Triangle Congruence Proof: Comparing Triangles with 30° Angles and X+2 Side Length

Triangle Congruence with Algebraic Side Lengths

Are the triangles in the drawing congruent?

303030303030X+2X+2X+23333332X+4

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Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

Are the triangles in the drawing congruent?

303030303030X+2X+2X+23333332X+4

2

Step-by-step solution

In order for triangles to be congruent, one must demonstrate that the S.A.S theorem is satisfied

We have a common side whose length in both triangles is equal to 3.

Now let's examine the lengths of the other sides:

2X+4=X+2 2X+4=X+2

We proceed with the sections accordingly:24=2XX 2-4=2X-X

2=X -2=X

We place this value in the right triangle we should find the length of the side:2+2=0 -2+2=0

However since it is not possible for the length of a side to be equal to 0, the triangles are not congruent.

3

Final Answer

No

Key Points to Remember

Essential concepts to master this topic
  • Congruence: All three sides must be equal for S.A.S theorem
  • Technique: Set equal sides: 2X+4 = X+2, solve X = -2
  • Check: Substitute X = -2: side length = 0, impossible in geometry ✓

Common Mistakes

Avoid these frequent errors
  • Assuming triangles are congruent based on one angle and one side
    Don't conclude congruence from just the 30° angles and one side of length 3! This ignores the algebraic expressions for other sides. Always solve the algebraic equation to check if all corresponding sides are equal and positive.

Practice Quiz

Test your knowledge with interactive questions

Look at the triangles in the diagram.

Which of the statements is true?

727272727272131313222131313222AAABBBCCCDDDEEEFFF

FAQ

Everything you need to know about this question

Why can't a side length be zero or negative?

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In geometry, side lengths represent distances, which must always be positive numbers. A side of length 0 means no side exists, and negative lengths are impossible in real triangles.

What if I got X = -2 but ignored the zero side length?

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Never ignore impossible geometric conditions! Even if your algebra is correct, you must check that your answer makes physical sense in the geometric context.

Could these triangles be congruent for a different value of X?

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No! For congruence, we need 2X+4=X+2 2X + 4 = X + 2 , which only has one solution: X = -2. Since this gives an impossible side length, the triangles can never be congruent.

What does S.A.S theorem require exactly?

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S.A.S means Side-Angle-Side: two triangles are congruent if they have two pairs of equal sides with the included angle between those sides also equal.

How do I know which theorem to use for triangle congruence?

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  • S.A.S: Two sides and included angle
  • A.S.A: Two angles and included side
  • S.S.S: All three sides equal

Choose based on what information is given in the problem.

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