Thread: Congruent Triangles

Originally Posted by magentarita

In the statement vs reasons chart, what is actually needed to prove that two triangles are congruent?

This is a pretty broad question. Based on whatever the "Given" conditions are, one can prove congruency of triangles by:

(1) SSS

(2) SAS

(3) ASA

(4) AAS

And if the triangles are right triangles you can throw in:

(5) HL

I'll leave you to look up the actual threorem (postulate) wording in conditional statement form.

Hello, magentarita!

In the statement vs reasons chart, what is actually needed
to prove that two triangles are congruent?

I refer to corresponding parts of two triangles.

Code:

          C                     R
          *                     *
         *  .                  *  .
        *     .               *     .
       * θ      .            * θ      .
    A *  *  *  *  * B     P *  *  *  *  * Q

If two sides and the included angle of one triangle
. . equals two sides and the included angle of another triangle,
. . the triangles are congruent.

This is called "side-angle-side" or s.a.s.

Code:

          .                     .
         .  .                  .  .
        *     *               *     *
       * α    β *            * α    β *
    A *  *  *  *  * B     P *  *  *  *  * Q

If two angles and the included side of one triangle
. . equals two angles and the included side of another triangle,
. . the triangles are congruent.

This is called "angle-side-angle" or a.s.a.

Code:

          C                     R
          *                     *
         *  *                  *  *
        *     *               *     *
       *        *            *        *
    A *  *  *  *  * B     P *  *  *  *  * Q

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

This is called "side-side-side" or s.s.s.

Originally Posted by masters

This is a pretty broad question. Based on whatever the "Given" conditions are, one can prove congruency of triangles by:

(1) SSS

(2) SAS

(3) ASA

(4) AAS

And if the triangles are right triangles you can throw in:

(5) HL

I'll leave you to look up the actual threorem (postulate) wording in conditional statement form.

You said:

"I'll leave you to look up the actual threorem (postulate) wording in conditional statement form." I thought a theorem is not a postulate, right?

Originally Posted by Soroban

Hello, magentarita!

I refer to corresponding parts of two triangles.

Code:

          C                     R
          *                     *
         *  .                  *  .
        *     .               *     .
       * θ      .            * θ      .
    A *  *  *  *  * B     P *  *  *  *  * Q

If two sides and the included angle of one triangle
. . equals two sides and the included angle of another triangle,
. . the triangles are congruent.

This is called "side-angle-side" or s.a.s.

Code:

          .                     .
         .  .                  .  .
        *     *               *     *
       * α    β *            * α    β *
    A *  *  *  *  * B     P *  *  *  *  * Q

If two angles and the included side of one triangle
. . equals two angles and the included side of another triangle,
. . the triangles are congruent.

This is called "angle-side-angle" or a.s.a.

Code:

          C                     R
          *                     *
         *  *                  *  *
        *     *               *     *
       *        *            *        *
    A *  *  *  *  * B     P *  *  *  *  * Q

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

This is called "side-side-side" or s.s.s.

This is fabulous information as always.

Originally Posted by magentarita

You said:

"I'll leave you to look up the actual threorem (postulate) wording in conditional statement form." I thought a theorem is not a postulate, right?

Actually, SSS, SAS, ASA, and HL are postulates, while AAS is a theorem.

And there are other theorems regarding right triangles that can easily be proven from the postulates; namely, LL, LA, and HA. That's why I didn't name them. If you remember those 5 I originally stated, you'll be ok.

Originally Posted by masters

Actually, SSS, SAS, ASA, and HL are postulates, while AAS is a theorem.

And there are other theorems regarding right triangles that can easily be proven from the postulates; namely, LL, LA, and HA. That's why I didn't name them. If you remember those 5 I originally stated, you'll be ok.

I love geometry. It is my favoriye math topic.