Congruent and Similar Triangles

Goals of the Lecture

• Determine congruent triangles
• Determine similar triangles
• Use similar triangles to solve problems

Congruent Triangles

• Definition: Same size, same shape
  • Corresponding sides have the same length
  • Corresponding angles are congruent
• Example: All sides and angles match one-to-one between the triangles

Similar Triangles

• Definition: Same shape, but not necessarily the same size
  • Corresponding angles must have the same measure
  • Corresponding sides are proportional (ratios of corresponding sides are equal)
• Example: Triangle with sides 8 inches and 4 inches corresponds to another with sides 10 inches and 5 inches (ratio 2:1)

Visualization

• Demonstrated using animated vertex to show similarity in shape but not size

Conditions for Similarity

1. AAA (Angle-Angle-Angle) or AA (Angle-Angle): Three or two pairs of corresponding angles are congruent
   • e.g., If two corresponding angles are congruent and sum of angles = 180°, the third pair must be congruent
2. SSS (Side-Side-Side): All three pairs of corresponding sides are proportional
   • e.g., If one side ratio is 9:6 (3:2), and other sides maintain this ratio
3. SAS (Side-Angle-Side): Two pairs of corresponding sides are proportional and the included angles are congruent
   • e.g., Sides with ratios 15:10 and 18:12 with an included angle confirm similarity

Example Problems

Problem 1: Determine Angles in Similar Triangles

• Given: Angle D = 35°, Angle F = 43°
• Corresponding angles C = 43°, Angle A = 35°,
• To find angle B:
  • Sum of angles in triangle = 180°
  • Angle B = 180° - 35° - 43° = 102°

Problem 2: Solve for Side Lengths in Similar Triangles

• Given: CB = 15, AC = 9, AB = 10, DF = 6
• Setup proportions due to similarity:
  • Ratio: 9:6 = 10:x and 9:6 = 15:y
  • Solve: x = 15, y = 10

Real-world Application

• Using similar triangles to solve practical problems:
  • Example: Calculate height of a tree using its shadow and a stick's shadow
  • Method: Form proportions between the corresponding sides of the triangles
  • Calculation: If a 3-foot stick casts a 5-foot shadow and a tree casts a 45-foot shadow, use proportion 3:5 = tree height:45 to find tree height
  • Result: Tree height = 27 feet