Math Problem Statement

solve

Determine which postulate can be used to prove that the triangles are congruent (based on the diagram where triangles ABD and CBD share side BD, and AB = BC).

Solution

The diagram provided shows two triangles, ABD\triangle ABD and CBD\triangle CBD, sharing a common side BDBD, with AB=BC\overline{AB} = \overline{BC} and BD\overline{BD} being a common side for both triangles. Additionally, ABD=CBD=90\angle ABD = \angle CBD = 90^\circ as they form right angles at point BB.

To prove that these triangles are congruent, we can use the SAS (Side-Angle-Side) Postulate. Here’s why:

  1. Side: BD\overline{BD} is shared by both triangles.
  2. Angle: ABD=CBD=90\angle ABD = \angle CBD = 90^\circ, both are right angles.
  3. Side: AB=BC\overline{AB} = \overline{BC}, as given by the tick marks showing they are equal in length.

Since we have two sides and the included angle congruent, the SAS (Side-Angle-Side) postulate applies.

The correct answer is SAS.

Would you like more details, or do you have any further questions?

Here are 5 related questions you can explore:

  1. What other methods can be used to prove triangle congruence?
  2. Can the AAS (Angle-Angle-Side) postulate apply here?
  3. Why is SSA (Side-Side-Angle) not a valid congruence postulate?
  4. How does the HL (Hypotenuse-Leg) theorem apply to right triangles?
  5. How do you differentiate between congruent and similar triangles?

Tip: Always check for shared sides or angles when proving triangle congruence, as these often provide key clues.

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Math Problem Analysis

Mathematical Concepts

Congruent Triangles
Geometric Postulates
Triangle Properties

Formulas

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Theorems

SAS (Side-Angle-Side) Postulate
Triangle Congruence

Suitable Grade Level

Grades 8-10