Similarity of Triangles Quiz
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Similarity of Triangles Quiz

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Questions and Answers

Which of the following statements is true about two triangles that have corresponding angles congruent and sides in proportion?

  • The triangles are similar. (correct)
  • They may or may not be similar.
  • The triangles are not similar.
  • They are right triangles only.

    • In the case of intersecting chords, if D is the midpoint of BC, which relationship must hold true about the segments formed?

  • The segments AB and AD are equal.
  • AD is half the length of DE.
  • (BD)^2 = AD × DE. (correct)
  • AE is equal to the length of BC.

    • What can be concluded when a line is drawn parallel to one side of a triangle and intersects the other two sides?

  • The resulting triangles are similar to the original triangle. (correct)
  • The triangles formed are not similar to the original triangle.
  • The triangles formed may have different proportions.
  • Only the angles of the triangles are equal.

    • If triangles ADB and ADC are similar to triangle ABC, what can be concluded about their altitude?

    <p>The altitude creates two triangles similar to the original triangle.</p> Signup and view all the answers

    What theorem confirms that two triangles are similar if their corresponding sides are in proportion?

    <p>S.S.S. similarity theorem.</p> Signup and view all the answers

    In the context of right triangles, what does the corollary state about the altitude drawn to the hypotenuse?

    <p>The altitude separates the triangle into two triangles similar to each other and to the original triangle.</p> Signup and view all the answers

    Given triangle DEO with DX = YO = 3 cm and DY = 5 cm, how would you find the length of XE?

    <p>Use the concept of similar triangles.</p> Signup and view all the answers

    When two triangles are similar, which statement about their angles is necessarily true?

    <p>All corresponding angles are equal.</p> Signup and view all the answers

    If BC is parallel to DE in triangle ABC, what can be concluded about the ratios of the sides?

    <p>AB/AC = AD/AE.</p> Signup and view all the answers

    Prove that triangles AADC and ABDE are similar when AE and BC intersect at midpoint D.

    <p>Triangles AADC and ABDE are similar by AA similarity criterion since they share angle A and angle D is equal in both triangles.</p> Signup and view all the answers

    Given DE // BC and intersects AB and AC at D and E, explain why ∆ABC ~ A ADE.

    <p>∆ABC is similar to ∆ADE by the basic proportionality theorem since DE is parallel to BC which creates proportional segments.</p> Signup and view all the answers

    If AD = 4.5 cm and DC = 8 cm in right triangle ABC with altitude BD, how do you compute values X and Y?

    <p>Using similar triangles, set up the proportion AD/DC = X/Y to find the values of X and Y.</p> Signup and view all the answers

    Define the SSS similarity theorem using an example from the content.

    <p>The SSS similarity theorem states that if the side lengths of two triangles are in proportion, the triangles are similar. For example, the colored triangles in the theorem described follow this property.</p> Signup and view all the answers

    What does the S.A.S similarity theorem state regarding triangles, and how can it be applied?

    <p>The S.A.S similarity theorem states that if an angle of one triangle is like an angle of another and the sides including those angles are in proportion, the triangles are similar. It can be applied in proving similarity of two triangles with a shared angle and proportional side lengths.</p> Signup and view all the answers

    In triangle DEO, if m(∠O) = m(∠DXY) and DX = YO = 3 cm, DY = 5 cm, how do you find XE?

    <p>Use the property of similar triangles to set the ratios: DY/DX = XE/YO, then solve for XE.</p> Signup and view all the answers

    What can be concluded about the segments BD and AC if BD bisects ∠ABE in the colored triangles?

    <p>If BD bisects ∠ABE, then AB/BE = AD/DB, indicating that the segments are proportional.</p> Signup and view all the answers

    When altitude AD is drawn to the hypotenuse in a right triangle, what similarity results?

    <p>Altitude AD creates two smaller triangles, ADB and ADC, which are similar to the original triangle ABC.</p> Signup and view all the answers

    Study Notes

    Similarity of Triangles

    • Two chords intersect within a circle, where one chord's midpoint is the intersection point.
      • Triangles formed by the chords and the circle's center are similar.
      • The square of half the chord length is equivalent to the product of the segments created by the intersection point.

    Similar Triangles with Parallel Lines

    • A line parallel to one side of a triangle intersects the other two sides, creating a smaller triangle.
      • The smaller triangle is similar to the original triangle.

    Right-Angled Triangle with Altitude

    • The altitude from the right angle of a right triangle to the hypotenuse divides the triangle into two smaller triangles.
      • All three triangles are similar.

    SSS Similarity Theorem

    • Two triangles with proportional side lengths are similar.
    • A line bisects an angle if it divides the opposite side into segments proportional to the adjacent sides.

    SAS Similarity Theorem

    • Two triangles with congruent angles and proportional sides including those angles are similar.

    Similar triangles

    • Two triangles are similar if their corresponding angles are congruent and their corresponding sides are in proportion.
    • When triangles are similar their corresponding sides are proportional.
    • Two triangles are similar if the ratio of the lengths of corresponding sides are equal.
    • Two triangles are similar if their corresponding angles are congruent.
    • Two triangles are similar if they have the same shape but not necessarily the same size.

    Proving similar triangles

    • AA postulate
      • Two triangles are similar if they have two pairs of congruent angles.
    • SSS similarity theorem
      • If the side lengths of two triangles are in proportion, then the triangles are similar.
    • SAS similarity theorem
      • If an angle of one triangle is congruent to an angle of another triangle and the lengths of the sides including those angles are in proportion, then the triangles are similar.

    Example of similar triangles

    • In the opposite figure: AE and BC are two intersecting chords at D in a circle where D is the midpoint of BC
      • Prove that: AADC~A BDE
        • Proof:
          • ∠ACD = ∠BDE (angles subtended by the same arc are equal)
          • ∠CAD = ∠DBE (angles subtended by the same arc are equal)
          • Therefore, ∆ADC ~ ∆BDE (AA similarity postulate)
    • In the opposite figure: DEO is a triangle, m (∠O) = m (∠DXY), DX = YO = 3 cm. and DY = 5 cm. Find the length of: XE
      • Proof:
        • ∆DXY ~ ∆EOY (AA similarity postulate)
        • Therefore, DX/EY = DY/OY
        • Substitute the values: 3/EY = 5/3
        • Solve for EY: EY = 9/5
        • XE = DX + EY
        • Substitute the values: XE = 3 + 9/5
        • Simplify: XE = 24/5 or 4.8 cm

    In the opposite figure:

    CE ∩ BD = {A}, BC // DE , BC = 5 cm.and DE = 2.5 cm. 1 Prove that: ∆ABC ~ AADE

    • ** Proof:**
      • ∠BAC = ∠DAE (common angle)
      • ∠ABC = ∠ADE (corresponding angles formed by parallel lines are congruent)
      • Therefore, ∆ABC ~ ∆ADE (AA similarity postulate)

    Corrolaries

    • Corollary 1: If a line is drawn parallel to one side of a triangle and intersects the other two sides or the lines containing them, then the resulting triangle is similar to the original triangle.
      • Proof:
        • Use AA similarity postulate
    • Corollary 2: In any right-angled triangle, the altitude to the hypotenuse separates the triangle into two triangles which are similar to each other and to the original triangle.
      • Proof:
        • Use AA similarity postulate

    In the opposite figure:

    ABC is a right-angled triangle at B and BD ⊥ AC If AD = 4.5 cm.and DC = 8cm.,

    • find the values of: X and y
      • Proof:
        • ∆DBA ~ ∆DAC ~ ∆ABC (Corollary 2)
        • Therefore, AD/DB = DB/DC
        • Substitute the values: 4.5/DB = DB/8
        • Solve for DB: DB = 6
        • Therefore, X = DB = 6 cm
        • Also, AC = AD + DC = 4.5 + 8 = 12.5 cm
        • Therefore, Y = AC = 12.5 cm

    Theorem 1: S.S.S.similarity theorem

    • In the opposite figure: Prove that:
      • 1 The two coloured triangles are similar.
      • 2 BD bisects ∠ ABE
        • Proof:
          • AB/AD = BE/DE = BC/DC (given)
          • Therefore, ∆ABC ~ ∆ADE (SSS similarity theorem)
          • ∠ABC = ∠ADE (corresponding angles of similar triangles)
          • ∠ABD = ∠DBE (vertical angles)
          • Therefore, ∠ABE = ∠ADB + ∠DBE = ∠ADB + ∠ABD = ∠BDE + ∠ABD = 180° - ∠BAD = 180° - ∠EAD = ∠EBD
          • Since ∠ABD = ∠DBE, then BD bisects ∠ABE

    Theorem 2: S.A.S.similarity theorem

    • If an angle of one triangle is congruent to an angle of another triangle and the lengths of the sides including those angles are in proportion, then the triangles are similar.
      • Proof:
        • Use SAS similarity postulate

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    Description

    Test your understanding of the concepts related to similar triangles, including their properties and theorems. This quiz covers topics such as the SSS and SAS similarity theorems, as well as the influence of parallel lines on triangle similarity. Dive in and assess your knowledge of these essential geometry concepts!

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