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 size — all corresponding sides and all corresponding angles are equal.

What does $\triangle ABC \equiv \triangle DEF$ mean?

Triangle $ABC$ is congruent to triangle $DEF$, with vertices in corresponding order, so $AB = DE$ and $\angle A = \angle D$.

True or false? A shape and its mirror image cannot be congruent.

False. Reflected and rotated shapes are still congruent — only size and shape matter, not orientation.

True or false? Two congruent shapes are always similar.

True. They are similar with scale factor $1$.

True or false? Two shapes with the same area must be congruent.

False. E.g. a $2\times6$ rectangle and a $3\times4$ rectangle have equal areas but different side lengths.

What are the four congruence criteria for triangles?

SSS, SAS, ASA and RHS.

What does the congruence criterion RHS stand for?

Right angle, Hypotenuse and one other Side equal in both triangles.

For the criterion SAS, where must the equal angle be?

Between (included by) the two pairs of equal sides.

True or false? Two sides and a non-included angle (SSA) prove two triangles are congruent.

False. SSA is not a congruence criterion — the only exception is RHS, where the angle is a right angle.

What are the steps to prove two triangles are congruent?

e.g. prove $\triangle ABC \equiv \triangle DCB$ given $AB = CD$ and $\angle ABC = \angle BCD$

1. State three equal pairs with reasons: $AB = CD$ (given), $\angle ABC = \angle BCD$ (given), $BC$ common.
2. Name the criterion: SAS.
3. Conclude: $\triangle ABC \equiv \triangle DCB$.

Once you have proved two triangles are congruent, how do you show two lengths are equal?

State that corresponding sides of congruent triangles are equal.
e.g. $\triangle ABC \equiv \triangle DCB$, so $AC = BD$.

What does it mean for two shapes to be similar?

Corresponding angles are equal and corresponding sides are all in the same ratio — one is an enlargement of the other.

What is the simplest condition that proves two triangles are similar?

AA — two pairs of corresponding angles are equal (the third pair is then equal too).

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

Corresponding sides are opposite equal angles (they join the same pairs of matching vertices).

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

False. Their angles match but the ratios of the sides need not be equal, e.g. $2\times3$ and $2\times5$.

True or false? AAA (three equal angles) proves two triangles are congruent.

False. Equal angles only prove the triangles are similar; the sides may be different sizes.

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

e.g. $\triangle ABC$ and $\triangle AED$, where $BC$ is parallel to $ED$

1. Find a pair of equal angles with a reason: $\angle A$ is common.
2. Find a second pair with a reason: $\angle ABC = \angle AED$ (corresponding angles, $BC \parallel ED$).
3. Conclude: the triangles are similar (AA).

In a 'split' triangle where $BE$ is parallel to $CD$ ($ABC$ and $AED$ straight lines), which sides correspond?

$AB$ with $AC$, $AE$ with $AD$, and $BE$ with $CD$ — the two similar triangles are $ABE$ and $ACD$.

True or false? In a split triangle with $BE$ parallel to $CD$, the side $AB$ corresponds to the part $BC$.

False. $AB$ corresponds to the whole side $AC$, measured from the common vertex $A$.

True or false? The scale factor between two similar shapes must be greater than 1.

False. Going from the larger shape to the smaller one gives a scale factor between $0$ and $1$, e.g. $0.25$.

For two similar solids, what is true of corresponding lengths?

Every corresponding length is in the same ratio — multiply by the linear scale factor.
e.g. with scale factor $3$, the height, width and radius all treble.

What is the standard reason quoted for the two angles opposite the equal sides of an isosceles triangle being equal?

Base angles of an isosceles triangle are equal.

State two properties of a parallelogram used in geometrical proofs.

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

Which two angle reasons can you quote for equal angles when a line crosses a pair of parallel lines?

Alternate angles are equal; corresponding angles are equal. (Co-interior angles sum to $180^\circ$.)

What are the steps to build a chain of reasoning for a proof?

e.g. prove the exterior angle $x$ of a triangle equals the sum of the two opposite interior angles $a$ and $b$

1. Write a fact with its reason: $x = 180^\circ - c$ (angles on a straight line).
2. Write the next fact with its reason: $a + b = 180^\circ - c$ (angles in a triangle sum to $180^\circ$).
3. Link them to the required conclusion: $x = a + b$.

True or false? Measuring the angles on an accurate diagram is enough to prove a geometrical result.

False. A proof must use known geometric facts, each with a reason, so it holds in every case.

True or false? The fact that the base angles of an isosceles triangle are equal can be proved using congruent triangles.

True. Splitting the triangle along its line of symmetry gives two congruent triangles (SSS or SAS), so the base angles are equal.