10/31/20151.4: geometry using paper folding 1.4: geometry using paper folding expectations: g1.1.3:...

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06/27/22 06/27/22 1.4: Geometry using Paper 1.4: Geometry using Paper Folding Folding 1.4: Geometry Using Paper 1.4: Geometry Using Paper Folding 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.

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Page 1: 10/31/20151.4: Geometry using Paper Folding 1.4: Geometry Using Paper Folding Expectations: G1.1.3: Perform and justify constructions, including midpoint

04/20/2304/20/23 1.4: Geometry using Paper Folding1.4: Geometry using Paper Folding

1.4: Geometry Using Paper 1.4: Geometry Using Paper Folding 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.

Page 2: 10/31/20151.4: Geometry using Paper Folding 1.4: Geometry Using Paper Folding Expectations: G1.1.3: Perform and justify constructions, including midpoint

04/20/2304/20/23 1.4: Geometry using Paper Folding1.4: Geometry using Paper Folding

ConstructionsConstructions

Diagrams created according to certain rules, using only a few specified geometric tools.

Page 3: 10/31/20151.4: Geometry using Paper Folding 1.4: Geometry Using Paper Folding Expectations: G1.1.3: Perform and justify constructions, including midpoint

04/20/2304/20/23 1.4: Geometry using Paper Folding1.4: Geometry using Paper Folding

Perpendicular LinesPerpendicular Lines

Defn: Two coplanar lines are perpendicular ( ) iff they intersect to form a ____________.⊥

l

m

l ⊥ m

Page 4: 10/31/20151.4: Geometry using Paper Folding 1.4: Geometry Using Paper Folding Expectations: G1.1.3: Perform and justify constructions, including midpoint

04/20/2304/20/23 1.4: Geometry using Paper Folding1.4: Geometry using Paper Folding

Parallel LinesParallel Lines

Defn: Two Defn: Two ____________ ____________ lines are parallel lines are parallel iff they do not intersect. iff they do not intersect.

m

nm || n

Page 5: 10/31/20151.4: Geometry using Paper Folding 1.4: Geometry Using Paper Folding Expectations: G1.1.3: Perform and justify constructions, including midpoint

04/20/2304/20/23 1.4: Geometry using Paper Folding1.4: Geometry using Paper Folding

Two Perpendiculars Theorem Two Perpendiculars Theorem

1. Fold the paper and crease it. Draw a line 1. Fold the paper and crease it. Draw a line down the crease and label it down the crease and label it l..

2. Fold 2. Fold ll onto itself and crease the onto itself and crease the paper. paper. Label this line Label this line m. .

3. What relationship exists between 3. What relationship exists between ll and and mm??

Page 6: 10/31/20151.4: Geometry using Paper Folding 1.4: Geometry Using Paper Folding Expectations: G1.1.3: Perform and justify constructions, including midpoint

04/20/2304/20/23 1.4: Geometry using Paper Folding1.4: Geometry using Paper Folding

Two Perpendiculars Theorem Two Perpendiculars Theorem

4. Mark a point on line 4. Mark a point on line ll other than the point other than the point where where ll and and mm intersect. Call this point intersect. Call this point P. Fold you paper through P such that P. Fold you paper through P such that ll lies on itself. Draw line lies on itself. Draw line nn through this through this crease.crease.

Page 7: 10/31/20151.4: Geometry using Paper Folding 1.4: Geometry Using Paper Folding Expectations: G1.1.3: Perform and justify constructions, including midpoint

04/20/2304/20/23 1.4: Geometry using Paper Folding1.4: Geometry using Paper Folding

Two Perpendiculars Theorem Two Perpendiculars Theorem

5. What relationship exists between 5. What relationship exists between ll and and nn??

6. What relationship exists between 6. What relationship exists between mm and and nn??

Your answers to 5 and 6 are called conjectures (statements you think are true based on observations).

Page 8: 10/31/20151.4: Geometry using Paper Folding 1.4: Geometry Using Paper Folding Expectations: G1.1.3: Perform and justify constructions, including midpoint

04/20/2304/20/23 1.4: Geometry using Paper Folding1.4: Geometry using Paper Folding

Two Perpendiculars Theorem Two Perpendiculars Theorem

If 2 coplanar lines are each perpendicular to If 2 coplanar lines are each perpendicular to the same line, then the lines are the same line, then the lines are ___________ to each other.___________ to each other.

Page 9: 10/31/20151.4: Geometry using Paper Folding 1.4: Geometry Using Paper Folding Expectations: G1.1.3: Perform and justify constructions, including midpoint

04/20/2304/20/23 1.4: Geometry using Paper Folding1.4: Geometry using Paper Folding

Challenge Challenge

Given a line and a point not on the line, Given a line and a point not on the line, determine a paper folding procedure that determine a paper folding procedure that will allow us to determine the shortest will allow us to determine the shortest distance between the line and the point.distance between the line and the point.

Page 10: 10/31/20151.4: Geometry using Paper Folding 1.4: Geometry Using Paper Folding Expectations: G1.1.3: Perform and justify constructions, including midpoint

04/20/2304/20/23 1.4: Geometry using Paper Folding1.4: Geometry using Paper Folding

Some new terms

Page 11: 10/31/20151.4: Geometry using Paper Folding 1.4: Geometry Using Paper Folding Expectations: G1.1.3: Perform and justify constructions, including midpoint

04/20/2304/20/23 1.4: Geometry using Paper Folding1.4: Geometry using Paper Folding

Segment Bisector Segment Bisector

Defn: A line, ray or segment is a Defn: A line, ray or segment is a segment segment bisectorbisector iff it splits the original segment iff it splits the original segment into 2 ____________________________.into 2 ____________________________.

A B

l l bisects AB

Page 12: 10/31/20151.4: Geometry using Paper Folding 1.4: Geometry Using Paper Folding Expectations: G1.1.3: Perform and justify constructions, including midpoint

04/20/2304/20/23 1.4: Geometry using Paper Folding1.4: Geometry using Paper Folding

Midpoint of a SegmentMidpoint of a Segment

Defn: Point M is the Defn: Point M is the midpointmidpoint of AB iff M is of AB iff M is

_____________ A and B and AM ____MB._____________ A and B and AM ____MB.

A M B

M is the midpoint of AB.

Page 13: 10/31/20151.4: Geometry using Paper Folding 1.4: Geometry Using Paper Folding Expectations: G1.1.3: Perform and justify constructions, including midpoint

04/20/2304/20/23 1.4: Geometry using Paper Folding1.4: Geometry using Paper Folding

Perpendicular Bisector Perpendicular Bisector

Defn: A bisector of a segment is a Defn: A bisector of a segment is a ____________________________________________________ of the of the segment iff it is perpendicular to the segment iff it is perpendicular to the segment.segment.

m

A B

m is the perp bis of AB.

Page 14: 10/31/20151.4: Geometry using Paper Folding 1.4: Geometry Using Paper Folding Expectations: G1.1.3: Perform and justify constructions, including midpoint

04/20/2304/20/23 1.4: Geometry using Paper Folding1.4: Geometry using Paper Folding

Angle Bisector Angle Bisector

Defn: A line (BD) or a ray (BD) is an Defn: A line (BD) or a ray (BD) is an angle angle bisectorbisector iff D is in the interior of the angle iff D is in the interior of the angle and it splits the given angle into and it splits the given angle into ____________________________.____________________________.

A CD

B BD bisects B∠

Page 15: 10/31/20151.4: Geometry using Paper Folding 1.4: Geometry Using Paper Folding Expectations: G1.1.3: Perform and justify constructions, including midpoint

04/20/2304/20/23 1.4: Geometry using Paper Folding1.4: Geometry using Paper Folding

Perpendicular Bisector Theorem 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 m point P.

Page 16: 10/31/20151.4: Geometry using Paper Folding 1.4: Geometry Using Paper Folding Expectations: G1.1.3: Perform and justify constructions, including midpoint

04/20/2304/20/23 1.4: Geometry using Paper Folding1.4: Geometry using Paper Folding

Perpendicular Bisector Theorem Perpendicular Bisector Theorem

4. What appears to be true about 4. What appears to be true about ll and and mm??

5. What is true about AP and BP?5. What is true about AP and BP?

6. Using your results from 4 and 5, how is 6. Using your results from 4 and 5, how is mm related to AB?related to AB?

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04/20/2304/20/23 1.4: Geometry using Paper Folding1.4: Geometry using Paper Folding

Perpendicular Bisector Theorem 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.

Page 18: 10/31/20151.4: Geometry using Paper Folding 1.4: Geometry Using Paper Folding Expectations: G1.1.3: Perform and justify constructions, including midpoint

04/20/2304/20/23 1.4: Geometry using Paper Folding1.4: Geometry using Paper Folding

Perpendicular Bisector Theorem 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?

Page 19: 10/31/20151.4: Geometry using Paper Folding 1.4: Geometry Using Paper Folding Expectations: G1.1.3: Perform and justify constructions, including midpoint

04/20/2304/20/23 1.4: Geometry using Paper Folding1.4: Geometry using Paper Folding

Perpendicular Bisector Theorem Perpendicular Bisector Theorem

If a point lies on the perpendicular bisector If a point lies on the perpendicular bisector of a segment, then it is of a segment, then it is

Page 20: 10/31/20151.4: Geometry using Paper Folding 1.4: Geometry Using Paper Folding Expectations: G1.1.3: Perform and justify constructions, including midpoint

04/20/2304/20/23 1.4: Geometry using Paper Folding1.4: Geometry using Paper Folding

Angle Bisector Theorem 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 l onto m.

Page 21: 10/31/20151.4: Geometry using Paper Folding 1.4: Geometry Using Paper Folding Expectations: G1.1.3: Perform and justify constructions, including midpoint

04/20/2304/20/23 1.4: Geometry using Paper Folding1.4: Geometry using Paper Folding

Angle Bisector Theorem Angle Bisector Theorem

3. Draw line q through the crease.

4. What relationship exists between q and APB?∠

5. Locate 3 points on q and label them C, D, and E.

Page 22: 10/31/20151.4: Geometry using Paper Folding 1.4: Geometry Using Paper Folding Expectations: G1.1.3: Perform and justify constructions, including midpoint

04/20/2304/20/23 1.4: Geometry using Paper Folding1.4: Geometry using Paper Folding

Angle Bisector Theorem Angle Bisector Theorem

6. Calculate the distances from C, D, and E 6. Calculate the distances from C, D, and E to to ll and and mm..

7. Make a conjecture about the relationship 7. Make a conjecture about the relationship between points on an angle bisector between points on an angle bisector and the sides of the angle.and the sides of the angle.

Page 23: 10/31/20151.4: Geometry using Paper Folding 1.4: Geometry Using Paper Folding Expectations: G1.1.3: Perform and justify constructions, including midpoint

04/20/2304/20/23 1.4: Geometry using Paper Folding1.4: Geometry using Paper Folding

Angle Bisector Theorem Angle Bisector Theorem

If a point lies on the bisector of an angle, If a point lies on the bisector of an angle, then it isthen it is

Page 24: 10/31/20151.4: Geometry using Paper Folding 1.4: Geometry Using Paper Folding Expectations: G1.1.3: Perform and justify constructions, including midpoint

Which statement is true about the figure shown Which statement is true about the figure shown below?below?

A.A.AB CD⊥AB CD⊥

B.B.AC || CDAC || CD

C.C.AD AB⊥AD AB⊥

D.D.AB AC⊥AB AC⊥

E.E.AC = CDAC = CD

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Page 25: 10/31/20151.4: Geometry using Paper Folding 1.4: Geometry Using Paper Folding Expectations: G1.1.3: Perform and justify constructions, including midpoint

The notation FG represents:The notation FG represents:

A.A.the length of a line.the length of a line.

B.B.the length of a segment.the length of a segment.

C.C.the length of a ray.the length of a ray.

D.D.two points.two points.

E.E.a plane.a plane.

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Page 26: 10/31/20151.4: Geometry using Paper Folding 1.4: Geometry Using Paper Folding Expectations: G1.1.3: Perform and justify constructions, including midpoint

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No assignment for section 1.4