Can someone help with deductive proofs please?

Answer 1

The idea is to show something must be true because when it is a special case of a general principle that is known to be true.

So say you know the general principle that the sum of the angles in any triangle is always 180 degrees, and you have a particular triangle in mind, you can then conclude that the sum of the angles in your triangle is 180 degrees.

So let's look at one you asked about so you get the idea.

The diagonals of a square are also angle bisectors.

Since we know a square is a rhombus with 90 angles, if we prove it for a rhombus in general, we have proved it for a square.

Let ABCD be a rhombus.

Segment AB is congruent to BC which is congruent to CD which is congruent to DA

Reason: Definition of Rhombus

Now Segment AC is congruent to itself. Reason Reflexive property

So Triangle ADC is congruent to triangle ABC by SSS postulate.

Next Angle DAC is congruent to angle BAC by CPCTC

And Angle DCA is congruent to angle BCA by the same reason.

We used the fact that corresponding parts of congruent triangles are congruent to prove that diagonals bisect the angles of the rhombus which proves it is true for a square.

The point being rhombus is a quadrilateral whose four sides are all congruent

Of course a square has 4 congruent sides, but also right angles. We don't need the right angle part to prove this, so we used a rhombus.

Every square is a rhombus, so if it is true for a rhombus it must be true for a square.