Mathematics
Grade 10
15 min
Proofs involving corresponding parts of congruent triangles
Proofs involving corresponding parts of congruent triangles
What you'll learn
- Solve division problems with decimal quotients to the hundredths place using a standard algorithm with at least 80% accuracy.
- Round decimal quotients to the nearest tenth and hundredth as instructed in word problems with 75% accuracy.
- Explain the process of rounding decimal quotients to a partner using mathematical vocabulary such as 'tenths place', 'hundredths place', and 'rounding rule'.
- Identify situations in word problems where rounding the decimal quotient is necessary to provide a reasonable answer.
Tutorial Preview
1
Introduction & Learning Objectives
Learning Objectives
Identify corresponding parts of congruent triangles from a congruence statement.
State the definition of CPCTC (Corresponding Parts of Congruent Triangles are Congruent).
Use SSS, SAS, ASA, AAS, and HL to prove two triangles are congruent as a prerequisite step.
Construct a formal two-column proof to demonstrate that a pair of sides or angles are congruent.
Apply CPCTC as the final reason in a proof to show corresponding parts are congruent.
Analyze a geometric diagram to identify given information and formulate a proof strategy.
Solve for unknown values in geometric figures by first proving triangles congruent and then applying CPCTC.
How can an architect be absolutely certain that two triangular support trusses in a bridge are perfect mirror images of...
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Key Concepts & Vocabulary
TermDefinitionExample
Congruent TrianglesTwo or more triangles that have the exact same size and shape. All three pairs of corresponding sides and all three pairs of corresponding angles are congruent.If ΔABC ≅ ΔDEF, it means AB ≅ DE, BC ≅ EF, AC ≅ DF, ∠A ≅ ∠D, ∠B ≅ ∠E, and ∠C ≅ ∠F.
Corresponding PartsThe specific sides and angles that are in the same position in two congruent figures. The order of the vertices in a congruence statement tells you which parts correspond.In the statement ΔPQR ≅ ΔXYZ, side PQ corresponds to side XY, and angle ∠Q corresponds to angle ∠Y.
CPCTCAn acronym for 'Corresponding Parts of Congruent Triangles are Congruent'. It is a theorem used as a reason in proofs *after* you have already proven that two triangles are congruent.Once you prove ΔCAT ≅ ΔDOG...
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Core Formulas
The CPCTC Principle
If ΔABC ≅ ΔXYZ, then it logically follows that: \overline{AB} ≅ \overline{XY}, \overline{BC} ≅ \overline{YZ}, \overline{AC} ≅ \overline{XZ}, ∠A ≅ ∠X, ∠B ≅ ∠Y, and ∠C ≅ ∠Z.
This is the core concept of this lesson. It is used as the final justification in a proof to show that a specific pair of sides or angles are congruent. You must first prove the triangles are congruent before you can use CPCTC.
Triangle Congruence Conditions
Δ₁ ≅ Δ₂ if any of the following are true: SSS, SAS, ASA, AAS, HL (for right triangles).
These are the essential tools needed to complete the first major step of a proof. You must use one of these five reasons to establish that the triangles are congruent before you can apply CPCTC.
4 more steps in this tutorial
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Challenging
Given isosceles trapezoid ABCD with overline{AB} || overline{DC} and overline{AD} ≅ overline{BC}. To prove that the diagonals are congruent (overline{AC} ≅ overline{BD}), which pair of overlapping triangles would be best to prove congruent first?
A.ΔADE and ΔBCE (where E is the intersection of diagonals)
B.ΔABD and ΔBAC
C.ΔADC and ΔBCD
D.Both B and C would work equally well.
Challenging
In quadrilateral ABCD, overline{AB} ≅ overline{AD} and overline{BC} ≅ overline{DC}. A proof is constructed to show that diagonal overline{AC} bisects ∠BAD. The proof first shows ΔABC ≅ ΔADC by SSS. It then uses CPCTC to state ∠BAC ≅ ∠DAC. A second step is to prove that overline{AC} ⊥ overline{BD}. What additional step is required?
A.Use CPCTC again on ΔABC and ΔADC to show ∠BCA ≅ ∠DCA.
B.Prove a second pair of triangles, ΔABE ≅ ΔADE (where E is the intersection), using the result from the first CPCTC step.
C.Use the property that diagonals of a kite are perpendicular.
D.Prove ΔBCD ≅ ΔDAB by SSS.
Challenging
A student's proof contains the following steps:
1. overline{AB} ≅ overline{DE} (Given)
2. overline{BC} ≅ overline{EF} (Given)
3. overline{AC} ≅ overline{DF} (Given)
4. ΔABC ≅ ΔDEF (SSS)
5. ∠A ≅ ∠F (CPCTC)
Identify the error in the student's reasoning.
A.The reason in Step 4 should be SAS.
B.There is no error in the proof.
C.The given information is not sufficient for SSS.
D.The CPCTC conclusion in Step 5 mismatches corresponding parts.
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Proofs involving corresponding parts of congruent triangles is a Grade 10 Mathematics lesson on ExcelOS.
What will I learn in Proofs involving corresponding parts of congruent triangles?
You'll be able to: Solve division problems with decimal quotients to the hundredths place using a standard algorithm with at least 80% accuracy; Round decimal quotients to the nearest tenth and hundredth as instructed in word problems with 75%….
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This lesson includes 24 practice questions across multiple difficulty levels, each with instant feedback and explanations.