Download proving triangle similarity

PROVING TRIANGLE SIMILARITY
Unit 2 lesson 2
OBJECTIVES
• determine whether two triangles are similar
• prove or disprove triangle similarity using
similarity shortcuts (AA, SSS, SAS)
SIMILAR TRIANGLES
Triangles are SIMILAR if their corresponding
sides are proportional and corresponding angles
are congruent. (same shape, different size)
Similarity Statement: ∆𝑷𝑷𝑷𝑷𝑷𝑷~∆𝑫𝑫𝑫𝑫𝑫𝑫
SIMILAR TRIANGLES
Congruent triangles have corresponding parts with both
angle measures and side lengths that are the same.
1. a) If 2 triangles are congruent, are they similar?
Explain.
b) If 2 triangles are similar, are they congruent?
Explain.
PROOFS
• Paragraph proofs are statements written out in complete
sentences in a logical order to show an argument
• Flow proofs are graphical methods of presenting logical
steps used to show an argument
• Two-Column Proofs are numbered statements and
corresponding reasons that show the argument in a logical
order. (This is the type of proof we will be working with)
EXAMPLE (TWO COLUMN PROOF)
Given: The figure to the right is a parallelogram,
Prove: 𝑚𝑚∠𝐷𝐷 = 𝑚𝑚𝑚𝑚𝑚.
statements
1.
is a
parallelogram
2. 𝑚𝑚∠𝐷𝐷 = 𝑚𝑚𝑚𝑚𝑚
reasons
1. Given
2. If a quadrilateral is a parallelogram, then
opposite sides are congruent
SIMILAR TRIANGLES
What is the minimum amount of information needed
to prove that two triangles are similar?
In other words, how much informations can I take
away before we no longer know they are similar?
ANGLE-ANGLE (AA) SIMILARITY SHORTCUT
If two angles of one triangle are congruent (equal) to
two angles of another triangle, then the triangles are
similar.
AA SIMILARITY
Using a two column proof, prove that ∆𝐴𝐴𝐴𝐴𝐴𝐴~∆𝑋𝑋𝑋𝑋𝑋𝑋
STATEMENT
REASON
SIDE-SIDE-SIDE SIMILARITY SHORTCUT
If the measure of the corresponding sides of two triangles
are proportional (Scale Factor), then the triangles are
similar.
SSS SIMILARITY
Using a two column proof, prove that the triangles are similar
STATEMENT
REASON
SIDE-ANGLE-SIDE SIMILARITY SHORTCUT
If the measures of two sides of a triangle are proportional
(scale factor) to the measures of two corresponding sides
of another triangle, and the included angles (between
these corresponding sides) are congruent, then the
triangles are similar.
SAS SIMILARITY
Prove that the triangles are similar.
STATEMENT
REASON
EXAMPLE PROBLEM 1
Determine whether the triangles are similar. Prove or Disprove.
STATEMENT
REASON
EXAMPLE PROBLEM 2
Determine whether the triangles are similar. Prove or disprove.
STATEMENT
REASON
OBJECTIVES
• determine whether two triangles are similar
• prove or disprove triangle similarity using
similarity shortcuts (AA, SSS, SAS)
IN CLASS ASSIGNMENT & HOMEWORK
