠A | The angle at vertex A. |
| Translation, reflection, rotation | A slide, a flip over a line, and a turn about a point. |
Top, front and side views
The idea: A view is what you see looking straight at one side of the object, drawn flat. Depth disappears: a view shows where cubes are, not how far back they sit.
- Front view: each column shows the tallest stack in that column, front to back. Here the columns are 3, 2 and 1 high.
- Top view: looking down with the front at the bottom, it shows every square with a cube on it — here an L of 4 squares — but no heights.
- Right side view: from the right, the front of the object is on your left. So the front row, 1 high, is on the left, and the back row, 3 high, is on the right.
Worked example: from views to the object. With only the three views, how many cubes are there?
- Top view: 4 squares, so 4 stacks — three across the back and one at the front left.
- Side view: the front row is 1 high, so the stack at the front left is 1 cube.
- Front view: the left column is 3 high, and the front-left stack is only 1, so the back-left stack is the 3. The other two columns have only back stacks: 2 and 1.
- Count: 3 + 2 + 1 + 1 = 7 cubes.
Turning the object. Give it a quarter turn so its right side faces you, and the old side view becomes the new front view. The top view turns a quarter turn with it.
Congruent polygons
The idea: Congruent polygons are exact copies: every side and every angle of one matches a side and an angle of the other. Work out which vertices match before comparing anything.
The letters make a claim. ABCD ≅ PQRS says A matches P, B matches Q, C matches R and D matches S. So it claims AB = PQ, BC = QR, ∠A = ∠P, and so on.
Worked example. Two quadrilaterals were measured, with angles to the nearest degree.
| ABCD | PQRS |
|---|---|
| AB = 8 cm, BC = 5 cm, CD = 5 cm, DA = 4 cm | PQ = 5 cm, QR = 8 cm, RS = 4 cm, SP = 5 cm |
| ∠A = 90°, ∠B = 53°, ∠C = 127°, ∠D = 90° | ∠P = 127°, ∠Q = 53°, ∠R = 90°, ∠S = 90° |
- Start with something only one vertex has. Each shape has just one 53° angle, so B matches Q. Each has just one 127° angle, so C matches P.
- Follow the sides. BC joins the 53° and 127° angles, and so does QP: both are 5 cm ✓. B's other side, BA, is 8 cm, and so is Q's other side, QR ✓, so A matches R. That leaves D matching S.
- Check the rest. ∠A and ∠R are both 90°, and so are ∠D and ∠S. AD and RS are both 4 cm; CD and PS are both 5 cm ✓.
- Write the letters in matching order: ABCD ≅ RQPS. Writing ABCD ≅ PQRS would claim AB = PQ, and 8 cm is not 5 cm.
Why sides and angles. A square and a rhombus with 5 cm sides match side for side, but the rhombus is pushed over, so its angles are not 90°: not congruent. Triangles are different, because three sides fix a triangle's shape. That is why a triangle of rods cannot be pushed over while a square of rods can, and why gates, bridges and roofs are braced with triangles.
Moves keep congruence. A translation, reflection or rotation changes no length or angle, so the image is always congruent to the original — even a mirror image.
What costs marks
The idea: Two are about views, and the rest are about matching the right parts.
- Showing depth in a view. A view is flat: squares only, no slanted lines.
- Drawing the side view back to front. From the right, the front of the object is on your left.
- Counting cubes from one view. A top view shows where the stacks are, not how tall they are.
- Matching vertices by letter order. Match by the measurements, then write the letters in that order.
- Checking sides only. A square and a rhombus can have the same sides.