Chapter Closure
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Can you spot any congruent triangles in the diagram? Start there.
See solution.
We have been given two pieces of information.
Let's add this information to the diagram.
Examining the diagram, we see that △ ABC and △ ADC have two pairs of congruent sides. Additionally, they also share AC as a side, which means this side of the triangles is congruent by the Reflexive Property. Therefore, △ ABC and △ ADC are congruent by the SSS (Side-Side-Side) Congruence Theorem.
Let's mark a pair of congruent corresponding angles, ∠ BCA and ∠ DCA, in these triangles.
Now we can identify a second pair of congruent triangles, △ BCE and △ DCE, by the SAS (Side-Angle-Side) Congruence Theorem.
Since ∠ BEC and ∠ DEC are corresponding angles and BD is a straight line, the measures of ∠ BEC and ∠ DEC must be half of a straight angle, which equals a right angle.
The definition of perpendicular lines is that they cut each other at a right angle. With this information we have proved that AC and BD are perpendicular. Let's show this as a two-column proof.
Statement
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Reason
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1. AB≅AD, BC≅DC
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1. Given
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2. AC≅AC
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2. Reflexive Property of Congruence
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3. △ ABC ≅ △ ADC
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3. SSS Congruence Theorem
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4. ∠ BCE ≅ ∠ DCE
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4. Corresponding parts of congruent triangles
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5. EC≅EC
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5. Reflexive Property of Congruence
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6. △ BCE ≅ △ DCE
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6. SAS Congruence Theorem
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7. ∠ BEC ≅ ∠ DEC
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7. Corresponding parts of congruent triangles
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8. ∠ BED is a straight angle
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8. As observed in the diagram
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9. m∠ BED=180^(∘)
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9. Definition of a straight angle
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10. m∠ BEC +m∠ DEC=180^(∘)
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10. Substitution Property of Equality
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11. m∠ BEC +m∠ BEC=180^(∘)
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11. Substitution Property of Equality
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12. 2m∠ BEC=180^(∘)
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12. Addition Property of Equality
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13. m∠ BEC=90^(∘)
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13. Divison Property of Equality
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14. AC⊥ BD
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14. Definition of perpendicular lines
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