Triangle Similarity Postulates Quiz
16 Questions
0 Views

Choose a study mode

Play Quiz
Study Flashcards
Spaced Repetition
Chat to Lesson

Podcast

Play an AI-generated podcast conversation about this lesson

Questions and Answers

What theorem can be used to prove that corresponding sides of similar triangles are proportional?

  • Means Extremes Products Theorem (correct)
  • Triangle Inequality Theorem
  • Pythagorean Theorem
  • Mean Value Theorem

    • Which postulate is used to prove that triangles are similar by comparing one side and the included angle?

  • Side-Angle-Side (SAS) Postulate (correct)
  • Angle-Side-Angle (ASA) Postulate
  • Side-Side-Side (SSS) Postulate
  • Angle-Angle (AA) Postulate

    • What property states that an angle is congruent to itself?

  • Symmetric Property
  • Congruent Angles Theorem
  • Reflexive Property (correct)
  • Transitive Property

    • If DE || AC, which statement can be correctly concluded?

    <p>AB × BE = BC × BD. (A)</p> Signup and view all the answers

    In the two-column proof steps for proving △BDC ~ △BEA, what is the reason for step 3?

    <p>AA similarity postulate (C)</p> Signup and view all the answers

    Which postulate can be used to prove that two triangles are similar if all three angles of one triangle are congruent to the corresponding angles of another triangle?

    <p>AAA Postulate (B)</p> Signup and view all the answers

    In the context of similar triangles, what are the extremes if the proportion is given as $10:5 = 6:3$?

    <p>$10$ and $3$ (C)</p> Signup and view all the answers

    What type of angles are formed by altitudes in triangles?

    <p>Right angles (C)</p> Signup and view all the answers

    If two triangles have two pairs of corresponding angles that are congruent, which postulate applies to prove their similarity?

    <p>AA Postulate (D)</p> Signup and view all the answers

    What is the conclusion of the Means Extremes Products Theorem in a proportional relationship?

    <p>The product of the means equals the product of the extremes. (C)</p> Signup and view all the answers

    When applying the SSS Postulate, what must be true about the sides of two triangles?

    <p>The ratios of corresponding sides must be equal. (C)</p> Signup and view all the answers

    What is the criteria for the SAS Postulate to apply in proving triangle similarity?

    <p>Two sides must be proportional, and the included angle must be congruent. (C)</p> Signup and view all the answers

    In the example using the SSS Postulate, what was the ratio of the corresponding sides found?

    <p>3:1 (B)</p> Signup and view all the answers

    For two triangles to be proven similar using the AA Postulate, how many pairs of corresponding angles must be congruent?

    <p>Two pairs (B)</p> Signup and view all the answers

    Which of the following does NOT contribute to the proof of triangle similarity using these postulates?

    <p>One side is longer than the corresponding side in another triangle. (D)</p> Signup and view all the answers

    In triangle similarity, which of the following combinations would definitely lead to a conclusion of similarity based on the SAS Postulate?

    <p>Two corresponding sides are proportional and the included angle is congruent. (D)</p> Signup and view all the answers

    Flashcards

    AAA Postulate

    If all three angles of one triangle are congruent to the corresponding angles of another triangle, then the triangles are similar.

    AA Postulate

    If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.

    SSS Postulate

    If the ratios of corresponding sides of two triangles are equal, then the triangles are similar.

    SAS Postulate

    If two sides of one triangle are proportional to two sides of another triangle and the included angles are congruent, then the triangles are similar.

    Signup and view all the flashcards

    Tilde (~)

    The symbol used to indicate that triangles are similar (e.g., △ABC ~ △DEF).

    Signup and view all the flashcards

    Isosceles Trapezoid

    A trapezoid with two congruent legs, like two identical legs of a table.

    Signup and view all the flashcards

    Alternate Interior Angles

    Angles formed when two parallel lines are intersected by a transversal. They are on opposite sides of the transversal and in corresponding positions.

    Signup and view all the flashcards

    Two-Column Proof

    A formal proof that uses a two-column format. One column lists statements and the other lists reasons supporting each statement.

    Signup and view all the flashcards

    Vertical angles

    When two angles are directly opposite each other and share the same vertex.

    Signup and view all the flashcards

    Similar triangles

    Two triangles are similar if their corresponding angles are congruent and their corresponding sides are proportional.

    Signup and view all the flashcards

    Altitude of a triangle

    An altitude of a triangle is a perpendicular segment from a vertex to the opposite side.

    Signup and view all the flashcards

    Means Extremes Products Theorem

    The Means Extremes Products Theorem states that if a/b = c/d, then b x c = a x d.

    Signup and view all the flashcards

    Corresponding angles of similar triangles

    Corresponding angles of similar triangles are congruent.

    Signup and view all the flashcards

    Study Notes

    Triangle Similarity Postulates

    • AAA Postulate: If all three angles of one triangle are congruent to the corresponding angles of another triangle, the triangles are similar. Less commonly used.
    • AA Postulate: If two angles of one triangle are congruent to two angles of another triangle, the triangles are similar. This is practical as the third angle must also be congruent. More frequently used.
    • SSS Postulate: If the ratios of corresponding sides of two triangles are equal, the triangles are similar. For example, if AB/DE = BC/EF = AC/DF, then △ABC ~ △DEF.
    • SAS Postulate: If two sides of one triangle are proportional to two sides of another triangle and the included angles are congruent, the triangles are similar. For example, if ∠A ≅ ∠D and AB/DE = AC/DF, then △ABC ~ △DEF.

    Example Applications

    • SSS Example: Given △ABC (AB = 12, BC = 15, AC = 18) and △DEF (DE = 4, EF = 5, DF = 6). The ratios of corresponding sides are equal (12/4 = 15/5 = 18/6 = 3). Therefore, △ABC ~ △DEF by the SSS Postulate.
    • SAS Example: Given △ABC and △DEC (D on AB, E on BC). DC = 7, EC = 12, AD = 14, BE = 24. AC = 21, BC = 36. Ratios of corresponding sides are equal (7/21 = 12/36 = 1/3), and ∠C ≅ ∠C. Thus, △ABC ~ △DEC by SAS Postulate.

    Proving Similarity (Two-Column Proofs)

    • Example 1: Given isosceles trapezoid ABCD. Prove △BEC ~ △DEA.
    • Statements: ABCD is an isosceles trapezoid, ∠EAD ≅ ∠ECB (alternate interior angles), ∠BEC ≅ ∠DEA (vertical angles).
    • Reason: Given, alternate interior angles, vertical angles.
    • Conclusion: △BEC ~ △DEA by AA Postulate.
    • Example 2: Given AE/AB = AD/AC. Prove △AED ~ △ABC.
    • Statements: AE/AB = AD/AC, ∠A ≅ ∠A (reflexive property).
    • Reason: Given, reflexive property.
    • Conclusion: △AED ~ △ABC by SAS Postulate.
    • Example 3: Given AE and CD are altitudes. Prove ∠BAE ≅ ∠BCD.
    • Statements: AE and CD are altitudes, ∠BDC and ∠BEA are right angles, ∠BDC ≅ ∠BEA, ∠B ≅ ∠B.
    • Reason: Given, definition of altitude, right angles are congruent, reflexive property.
    • Conclusion: △BDC ~ △BEA by AA Postulate and ∠BAE ≅ ∠BCD (corresponding angles).
    • Example 4: Given DE || AC. Prove AB × BE = BC × BD.
    • Reason: Given, Corresponding angles (parallel lines), Reflexive Property, AA Postulate, Corresponding sides are proportional and Means/Extremes Product Theorem.

    Means-Extremes Product Theorem

    • If a/b = c/d, then b × c = a × d (product of means = product of extremes).
    • This theorem is useful for working with similar triangles. If two triangles are similar, proportions can be established between corresponding sides, and this theorem can be applied.

    Studying That Suits You

    Use AI to generate personalized quizzes and flashcards to suit your learning preferences.

    Quiz Team

    Description

    Test your knowledge on the various postulates that determine triangle similarity, including the AAA, AA, SSS, and SAS postulates. This quiz includes examples and applications to help solidify your understanding of these important concepts in geometry.

    More Like This

    Use Quizgecko on...
    Browser
    Open
    Browser
    Browser