Triangle Similarity Postulates Quiz

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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 â–³BDC ~ â–³BEA, 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.

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Tilde (~)

The symbol used to indicate that triangles are similar (e.g., â–³ABC ~ â–³DEF).

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Isosceles Trapezoid

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

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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.

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Two-Column Proof

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

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Vertical angles

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

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Similar triangles

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

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Altitude of a triangle

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

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Means Extremes Products Theorem

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

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Corresponding angles of similar triangles

Corresponding angles of similar triangles are congruent.

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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 â–³ABC ~ â–³DEF.
  • 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 â–³ABC ~ â–³DEF.

Example Applications

  • SSS Example: Given â–³ABC (AB = 12, BC = 15, AC = 18) and â–³DEF (DE = 4, EF = 5, DF = 6). The ratios of corresponding sides are equal (12/4 = 15/5 = 18/6 = 3). Therefore, â–³ABC ~ â–³DEF by the SSS Postulate.
  • SAS Example: Given â–³ABC and â–³DEC (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, â–³ABC ~ â–³DEC by SAS Postulate.

Proving Similarity (Two-Column Proofs)

  • Example 1: Given isosceles trapezoid ABCD. Prove â–³BEC ~ â–³DEA.
  • Statements: ABCD is an isosceles trapezoid, ∠EAD ≅ ∠ECB (alternate interior angles), ∠BEC ≅ ∠DEA (vertical angles).
  • Reason: Given, alternate interior angles, vertical angles.
  • Conclusion: â–³BEC ~ â–³DEA by AA Postulate.
  • Example 2: Given AE/AB = AD/AC. Prove â–³AED ~ â–³ABC.
  • Statements: AE/AB = AD/AC, ∠A ≅ ∠A (reflexive property).
  • Reason: Given, reflexive property.
  • Conclusion: â–³AED ~ â–³ABC 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: â–³BDC ~ â–³BEA 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.

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