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Unit 8: Geometry and Trigonometry

Topic 3

Similar Triangles and Congruence

Two triangles are similar if they have the same shape but not necessarily the same size. Formally, similar triangles have all three pairs of corresponding angles equal, and all three pairs of corresponding sides in the same ratio (called the scale factor). On the SAT, similar triangles almost always show up through one of two shortcuts for proving similarity:
  • AA (Angle-Angle): If two angles of one triangle equal two angles of another triangle, the triangles are similar. (The third pair of angles must also be equal, since all three add to 180∘180^\circ.)
  • Parallel lines cutting a transversal: When a line parallel to one side of a triangle intersects the other two sides, it creates a smaller triangle that is similar to the original. This is the most common SAT setup.
Once you know two triangles are similar, you can set up a proportion to find unknown side lengths. If triangle ABCABC is similar to triangle DEFDEF (written △ABC∼△DEF\triangle ABC \sim \triangle DEF), then:
ABDE=BCEF=ACDF\frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF}
The key skill is matching corresponding sides correctly. Corresponding sides are opposite the equal angles. Writing the similarity statement with vertices in the correct order keeps things straight: the first vertex of each triangle corresponds, the second corresponds, and so on.
When similar triangles appear in word problems, the most common pattern is a shadow or height problem: two objects and their shadows form similar right triangles because sunlight arrives at the same angle.
Two triangles are congruent if they have the same shape and the same size. Congruence is a special case of similarity where the scale factor is 1. The SAT occasionally tests congruence criteria:
  • SSS (Side-Side-Side): All three pairs of sides are equal.
  • SAS (Side-Angle-Side): Two pairs of sides and the included angle are equal.
  • ASA (Angle-Side-Angle): Two pairs of angles and the included side are equal.
  • AAS (Angle-Angle-Side): Two pairs of angles and a non-included side are equal.
Note that SSA (two sides and a non-included angle) does not guarantee congruence. Also, AAA proves similarity but not congruence, since the triangles could be different sizes.

Core Skills

  • Identify similar triangles using the AA criterion, especially when parallel lines or shared angles are present.
  • Set up and solve proportions using corresponding sides of similar triangles.
  • Find unknown lengths using a known scale factor between similar triangles.
  • Use properties of congruent triangles to determine unknown sides or angles.
  • Solve real-world similarity problems such as shadow/height and scale-model scenarios.

Example 1: Setting Up a Proportion

In △ABC∼△DEF\triangle ABC \sim \triangle DEF, AB=6AB = 6, BC=9BC = 9, AC=12AC = 12, and DE=4DE = 4. What is the length of EFEF? Step 1: Identify corresponding sides. Since △ABC∼△DEF\triangle ABC \sim \triangle DEF, side ABAB corresponds to DEDE and side BCBC corresponds to EFEF. Step 2: Set up the proportion.
ABDE=BCEF  ⟹  64=9EF\frac{AB}{DE} = \frac{BC}{EF} \implies \frac{6}{4} = \frac{9}{EF}
Step 3: Cross-multiply and solve.
6⋅EF=4⋅9=36  ⟹  EF=66 \cdot EF = 4 \cdot 9 = 36 \implies EF = 6
The length of EFEF is 6\boxed{6}.

Example 2: Parallel Line Creating Similar Triangles

In triangle PQRPQR, line segment STST is parallel to QRQR, where SS is on PQPQ and TT is on PRPR. If PS=4PS = 4, SQ=6SQ = 6, and QR=15QR = 15, what is the length of STST? Step 1: Since ST∥QRST \parallel QR, triangle PSTPST is similar to triangle PQRPQR by AA (the parallel lines create equal corresponding angles). Step 2: Find the scale factor. PQ=PS+SQ=4+6=10PQ = PS + SQ = 4 + 6 = 10.
PSPQ=410=25\frac{PS}{PQ} = \frac{4}{10} = \frac{2}{5}
Step 3: Corresponding sides are in the same ratio.
STQR=25  ⟹  ST15=25  ⟹  ST=6\frac{ST}{QR} = \frac{2}{5} \implies \frac{ST}{15} = \frac{2}{5} \implies ST = 6
The length of STST is 6\boxed{6}.

Example 3: Shadow Problem

A 6-foot-tall person casts a 4-foot shadow at the same time that a nearby tree casts a 22-foot shadow. How tall is the tree, in feet? Step 1: The sun hits both the person and the tree at the same angle, creating two similar right triangles (one for the person and one for the tree). The heights correspond, and the shadow lengths correspond. Step 2: Set up the proportion.
person’s heightperson’s shadow=tree’s heighttree’s shadow\frac{\text{person's height}}{\text{person's shadow}} = \frac{\text{tree's height}}{\text{tree's shadow}}
64=h22\frac{6}{4} = \frac{h}{22}
Step 3: Cross-multiply and solve.
4h=6⋅22=132  ⟹  h=334h = 6 \cdot 22 = 132 \implies h = 33
The tree is 33\boxed{33} feet tall.

Example 4: Finding the Scale Factor from Areas

Two similar triangles have areas of 18 square centimeters and 50 square centimeters. If the shorter triangle has a base of 6 cm, what is the base of the larger triangle? Step 1: The ratio of the areas of similar figures equals the square of the scale factor.
A1A2=(s1s2)2  ⟹  1850=(s1s2)2\frac{A_1}{A_2} = \left(\frac{s_1}{s_2}\right)^2 \implies \frac{18}{50} = \left(\frac{s_1}{s_2}\right)^2
Step 2: Take the square root to get the scale factor.
s1s2=1850=925=35\frac{s_1}{s_2} = \sqrt{\frac{18}{50}} = \sqrt{\frac{9}{25}} = \frac{3}{5}
Step 3: Apply the scale factor to the base.
6b=35  ⟹  3b=30  ⟹  b=10\frac{6}{b} = \frac{3}{5} \implies 3b = 30 \implies b = 10
The base of the larger triangle is 10\boxed{10} cm.

Example 5: Using Congruence

Triangles ABCABC and DEFDEF are congruent. If AB=5AB = 5, BC=7BC = 7, ∠B=60∘\angle B = 60^\circ, and DE=5DE = 5 and ∠E=60∘\angle E = 60^\circ, what is the length of EFEF? Step 1: The congruence △ABC≅△DEF\triangle ABC \cong \triangle DEF means corresponding parts are equal. Vertex BB corresponds to vertex EE (both have the 60∘60^\circ angle), ABAB corresponds to DEDE (both equal 5), and BCBC corresponds to EFEF. Step 2: Since corresponding sides of congruent triangles are equal:
EF=BC=7EF = BC = 7
The length of EFEF is 7\boxed{7}.

Key Takeaways

  • The most common way the SAT establishes similarity is through AA: shared angles plus parallel lines. Always look for parallel lines or right angles that give you two matching angle pairs.
  • When setting up proportions, write the similarity statement carefully and match corresponding sides. A mismatch flips the proportion and gives a wrong answer.
  • Shadow and scale-model problems are the standard real-world application of similar triangles. The setup is always the same: two proportional right triangles.
  • The ratio of areas of similar figures is the square of the scale factor. If you are given areas and need a length, take the square root of the area ratio to get the length ratio.