Proving - All the angles of a rectangle are right angles.

Hey Guys,

I need some help trying to prove that all the angles of a rectangle are right angles.

My teacher asks us to go in depth so we have to list a ton of reasons.

So first let's have the Given

Given: Rectangle ABCD with right angle A

Reason: Given, & the it's also the definition of a rectangle.

1) Statement: Diagonal line segment AC and line segment DB

1)Reason: I constructed a diagonal line across the rectangle

2)Statement: DA || BC and AB || DC

2)Reason: Definition of parallelogram

3)Statement: AC is congruent to AC and BD is congruent to BD

3)Reason: Identity

4)Statement: (M) is for midpoint where the two diagonals meet. So

Triangle DMA is congruent to Triangle MCB

4)Reason: SSS

and here I am lost. I'm not sure if my statements and reasons are correct and in order. I will appreciate the help. Thank you, forum.

Re: Proving - All the angles of a rectangle are right angles.

A rectangle is DEFINED to be a four sided polygon with all angles congruent.

The angle sum of a polygon of "n" sides is $\displaystyle \displaystyle \begin{align*} (n - 2) \cdot 180^{\circ} \end{align*}$. So for a quadrilateral, the angle sum is $\displaystyle \displaystyle \begin{align*} (4 - 2)\cdot 180^{\circ} = 360^{\circ} \end{align*}$.

Since all the angles are equal, that means if "a" is one of the angles, then $\displaystyle \displaystyle \begin{align*} 4a= 360^{\circ} \implies a = 90^{\circ} \end{align*}$.

Therefore, the angles in a rectangle are all right angles.

Re: Proving - All the angles of a rectangle are right angles.

Hey Cake.

You might want to give us an indication of how much detail is required from your teacher.

The reason I ask is that many people would just take this fact as a given where-as you are on the other hand, required to go into a lot of detail.

Do you have a list of axioms that you need to use and if so, can you list them here?

Re: Proving - All the angles of a rectangle are right angles.

Quote:

Originally Posted by

**Prove It** A rectangle is DEFINED to be a four sided polygon with all angles congruent.

The angle sum of a polygon of "n" sides is $\displaystyle \displaystyle \begin{align*} (n - 2) \cdot 180^{\circ} \end{align*}$. So for a quadrilateral, the angle sum is $\displaystyle \displaystyle \begin{align*} (4 - 2)\cdot 180^{\circ} = 360^{\circ} \end{align*}$.

Since all the angles are equal, that means if "a" is one of the angles, then $\displaystyle \displaystyle \begin{align*} 4a= 360^{\circ} \implies a = 90^{\circ} \end{align*}$.

Therefore, the angles in a rectangle are all right angles.

Hey Prove it,

Yes, I am aware of all the facts you have listed. I just have to list the in between theorems and postulates.

Re: Proving - All the angles of a rectangle are right angles.

Quote:

Originally Posted by

**chiro** Hey Cake.

You might want to give us an indication of how much detail is required from your teacher.

The reason I ask is that many people would just take this fact as a given where-as you are on the other hand, required to go into a lot of detail.

Do you have a list of axioms that you need to use and if so, can you list them here?

Hey Chiro,

Well, I have to list a bunch of in between theorems and postulates. Right triangle theorems as well as parallelogram theorems, and finish it off with a rectangle theorem stating all angles of a rectangle are right angles. It's sort of answering a proof in paragraph form except they just have to be bullet points. I think it's way too much to answer online, considering the amount of exact detail that my teacher goes by.