Congruent Shapes

Worked example

1. 1

   $\text{given: }3\text{ across},2\text{ up}$ — Count the squares along the two short sides of the shaded triangle. These two numbers are what we compare.

2. 2

   $A:2\text{ across},3\text{ up}$ — The same two numbers, the other way round. A is the shaded triangle turned on its side, so it is congruent.

3. 3

   $B:6\text{ across},4\text{ up}$ — Both numbers are doubled, so B is the same shape but bigger. Different size means similar only, not congruent.

4. 4

   $D:3\text{ across},2\text{ up}$ — The same two numbers in the same places. D is the shaded triangle flipped over, so it is congruent.

Worked example

1. 1

   $\text{bottom: }4\text{ across}$ — Start at any grid point and copy the first side exactly. Count 4 squares along.

2. 2

   $\text{right side: }2\text{ up}$ — From the end of that side, go 2 squares up to copy the next side.

3. 3

   $\text{slope: }2\text{ left},1\text{ up}$ — Copy a sloping side by counting squares across and up. Never judge its length by eye.

4. 4

   $\text{top: }2\text{ left}$ — Copy the flat top side: 2 squares to the left.

5. 5

   $\text{last side: }3\text{ down}$ — Go 3 squares down. We are back at the start, so the shape closes. If it does not close, one count is wrong.

Worked example

1. 1

   $AB=PQ=5\text{ cm}$ — A first pair of equal sides. We mark it S, for side.

2. 2

   $BC=QR=7\text{ cm}$ — A second pair of equal sides. Another S.

3. 3

   $\angle ABC=\angle PQR=40^{\circ }$ — This angle sits between the two equal sides in each triangle. We mark it A, for angle.

4. 4

   $△ABC\equiv △PQR$ — Side, angle, side is the SAS test, so the triangles are congruent. Write the letters in matching order: A with P, B with Q, C with R.

Worked example

1. 1

   $\angle A=\angle P=90^{\circ }$ — Both triangles have a right angle. That is the R in RHS.

2. 2

   $BC=QR=10\text{ cm}$ — These sides are opposite the right angles, so they are the hypotenuses, the longest sides. They are equal. That is the H.

3. 3

   $AB=PQ=6\text{ cm}$ — One more pair of equal sides. That is the S.

4. 4

   $△ABC\equiv △PQR$ — Right angle, hypotenuse, side is the RHS test, so the triangles are congruent.

Worked example

1. 1

   $AB=AC$ — Given in the question, because the triangle is isosceles.

2. 2

   $BM=CM$ — $M$ is the midpoint of $BC$, so it cuts $BC$ into two equal parts.

3. 3

   $AM=AM$ — $AM$ is a side of both triangles, so it is equal to itself. We call it a common side.

4. 4

   $△ABM\equiv △ACM$ — Three pairs of equal sides is the SSS test, so the triangles are congruent.

Worked example

1. 1

   $AX=DX$ — $X$ is the midpoint of $AD$.

2. 2

   $\angle AXB=\angle DXC$ — The lines cross at $X$, so the angles facing each other are vertically opposite angles. They are equal.

3. 3

   $BX=CX$ — $X$ is the midpoint of $BC$.

4. 4

   $△ABX\equiv △DCX$ — Side, angle, side with the angle between the two sides is the SAS test, so the triangles are congruent. $A$ matches $D$ and $B$ matches $C$.