Proving Triangle Congruence: Angle ABC & ABD

In summary, to prove that triangle ACB is congruent to triangle ADB, you can use the angle-side-angle theorem and the knowledge that the sum of interior angles in a triangle is constant. By substituting the given congruent angles, ABC and BCA, into the equation, you can cancel them out and prove that CAB is equal to DAC.
  • #1
hecate
12
0
can anyone explain how to do this proof?
Given: angle ABC is congruent to angle ABD, angle ACB is congruent to angle ADB
Prove: triangle ACB is congruent to triangle ADB.
thank you!
 
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  • #2
Do you have any theorems that you can use? Like the angle-side-angle theorem?

If you have that, and you can show that BAD and BAC are the same, then you're done.
 
  • #3
yeah...i think i could use the ASA theorem...but i don't know how to explain it and when to use it.
 
  • #4
You know that the sum of the interior angles of a triangle is always the same, right?
 
  • #5
yes.
 
  • #6
So you know that:
ABC + BCA + CAB = ABD + BDA + DAC

from the problem you have:
ABC=ABD
and
BCA=BDA

so if you make the substitutions you get
ABC+BCA+CAB=ABC+BCA+DAC

ABD+BCA can be cancellet from both sides, so you get
CAB=DAC
 
  • #7
oh...i get it now! thanx!
 

1. What is Triangle Congruence?

Triangle congruence is a geometric concept that states two triangles are congruent if they have the same size and shape. This means that all corresponding angles and sides of the two triangles are equal.

2. How can you prove Triangle Congruence?

Triangle congruence can be proven using various methods, such as Side-Angle-Side (SAS), Side-Side-Side (SSS), Angle-Side-Angle (ASA), and Angle-Angle-Side (AAS). These methods involve comparing the corresponding sides and angles of the two triangles to determine if they are equal.

3. What is the difference between ASA and AAS congruence?

ASA (Angle-Side-Angle) congruence states that if two angles and the included side of one triangle are equal to two angles and the included side of another triangle, then the triangles are congruent. AAS (Angle-Angle-Side) congruence states that if two angles and a non-included side of one triangle are equal to two angles and the corresponding non-included side of another triangle, then the triangles are congruent. The main difference between ASA and AAS is the position of the included side or non-included side in relation to the given angles.

4. Can two triangles be congruent if only their angles are equal?

No, two triangles cannot be congruent if only their angles are equal. In order for two triangles to be congruent, their corresponding sides must also be equal. This is known as the Side-Angle-Side (SAS) congruence criterion.

5. Why is proving Triangle Congruence important?

Proving Triangle Congruence is important because it allows us to confidently determine that two triangles are identical in size and shape. This concept is essential in many areas of mathematics, such as geometry, trigonometry, and calculus, and also has real-world applications in fields like engineering, architecture, and physics.

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