Triangles and Similarity Checkpoint
A free Geometry lesson from the “Triangles, Similarity, and Trigonometry” unit, with a worked example and practice problems including step-by-step solutions.
This checkpoint blends congruence, similarity, scale factors, and right triangles. The goal is to decide whether the problem is about equal parts, proportional parts, or a right-triangle relationship.
What you'll learn
- Use congruent triangle relationships
- Choose congruence shortcuts
- Use similarity and right-triangle tools
Worked example
Problem. Two similar triangles have matching sides 6 and 15. A smaller side is 8. Find the larger matching side.
- The scale factor is 15/6 = 2.5.
- Multiply 8 by 2.5.
- The larger side is 20.
Answer: 20
Practice problems
1. Triangles ABC and DEF are congruent. If AC = 18, what is DF?
Show solution
- Congruence: First identify exactly what the question is asking: Triangles ABC and DEF are congruent. If AC = 18, what is DF?
- Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
- AC corresponds to DF.
- Check the result by substituting or estimating: the response should match 18 and make sense in the original problem.
Answer: 18
2. Which shortcut uses two sides and the included angle?
Choices: SAS · SSA · AAA · AAS
Show solution
- Congruence: First identify exactly what the question is asking: Which shortcut uses two sides and the included angle?
- Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
- Side-angle-side.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: SAS
3. Which shortcut is not valid for general congruence?
Choices: SSA · SSS · SAS · ASA
Show solution
- Congruence: First identify exactly what the question is asking: Which shortcut is not valid for general congruence?
- For congruence, match corresponding sides and angles in the same order before choosing a theorem or equation.
- SSA can be ambiguous.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: SSA
4. A side of 7 scales by factor 4. What is the new length?
Show solution
- Similarity: First identify exactly what the question is asking: A side of 7 scales by factor 4. What is the new length?
- Use the structure of the expression to choose a factoring pattern, then check that the factors multiply back to the original expression.
- 7 x 4 = 28.
- Check the result by substituting or estimating: the response should match 28 and make sense in the original problem.
Answer: 28
5. Matching sides are 5 and 20. What is the scale factor from smaller to larger?
Show solution
- Similarity: First identify exactly what the question is asking: Matching sides are 5 and 20. What is the scale factor from smaller to larger?
- Use the structure of the expression to choose a factoring pattern, then check that the factors multiply back to the original expression.
- 20/5 = 4.
- Check the result by substituting or estimating: the response should match 4 and make sense in the original problem.
Answer: 4
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