Congruence, Similarity & Geometrical Proof Flashcards

All 27 cards in this deck

What does it mean for two shapes to be congruent?

They are exactly the same shape and the same size — all corresponding sides and corresponding angles are equal.

What is the difference between congruent shapes and similar shapes?

Congruent: same shape and same size. Similar: same shape, one is an enlargement of the other.

True or false? If a shape is reflected or rotated it is no longer congruent to the original.

False. Reflecting or rotating does not change size or shape, so the shapes are still congruent.

True or false? Congruent shapes always have the same area and the same perimeter.

True. All corresponding sides are equal, so area and perimeter match.

What are the four congruence criteria for triangles?

SSS, SAS, ASA and RHS.

In triangle congruence, what does SAS stand for?

Two sides and the included angle (the angle between those two sides) are equal.

In triangle congruence, what does ASA stand for?

Two angles and a corresponding side are equal.

In triangle congruence, what does RHS stand for?

Right angle, hypotenuse and one other side are equal.

True or false? If two triangles have all three angles equal (AAA), they are congruent.

False. Equal angles make them similar, but one could be an enlargement of the other.

What are the steps to show two triangles are congruent?

e.g. triangles $ABC$ and $DEF$ with $AB=DE$, $BC=EF$ and angle $B=$ angle $E$

1. State the equal pairs of sides: $AB=DE$, $BC=EF$
2. State the equal angle and that it is included: angle $B=$ angle $E$
3. Name the criterion: SAS, so the triangles are congruent.

Once you have proved two triangles are congruent, what can you say about their other sides and angles?

e.g. triangle $ABC$ congruent to triangle $DEF$

All corresponding sides and angles are equal, e.g. $AC=DF$ and angle $C=$ angle $F$.

What does it mean for two shapes to be mathematically similar?

One is an enlargement of the other: corresponding angles are equal and all corresponding sides are in the same ratio.

What is the scale factor between two similar shapes?

The number every length is multiplied by: a length on the second shape $\div$ the corresponding length on the first.

Two triangles have the same three angles. Are they mathematically similar?

Yes. Equal corresponding angles is enough for triangles to be similar.

True or false? All rectangles are mathematically similar to each other.

False. Their angles match, but the sides need not be in the same ratio.

True or false? In two similar shapes the corresponding angles are equal.

True. Enlargement changes lengths but not angles.

What are the steps to show two triangles are mathematically similar?

e.g. sides $3, 4, 5$ cm and $7.5, 10, 12.5$ cm

1. Pair up corresponding sides (smallest with smallest, etc.)
2. Divide each pair: $7.5\div3=2.5$, $10\div4=2.5$, $12.5\div5=2.5$
3. State all sides are enlarged by the same scale factor, so the triangles are similar.

What are the steps to find a missing length on the larger of two similar triangles?

e.g. $5$ cm corresponds to $20$ cm; find the side matching $4$ cm

1. Divide a pair of corresponding sides: $20\div5=4$
2. Multiply the known side by the scale factor: $4\times4=16$ cm.

How do you find a missing length on the smaller of two similar shapes?

e.g. scale factor $1.5$, corresponding length on the larger shape $15$ cm

Divide by the scale factor: $15\div1.5=10$ cm.

In two similar triangles, how do you decide which sides correspond?

e.g. triangles $ABC$ and $ADE$ with $BC$ parallel to $DE$

Sides in the same position relative to the equal angles correspond: $AB$ with $AD$, $AC$ with $AE$, $BC$ with $DE$.

True or false? To find a missing length in similar shapes you add on the difference between a pair of corresponding sides.

False. You multiply (or divide) by the scale factor, because similar lengths are in the same ratio.

True or false? The scale factor from the larger shape to the smaller similar shape is less than $1$.

True. It is a reduction, e.g. a scale factor of $0.25$ instead of $4$.

In a geometrical proof, what must you write alongside each statement?

A reason, e.g. "angles on a straight line add to $180^\circ$" or "base angles of an isosceles triangle are equal".

What are the steps to set out a chain of reasoning in a geometrical proof?

e.g. isosceles triangle $ABC$ with $AB=AC$ and $D$ the midpoint of $BC$: show angle $B=$ angle $C$

1. State the facts with reasons: $AB=AC$ (isosceles), $BD=DC$ ($D$ is the midpoint), $AD$ is common
2. Name the criterion: SSS, so triangles $ABD$ and $ACD$ are congruent
3. Conclude: corresponding angles are equal, so angle $B=$ angle $C$.

True or false? Checking that a geometrical result works for one particular triangle proves it.

False. A proof must use general facts and reasons that hold for every case.

Which known result links the three sides of a right-angled triangle and can be quoted in a proof?

Pythagoras' theorem: $a^2+b^2=c^2$, where $c$ is the hypotenuse.

Which properties of a parallelogram can you quote as reasons in a proof?

Opposite sides are equal and parallel, and opposite angles are equal.