Math Problem Statement

Determine if triangle CDE and triangle FGH are or are not similar, and if they are, state how you know. (Note that figures are NOT necessarily drawn to scale.)

Solution

To determine whether triangles CDE\triangle CDE and FGH\triangle FGH are similar, we check if their corresponding angles are equal or their sides are proportional. Triangles are similar if they meet either the AA (Angle-Angle) similarity criterion or the SSS (Side-Side-Side) or SAS (Side-Angle-Side) proportionality criteria.

Given Information:

  • For CDE\triangle CDE:

    • CDE=50\angle CDE = 50^\circ,
    • Side lengths: CD=20CD = 20, DE=28DE = 28.
  • For FGH\triangle FGH:

    • GFH=50\angle GFH = 50^\circ, FGH=82\angle FGH = 82^\circ,
    • Side lengths: GF=60GF = 60, FH=81FH = 81, GH=63GH = 63.

Step 1: Check angles for CDE\triangle CDE and FGH\triangle FGH

  1. Both triangles have CDE=GFH=50\angle CDE = \angle GFH = 50^\circ.
  2. FGH\triangle FGH has FGH=82\angle FGH = 82^\circ, which makes the third angle FGH=180(82+50)=48\angle FGH = 180^\circ - (82^\circ + 50^\circ) = 48^\circ.
  3. CDE\triangle CDE has CDE=50\angle CDE = 50^\circ, so the remaining angle is 180(50+50)=80180^\circ - (50^\circ + 50^\circ) = 80^\circ.

Step 2: Compare side proportions

We calculate the side ratios for corresponding sides:

  1. CDGF=2060=13\frac{CD}{GF} = \frac{20}{60} = \frac{1}{3},
  2. DEFH=28810.345\frac{DE}{FH} = \frac{28}{81} \approx 0.345,
  3. CEGH\frac{CE}{GH} (needs to be determined from the triangle).

The sides are not proportional.

Conclusion:

The triangles are not similar because they fail the angle similarity criterion (AA) and the sides are not proportional (SSS or SAS). Let me know if you'd like a more detailed explanation!

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Math Problem Analysis

Mathematical Concepts

Geometry
Triangle Similarity
Proportionality
Angle-Angle Criterion
Side-Side-Side Criterion
Side-Angle-Side Criterion

Formulas

Angle sum of a triangle = 180°
Proportionality: a/b = c/d

Theorems

Angle-Angle (AA) Similarity Theorem
Side-Side-Side (SSS) Similarity Theorem
Side-Angle-Side (SAS) Similarity Theorem

Suitable Grade Level

Grades 8-10