Math Problem Statement
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End of Semester Test: Geometry, Semester A
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7
Consider the following proof.
Given: ΔABC is a right triangle and CD ⊥ AB.
Prove: AC2 + BC2 = AB2
Proof:
StatementsReasons
1. ∠ACB ≅ ∠ADC ≅ ∠CDB
Angles with the same measure are congruent
2. ∠CAB ≅ ∠DAC ∠ABC ≅ ∠CBD
Reflexive property
3. ΔABC ~ ΔACD ΔABC ~ ΔCBD
AA similarity criteria
4. ACAB = ADAC and BCAB = DBBC
?
5. AC2 = (AB)(AD) BC2 = (AB)(DB)
Cross-multiplication
6. AC2 + BC2 = (AB)(AD) + (AB)(DB)
Addition
7. AC2 + BC2 = AB(AD + DB)
Distributive property
8. AB = AD + DB
Segment addition
9. AC2 + BC2 = (AB)(AB)
Substitution
10. AC2 + BC2 = AB2
Multiplication
Which reason completes the proof?
A.
Corresponding parts of congruent triangles are congruent.
B.
Corresponding sides of congruent triangles are proportional.
C.
Corresponding parts of similar triangles are congruent.
D.
Corresponding sides of similar triangles are proportional.
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Solution
The proof is using properties of similar triangles to establish relationships between the sides of the triangles, particularly the proportions between them.
In step 4, the statement involves the ratios of the sides of similar triangles. When two triangles are similar, their corresponding sides are proportional. This is a key feature of similar triangles and is essential for setting up the correct proportions needed for the proof.
Thus, the reason that completes the proof in step 4 is:
D. Corresponding sides of similar triangles are proportional.
Would you like further details or have any questions? Here are some related questions you might find useful:
- What is the AA similarity criterion for triangles?
- How do similar triangles help in proving the Pythagorean theorem?
- Can you explain the reflexive property in geometry?
- What is the distributive property used for in this proof?
- How does segment addition contribute to geometric proofs?
Tip: Similar triangles are a powerful tool in geometry, often used to establish ratios and solve for unknown lengths. Understanding their properties can simplify complex problems.
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Math Problem Analysis
Mathematical Concepts
Geometry
Similar Triangles
Pythagorean Theorem
Formulas
AC^2 + BC^2 = AB^2
Cross-multiplication
Segment addition
Theorems
Pythagorean Theorem
AA Similarity Criterion
Suitable Grade Level
Grades 9-10