4.3 Similarity Proofs & Proportional Relationships

Proving Two Triangles Similar

Constructed-response questions in Parts II-IV often ask you to prove a similarity and then use it. Because AA is the easiest criterion to satisfy, most similarity proofs come down to finding two pairs of congruent angles. Standard G-SRT.B.5 expects you to use similarity criteria both to solve problems and to justify relationships.

Reliable sources of congruent angles include:

• A shared (common) angle belonging to both triangles.
• Parallel lines cut by a transversal, giving congruent corresponding or alternate interior angles.
• Vertical angles where two lines cross.
• Right angles, all congruent to each other.

Once you have AA, state the similarity in matching vertex order, then write the proportion of corresponding sides. That proportion is the tool for finding a missing length.

A Model AA Proof

Suppose DE is parallel to BC with D on AB and E on AC, and you must prove triangle ADE ~ triangle ABC.

1. Angle A is congruent to angle A (shared angle).
2. Because DE is parallel to BC, angle ADE is congruent to angle ABC (corresponding angles).
3. Therefore triangle ADE ~ triangle ABC by AA.
4. Corresponding sides are proportional: AD/AB = AE/AC = DE/BC.

The Side-Splitter (Triangle Proportionality) Theorem

Side-Splitter Theorem (G-SRT.B.4): If a line is parallel to one side of a triangle and intersects the other two sides, it divides those two sides proportionally.

In triangle ABC with the parallel line meeting AB at D and AC at E:

AD/DB = AE/EC

Converse: If a line divides two sides of a triangle proportionally, then it is parallel to the third side. The converse is what lets you prove lines parallel from a proportion.

Worked example. A line parallel to one side cuts the other two sides so that AD = 4, DB = 6, and AE = 6. Find EC. By the side-splitter theorem, AD/DB = AE/EC, so 4/6 = 6/EC. Cross-multiply: 4 * EC = 36, giving EC = 9.

Be careful which segments you use. The side-splitter theorem compares the two pieces of each cut side (AD to DB), not a piece to the whole. If a question gives the whole side instead, switch to the similar-triangle ratio AD/AB = DE/BC.

Worked example (whole-side ratio). DE is parallel to BC, AD/AB = 3/5, and BC = 20. Since triangle ADE ~ triangle ABC, DE/BC equals the scale factor AD/AB = 3/5, so DE = (3/5)(20) = 12.

The Angle-Bisector Theorem

Angle-Bisector Theorem: In a triangle, the bisector of an angle divides the opposite side into two segments proportional to the other two sides.

If AD bisects angle A in triangle ABC and meets BC at D, then:

AB/AC = BD/DC

Worked example. AD bisects angle A and meets BC at D, with BD = 6, DC = 10, and AB = 9. Then AB/AC = BD/DC, so 9/AC = 6/10. Cross-multiply: 6 * AC = 90, giving AC = 15.

A memory hook: the two segments the bisector creates on the far side (BD and DC) are "pulled" toward the longer of the two sides. Since AC (15) is longer than AB (9), the segment DC (10) next to it is longer than BD (6).

Finding Missing Sides from Similarity

The general procedure ties this chapter together:

1. Identify or prove the triangles similar (usually AA).
2. Write the similarity statement in correct vertex order.
3. Build a proportion from corresponding sides.
4. Cross-multiply and solve.

Keeping these four setups straight prevents the most common error: pairing a segment with the wrong partner and cross-multiplying a false proportion.

The Midsegment: A Special Side-Splitter

A midsegment of a triangle joins the midpoints of two sides. It is the side-splitter case where the parallel line cuts both sides in a 1:1 ratio, so it is parallel to the third side and exactly half its length. If a midsegment measures 7, the side it faces measures 14. This is a fast, frequently tested consequence of the side-splitter theorem.

Using the Converse to Prove Parallel

The converse turns a proportion into a parallel conclusion. Suppose in triangle ABC that AD = 3 and DB = 9 on side AB, while AE = 4 and EC = 12 on side AC. Compare the ratios: AD/DB = 3/9 = 1/3 and AE/EC = 4/12 = 1/3. Because the two sides are divided in the same ratio, the converse of the side-splitter theorem guarantees that DE is parallel to BC. On a constructed-response item, cite the theorem by name as your reason.

Writing the Reason Column

Parts II-IV award credit for reasons, not just answers. A similarity proof typically chains together: "Given," "Reflexive property" for a shared angle or side, "Parallel lines form congruent corresponding angles," "AA," and finally "Corresponding sides of similar triangles are proportional." Naming each theorem earns the justification credit that separates a full-credit response from a partial score.