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Understanding Perpendicular Lines and Biconditional Statements in Geometry

This lesson covers the definitions of perpendicular lines, their geometric relationships, and biconditional statements. A line is perpendicular to another if they intersect at a right angle. Furthermore, we explore biconditional statements through examples, explaining that for a statement to be biconditional, it must be true in both directions. Through various exercises, learners will determine the truth of statements about geometric figures, understand the relationship between conditional statements and their converses, and provide counterexamples to demonstrate falsehood in certain cases.

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Understanding Perpendicular Lines and Biconditional Statements in Geometry

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  1. 2.2 Definitions and Biconditional Statements

  2. Definition • Two lines are called perpendicular lines if they intersect to form a right angle. • A line perpendicular to a plane is a line that intersects the plane in a point and is perpendicular to every line in the plane that intersects it.

  3. Exercise • Decide whether each statement about the diagram is true. Explain your answer using the definitions you have learned. • Points D, X, and B are collinear. • AC is perpendicular to DB. • <AXB is adjacent to <CXD. . A . . D X B . C

  4. Biconditional Statement • Biconditional Statement • It is Saturday, only ifI am working at the restaurant. • Conditional Statement • If it is Saturday, then I am working at the restaurant.

  5. Consider the following statement x = 3 if and only if x2 = 9. • Is this a biconditional statement? Yes • Is the statement true? No, because x also can be -3.

  6. Rewrite the biconditional as conditional statement and its converse. • Two angles are supplementary if and only if the sum of their measures is 180°. • Conditional: If two angles are supplementary, then the sum of their measures is 180°. • Converse: If the sum of two angles measure 180°, then they are supplementary.

  7. State a counterexample that demonstrates that the converse of the statement is false. • If three points are collinear, then they are coplanar. • If an angle measures 48°, then it is acute.

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