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1.4: Geometry Using Paper Folding. Expectations: G1.1.3: Perform and justify constructions, including midpoint of a line segment and bisector of an angle, using straightedge and compass.

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## 1.4: Geometry Using Paper Folding

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**1.4: Geometry Using Paper Folding**• Expectations: • G1.1.3: Perform and justify constructions, including midpoint of a line segment and bisector of an angle, using straightedge and compass. • G1.1.6: Recognize Euclidean geometry as an axiom system. Know the key axioms and understand the meaning of and distinguish between undefined terms (e.g., point, line, and plane), axioms, definitions, and theorems. 1.4: Geometry using Paper Folding**Constructions**Diagrams created according to certain rules, using only a few specified geometric tools. 1.4: Geometry using Paper Folding**m**l ⊥ m l Perpendicular Lines Defn: Two coplanar lines are perpendicular (⊥) iff they intersect to form a ____________. 1.4: Geometry using Paper Folding**m**n m || n Parallel Lines • Defn: Two ____________ lines are parallel iff they do not intersect. 1.4: Geometry using Paper Folding**Two Perpendiculars Theorem**1. Fold the paper and crease it. Draw a line down the crease and label it l. 2. Fold lonto itself and crease the paper. Label this line m. 3. What relationship exists between l and m? 1.4: Geometry using Paper Folding**Two Perpendiculars Theorem**4. Mark a point on line l other than the point where l and mintersect. Call this point P. Fold you paper through P such that llies on itself. Draw line nthrough this crease. 1.4: Geometry using Paper Folding**Two Perpendiculars Theorem**5. What relationship exists between land n? 6. What relationship exists between mand n? Your answers to 5 and 6 are called conjectures (statements you think are true based on observations). 1.4: Geometry using Paper Folding**Two Perpendiculars Theorem**• If 2 coplanar lines are each perpendicular to the same line, then the lines are ___________ to each other. 1.4: Geometry using Paper Folding**Challenge**Given a line and a point not on the line, determine a paper folding procedure that will allow us to determine the shortest distance between the line and the point. 1.4: Geometry using Paper Folding**Some new terms**1.4: Geometry using Paper Folding**l bisects AB**l A B Segment Bisector • Defn: A line, ray or segment is a segment bisector iff it splits the original segment into 2 ____________________________. 1.4: Geometry using Paper Folding**M is the midpoint of AB.**A B M Midpoint of a Segment • Defn: Point M is the midpoint of AB iff M is • _____________ A and B and AM ____MB. 1.4: Geometry using Paper Folding**Perpendicular Bisector**• Defn: A bisector of a segment is a __________________________of the segment iff it is perpendicular to the segment. m mis the perp bis of AB. A B 1.4: Geometry using Paper Folding**D**C A BD bisects ∠B B Angle Bisector • Defn: A line (BD) or a ray (BD) is an angle bisector iff D is in the interior of the angle and it splits the given angle into ____________________________. 1.4: Geometry using Paper Folding**Perpendicular Bisector Theorem**1. Fold your paper. Label the crease line l. Label 2 points on l, A and B. 2. Fold A onto B. Call this line m. 3. Label the intersection of l and mpoint P. 1.4: Geometry using Paper Folding**Perpendicular Bisector Theorem**4. What appears to be true about l and m? 5. What is true about AP and BP? 6. Using your results from 4 and 5, how is mrelated to AB? 1.4: Geometry using Paper Folding**Perpendicular Bisector Theorem**7. Identify 4 other points on m. Label these points Q, R, S, T. 8. Determine AQ and BQ; AR and BR; AS and BS; and AT and BT. 1.4: Geometry using Paper Folding**Perpendicular Bisector Theorem**9. What is true about the distance between any point on the perpendicular bisector of a segment and the endpoints of the segment? 1.4: Geometry using Paper Folding**Perpendicular Bisector Theorem**• If a point lies on the perpendicular bisector of a segment, then it is 1.4: Geometry using Paper Folding**Angle Bisector Theorem**1. Fold two intersecting lines, l and m. Label the point of intersection P and one point on each line such that the lines form ∠APB. 2. Fold lonto m. 1.4: Geometry using Paper Folding**Angle Bisector Theorem**3. Draw line q through the crease. 4. What relationship exists between q and ∠APB? 5. Locate 3 points on qand label them C, D, and E. 1.4: Geometry using Paper Folding**Angle Bisector Theorem**6. Calculate the distances from C, D, and E to l and m. 7. Make a conjecture about the relationship between points on an angle bisector and the sides of the angle. 1.4: Geometry using Paper Folding**Angle Bisector Theorem**• If a point lies on the bisector of an angle, then it is 1.4: Geometry using Paper Folding**Which statement is true about the figure shown below?**AB ⊥ CD AC || CD AD ⊥ AB AB ⊥ AC AC = CD 1.4: Geometry using Paper Folding**The notation FG represents:**the length of a line. the length of a segment. the length of a ray. two points. a plane. 1.4: Geometry using Paper Folding**No assignment for section 1.4**1.4: Geometry using Paper Folding

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