MATH

• SSS similarity


  If the three sides of one triangle are proportional to the three sides of another triangle, then the two triangles are similar
• SAS similarity


  If one angle of a triangle is congruent to one angle of another triangle, and the sides including these angles are proportional, then the two triangles are similar
• AA similarity


  If two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar
• Proving AA similarity

  1. Angle A is congruent to Angle D

  2. Angle B is congruent to Angle E

  3. Measure of Angle A + Measure of Angle B = Measure of Angle D + Measure of Angle E

  4. Measure of Angle C is equal to Measure of Angle F

  5. Angle C is congruent to Angle F

  6. Therefore, Triangle ABC is similar to Triangle DEF
• Determining similar triangles


  • Triangles ABC and PQR are similar by AA similarity
  • Triangles ABC and DEF are not similar as there are not two congruent angles
• Corresponding angles


  Angles that are in the same relative position in two similar triangles
• Corresponding sides


  Sides that are in the same relative position in two similar triangles
• Proving SAS similarity

  1. Construct point X on segment QR such that AB is congruent to QX

  2. Segment EB is congruent to segment QR by construction

  3. Angle X is congruent to Angle R by parallel postulate

  4. Angle A is congruent to Angle Q by given

  5. Therefore, Triangle QXY is similar to Triangle QRS
• When a line is drawn through x parallel to segment rs


  Triangle qxy is congruent to triangle qrs
• Parallel lines cut by a transversal

  • Corresponding angles are congruent
• When triangle qxy is similar to triangle qrs

  Proportional sides: qx/qr = qy/qs
• Segment ab is congruent to segment qx
• Segment ab/qr = ec/qs
• Segment qy is equal to segment ac


  By multiplication property of equality
• Triangle abc is congruent to triangle qxy
• Angle b is congruent to angle x
• Triangle abc is similar to triangle qrs
• Sas similarity theorem


  If two triangles have two angles and one side congruent, then the triangles are similar
• If angle q is congruent to angle e, then the proportion pq/qr = de/ef must be true for triangle pqr to be similar to triangle def
• Proving two triangles are similar

  1. Show that two sides are proportional

  2. Show that the included angle is congruent
• Sss similarity theorem


  If three sides of one triangle are proportional to the three sides of another triangle, then the triangles are similar
• Triangle abc is similar to triangle def
• Aa similarity theorem


  If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar
• Triangle abc is similar to triangle ghi
• Triangle def is similar to triangle ghi
• Solving for unknown sides/angles in similar triangles

  1. Set up proportions using the given information

  2. Solve for the unknown values
• The value of x is 9
• The value of y is 12
• The value of x is 10