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MathChapter 17: Triangles

Notes

Similar Triangles and Proportional Parts Notes

Establish AA similarity, preserve vertex correspondence, and solve side, perimeter, proportionality, and midsegment problems.

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Similarity preserves shape, not size. Corresponding angles match and every corresponding side uses one common scale factor.

Correspondence comes from the order

Corresponding similar trianglesTriangle ABC with sides 6, 8, and 10 is similar to triangle DEF with corresponding sides 9, 12, and 15. Matching arcs identify vertex correspondence A-D, B-E, C-F.
Corresponding similar trianglesTriangle ABC with sides 6, 8, and 10 is similar to triangle DEF with corresponding sides 9, 12, and 15. Matching arcs identify vertex correspondence A-D, B-E, C-F.ABCDEF681091215△ABC ∼ △DEF
Similarity correspondence
Triangle ABCTriangle DEF
AD
BE
CF
ABDE
BCEF
ACDF
Corresponding-side proportion
\[\frac{AB}{DE}=\frac{BC}{EF}=\frac{AC}{DF}\]

AA Similarity Postulate

AA SimilarityTwo triangles have one pair of angles marked with single arcs and another pair marked with double arcs, establishing similarity by AA.
AA SimilarityTwo triangles have one pair of angles marked with single arcs and another pair marked with double arcs, establishing similarity by AA.ABCDEFABBCACDEEFDFtwo angle pairs → AA similarity
Worked example

Scale factor and perimeter

Corresponding sides are 8 and 14. The smaller perimeter is 30. Find the larger perimeter.

  1. Scale factor

    Larger/smaller is (14/8=7/4).

  2. Apply

    Perimeters use the same factor: (30(7/4)=52.5).

The larger perimeter is (52.5).

Parallel segments create proportional parts

Triangle Proportionality TheoremTriangle ABC contains segment DE parallel to BC, creating smaller triangle ADE similar to triangle ABC and dividing sides AB and AC proportionally.
Triangle Proportionality TheoremTriangle ABC contains segment DE parallel to BC, creating smaller triangle ADE similar to triangle ABC and dividing sides AB and AC proportionally.BACDEDE ∥ BC; △ADE ∼ △ABC

Figure not drawn to scale.

A valid part-to-part proportion
\[\frac{AD}{DB}=\frac{AE}{EC}\]
Triangle midsegmentD and E are marked midpoints of sides AB and AC. Segment DE is parallel to BC and has half the length of BC.
Triangle midsegmentD and E are marked midpoints of sides AB and AC. Segment DE is parallel to BC and has half the length of BC.BACDED,E midpoints; DE ∥ BC; DE = ½BC

Figure not drawn to scale.

Intersecting similar trianglesTwo triangles meet at R. Vertical angles at R and a second angle pair from parallel outer sides establish AA similarity.
Intersecting similar trianglesTwo triangles meet at R. Vertical angles at R and a second angle pair from parallel outer sides establish AA similarity.Rvertical angles + parallel-line angles → AA

Figure not drawn to scale.

Mini check

Area factor is different

Similar triangles have side scale factor 3. What is their perimeter scale factor?

Show answer and explanation

Answer: 3.

Perimeter is linear. Only area uses the square of the side factor.

Key takeaways

Key takeaways

  • Similarity notation fixes vertex correspondence.
  • AA requires two matching angle pairs.
  • Corresponding sides and perimeters share one scale factor.
  • A parallel segment creates smaller similar triangles and proportional side pieces.
  • A midpoint segment is parallel to the third side and half as long.
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Put these notes into practice

Apply the ideas with SAT-style questions, then reinforce key details with flashcards.