Conditions for Similarity of Triangles

• Similarity


  Corresponding angles are congruent, corresponding sides are proportional
• Congruence


  All corresponding parts (angles and sides) are equal
• Proving triangle similarity


  1. Identify congruent angles

  2. Identify proportional sides

  3. Apply similarity theorem
• Triangle similarity theorems


  • SSS similarity
  • SAS similarity
  • AA similarity
• Proving SSS similarity


  1. Identify 3 pairs of proportional sides

  2. Conclude triangles are similar
• Proving SAS similarity


  1. Identify 1 pair of congruent angles

  2. Identify 2 pairs of proportional sides

  3. Conclude triangles are similar
• Proving AA similarity


  1. Identify 2 pairs of congruent angles

  2. Conclude triangles are similar
• Identifying similar triangles


  • Check for congruent angles
  • Check for proportional sides
• The sum of the interior angles of a triangle is always 180 degrees
• Corresponding angles of parallel lines cut by a transversal are congruent
• Construction steps


  1. Put io by construction

  2. Opposite point x d to sub segment qr

  3. Segment eb at the opposite segment eb is congruent to segment qr

  4. Draw a line through x parallel to segment rs
• If two lines are cut by a transversal


  Corresponding angles are congruent
• Proving triangle similarity


  1. Angle x is congruent to angle r

  2. Angle a is congruent to angle q

  3. Triangle qxy is similar to triangle qrs

  4. qx/qr = qy/qs

  5. ab/qr = ec/qs

  6. qy = ac
• Proving triangle congruence


  1. Triangle abc is congruent to triangle qxy

  2. Angle b is congruent to angle x
• Proving triangle similarity


  Triangle abc is similar to triangle qrs
• If angle q is congruent to angle e, the proportion pq/qr = de/ef must be true for triangle pqr to be similar to triangle def
• Proving triangle similarity


  1. Segment bi = 18

  2. Segment ij = 12

  3. Hi = 24

  4. Segment ir = 16

  5. Triangle big is similar to triangle hir
• SSS similarity theorem


  • Sides are proportional
• Proving triangle similarity


  1. Assume ab <= de

  2. Take point x on de

  3. Triangle abc is congruent to triangle dxy

  4. Triangle dxy is similar to triangle def
• Triangle abc is similar to triangle def
• Triangle abc is not similar to triangle def
• Triangle def is similar to triangle ghi
• Similarity theorems


  • AA
  • SAS
  • SSS
• Solving for x and y

  1. Set up proportions

  2. Solve for x = 9

  3. Solve for y = 12
• Solving for angle x

  180 - (40 + 35) = 105 degrees
• Solving for x and y

  1. Set up proportions

  2. Solve for x = 9

  3. Solve for y = 8