Are there any theorems/postulates in geometry that can help me with this proof? Open for a better explanation.

Question

I was recently given back a math test for geometry today and my teacher took off something that I believe is right. I tried to prove that two triangles were congruent in a novel way (as she describes it.) Instead of using SAS or SSS or AAS or any other postulates/theorems, I did something that my teacher said doesn't exist in geometric theorems.

Here is the picture of the triangle from the test that I drew for you guys:
http://img413.imageshack.us/img413/1027/mathrl9.png

I am supposed to prove that the two trianlges in that picture are congruent. Everything in black was given to me in the illustration in the test. I drew in the red marks. Is there any theorem/postulate that can prove to me that the red marks CAN be there? Or is it just wrong all together? My teacher said that if I can find a postulate/theorem for this, she'll give me the points.

I searched through the entire list of theorems and postulates of geometry and really found nothing (at least in my book.)

Thanks for all your help
I found the theorem i needed

(or rather converse)

its called the Converse of the Angle Bisector Theorem...but regardless

you guys were great!

Answer 1

You basically have two right triangles whose legs are congruent (the horizontal legs are equal because you were told they are; the vertical legs are equal because they are the same line segment).

Because the legs are equal (and the right angles are equal), the two triangles are congruent (by SAS).

Once you prove that they are congruent, you can mark the corresponding acute angles equal. However, you first have to prove congruence. Because you knew from experience that the two were congruent, you jumped the gun and marked the angles equal. But in geometry, you have to prove everything, not simply report that "you can see" that it's true.

Maybe there is a theorem that will come to your rescue (certainly not an axiom or postulate), but my best guess is that there is no such theorem. If there were, it would read somewhat as follows:
If the legs of a right triangle are congruent to the corresponding legs of another right triangle, then the triangles are congruent.

Answer 2

Take it in steps. It's not just one postulate or theorem.

I'm not quite fond of what they are called, so I'll explain it - and you can take it from there...

The red lines are correct because of the congruent marks ( | ) on the bottom part of the triangle. Anything across from each other (whether it be angle to side, or vise versa) will be congruent to each other.

Next off, since you have the 90 degrees mark on the right triangle, the other triangle will also be 90 degrees (it will make a straight angle, 180 degrees).

From what I said above, lines AB and BC are congruent because of the 90 degree angles.

After that, you have a congruent angle and a congruent side of both triangles. What's last?

The third side has to be equivalent to each other if the others are congruent. It only makes sense.

I hope this helps you!

Answer 3

yes it is possible


To prove that they are congruent by SAS, you use the right angle and the given tick marks, they tell you one side and one angle is already congruent so you just need to find the other side
The other side is the perpendicular bisector(the line in the middle of the trangle.), the perpendicular bisector is congruent to itself, this triangle is missing something, cause I'm not sure what you would call this perpendicular bisector in your proof. Now you can use this to prove that the angles are congruent.

You can prove that the triangles are congurent with SAS and then put as your last steo in your proof, that the angles are congruent by CPCTC, this means that once you know the traingle is congruent by SSS,SAS or whatever, you can prove that everything else in the traingle is congruent with CPCTC




hope this helps ^_^

Answer 4

Let us go into what it is that you are trying to do here. You want to learn how to prove that TWO triangles are congruent, right?

Let me you a definition first.

Definition: Two triangles are congruent if all pairs of corresponding sides are congruent, and all pairs of corresponding angles are congruent.

Got it?

In your picture, you need to label this triangle with letters for easy reading.

You have a triangle with no letters.

I will call the center top B
The lower left corner A
The lower right corner C.
The center BETWEEN A and C, I will call D.

Go to your triangle and label the parts as I showed you.

With me so far?

By putting in the letters, I divided triangle ABC into two triangles and they are:

triangle ABD and triangle CBD. See it?

We can prove that two triangles are congruent using ONE of the following methods:

SSS :If three sides of one triangle are congruent to three sides of another triangle, the triangles are congruent.

SAS: If two sides and the included angle of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent.

ASA: If two angles and the included side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent.

AAS: If two angles and the non-included side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent.

HL: If the hypotenuse and leg of one right triangle are congruent to the corresponding parts of another right triangle, the right triangles are congruent.

Going back to your triangle and the letters applied, we have this:

side AB = side BC
BD is equal to itself
AD = CD

What method did I use from the ones listed above?

I used SSS method to prove that two triangles given are congruent.

Guido

Answer 5

Geometry Theorems
All right angles are congruent.
Vertical Angles Theorem : Vertical angles are congruent.

Congruent Supplements Theorem
Angles supplementary to the same angle are congruent.
Angles supplementary to congruent angles are congruent.

Congruent Complements Theorem
Angles complementary to the same angle are congruent.
Angles complementary to congruent angles are congruent.


If 2 angles are congruent and supplementary, then each angle is a right angle.

Alternate Interior Angles Theorem : If 2 parallel lines are cut by a transversal, then each pair of alternate interior angles is congruent.

Same-Side Interior Angles Theorem : If 2 parallel lines are cut by a transversal, then each pair of same-side interior angles is supplementary.

Converse of Alternate Interior Angles Theorem : If two lines are cut by a transversal so that a pair of alternate interior angles is congruent, then the lines are parallel.

Converse of Same-Side Interior Angles Theorem : If two lines are cut by a transversal so that a pair of same-side interior angles is supplementary, then the lines are parallel.


Triangle Angle Sum Theorem : The sum of the measures of the interior angles of a triangle is 180.

Triangle Exterior Angle Theorem : The measure of each exterior angle of a triangle is equal to the sum of the measures of its two remote interior angles.
Corollary: The measure of an exterior angle of a triangle is greater than the measure of either of its remote interior angles.

The acute angles of a right triangle are complementary.
Third Angle Theorem If 2 angles of one triangle are congruent to 2 angles of another triangle, then the 3rd angles are congruent.
Polygon Interior Angle Sum Theorem: The sum of the measures of the interior angles of a convex polygon is (n - 2) 180.


Polygon Exterior Angles Sum Theorem: In a convex polygon, the sum of the exterior angles, one at each vertex, is 360.

Isosceles Triangle Theorem : Base angles of an isosceles triangle are congruent.
Corollary: An equilateral triangle is also equiangular.
The bisector of the vertex angle of an isosceles triangle is the perpendicular bisector of the base.
Converse of the Isosceles Triangle Theorem : If two angles of a triangle are congruent then the sides opposite those angles are congruent.
Corollary: An equiangular triangle is also equilateral.


AAS Theorem : If two angles and the nonincluded side of one triangle are congruent to the corresponding parts of another triangle, then the triangles are congruent.
HL Theorem : If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and corresponding leg of another right triangle, then the triangles are congruent.

Pythagorean Theorem : In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
Converse of the Pythagorean Theorem : If the square of the longest side of a triangle is equal to the sum of the squares of the other two sides, the the triangle is a right triangle.
Corollary 1: If the square of the longest side of a triangle is greater than the sum of the squares of the other two sides, then the triangle is an obtuse triangle.
Corollary 2: If the square of the longest side of a triangle is less than the sum of the squares of the other two sides, then the triangle is an acute triangle.
In a 45-45-90 triangle, the hypotenuse is square root of 2 times as long as a leg.
In a 30-60-90 triangle, the hypotenuse is twice as long as the shorter leg and the longer leg is square root of 3 times as long as the shorter leg.

Opposite sides of a parallelogram are congruent.
Opposite angles of a parallelogram are congruent.
Consecutive angles in a parallelogram are supplementary.
The diagonals of a parallelogram bisect each other.
If three or more parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal.
A diagonal of a parallelogram divides the parallegram into two congruent triangles.
If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.
If one pair of opposite sides of a quadrilateral are both parallel and congruent, then the quadrilateral is a parallelogram.
If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.
If both pairs of opposite angles in a quadrilateral are congruent, then the quadrilateral is a parallelogram.

Each diagonal of a rhombus bisects a pair of opposite angles.
The diagonals of a rhombus are perpendicular.
The area of a rhombus is equal to half the product of the lengths of its diagonals.
The diagonals of a rectangle are congruent.

If one diagonal of a parallelogram bisects two angles of the parallelogram, then the parallelogram is a rhombus.
If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus.
If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle.

Base angles of an isosceles trapezoid are congruent.
The diagonals of an isosceles trapezoid are congruent.
The diagonals of a kite are perpendicular.