Grade 8 Geometry of 2D shapes

1. Two sides are equal, so the triangle is isosceles.
2. Angle $B$ lies opposite $AC$, and angle $C$ lies opposite $AB$.
3. Those are the base angles, so they are equal.
4. So angle $C$ is $65^{\circ }$ too.

Answer: $65^{\circ }$

The base angles sit opposite the equal sides, not at the bottom of the page.

1. All four sides of a rhombus are equal, so this is not one.
2. In a parallelogram the equal sides face each other.
3. Here the equal sides sit next to each other, and that is a kite.

Answer: a kite

Next to each other, not opposite, is what settles it.

1. Four angles of $90^{\circ }$ make it a rectangle.
2. All four sides are equal too, so it is also a rhombus.
3. A shape that is both is a square, and square is the fuller name.

Answer: a square, and yes it is a rectangle

The families sit inside one another, so one shape carries several names.

1. Match the sides: $5$ with $5$, $7$ with $7$, $9$ with $9$.
2. Three matching sides fix a triangle completely.
3. Turning a figure changes no side and no angle.
4. So the two triangles are congruent.

Answer: yes, they are congruent

Congruence is about shape and size, not about which way a figure faces.

1. Every side matches, so the sides alone would say yes.
2. But the square's angles are all $90^{\circ }$.
3. The rhombus has $60^{\circ }$ and $120^{\circ }$ instead.
4. The corresponding parts do not all match, so they are not congruent.

Answer: no, the angles do not match

Congruence needs the angles to match as well as the sides.

1. Divide the matching sides instead of subtracting them.
2. $40÷30=\frac{4}{3}$ and $60÷45=\frac{4}{3}$.
3. Both flags are rectangles, so every angle is $90^{\circ }$.
4. Equal angles and one ratio, so the flags are similar.

Answer: yes, they are similar

Subtracting would have given $10$ and $15$, and answered no by mistake.

1. The angle sum of a quadrilateral is $360^{\circ }$, not $180^{\circ }$.
2. Add the three you have: $80+95+110=285$.
3. Take that from the total: $360-285=75$.
4. Reason: angle sum of a quadrilateral.

Answer: $75^{\circ }$

Choosing the right total first is what made the subtraction right.

1. Two sides are equal, so the triangle is isosceles.
2. That makes the base angles $L$ and $M$ equal.
3. The angle sum is $180^{\circ }$, so $180-40=140$ is left for the two of them.
4. Share it out: $140÷2=70$.

Answer: $70^{\circ }$

The equal sides were given, so the equal angles were allowed.

1. Angle $C$ faces angle $A$, and opposite angles of a parallelogram are equal.
2. So angle $C$ is $115^{\circ }$.
3. Side $CD$ faces $AB$, and opposite sides of a parallelogram are equal.
4. So $CD$ is $7$ cm, matching $AB$ and not the $4$ cm of $BC$.

Answer: angle $C$ is $115^{\circ }$, and $CD$ is $7$ cm

Opposite parts of a parallelogram match, while adjacent ones need not.