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UNIT 1 • SIMILARITY, CONGRUENCE, AND PROOFSLesson 7: Proving Similarity
IntroductionArchaeologists, among others, rely on the Angle-Angle (AA), Side-Angle-Side (SAS), and Side-Side-Side (SSS) similarity statements to determine actual distances and locations created by similar triangles. Many engineers, surveyors, and designers use these statements along with other properties of similar triangles in their daily work. Having the ability to determine if two triangles are similar allows us to solve many problems where it is necessary to find segment lengths of triangles.
Key Concepts
• If a line parallel to one side of a triangle intersects the other two sides of the triangle, then the parallel line divides these two sides proportionally.
• This is known as the Triangle Proportionality Theorem.
TheoremTriangle Proportionality Theorem
If a line parallel to one side of a triangle intersects the other two sides of the triangle, then the parallel line divides these two sides proportionally.
B
A
C
D
E
• In the figure above, AC DE ; therefore, =
AD
DB
CE
EB.
Prerequisite Skills
This lesson requires the use of the following skills:
• creating ratios
• solving proportions
• identifying both corresponding and congruent parts of triangles
• understanding angle bisectors
UNIT 1 • SIMILARITY, CONGRUENCE, AND PROOFSLesson 7: Proving Similarity
• This information is also helpful when determining segment lengths and proving statements.
• If one angle of a triangle is bisected, or cut in half, then the angle bisector of the triangle divides the opposite side of the triangle into two segments that are proportional to the other two sides of the triangle.
• This is known as the Triangle Angle Bisector Theorem.
TheoremTriangle Angle Bisector Theorem
If one angle of a triangle is bisected, or cut in half, then the angle bisector of the triangle divides the opposite side of the triangle into two segments that are proportional to the other two sides of the triangle.
CD
A
B
• In the figure above, ∠ ≅∠ABD DBC ; therefore, =AD
DC
BA
BC.
• These theorems can be used to determine segment lengths as well as verify that lines or segments are parallel.
Common Errors/Misconceptions
• assuming a line parallel to one side of a triangle bisects the remaining sides rather than creating proportional sides
• interchanging similarity statements with congruence statements
UNIT 1 • SIMILARITY, CONGRUENCE, AND PROOFSLesson 7: Proving Similarity