Warm-up 1.5

1. Warm-up 1.5

2. 1.5 Constructions

3. Definitions • Construction • Mathematically precise figure • Use a straightedge and a compass to draw a geometric figure • Straightedge • Ruler with no markings on it. • Compass • Geometric tool used to draw circles and parts of circles called arcs.

4. Given: A B Construction #1 Construct a segment congruent to a given segment. This is our compass. Procedure: 1. Use a straightedge to draw a line. Call it l. Construct: XY = AB Don’t change your radius! 2. Choose any point on l and label it X. 3. Set your compass for radius AB and make a mark on the line where B lies. Then, move your compass to line l and set your pointer on X. Make a mark on the line and label it Y. l X Y

5. Construction #2 A 1) Draw a ray. Label it RY. 2) Using B as center and any radius, draw an arc intersecting BA and BC. Label the points of intersection D and E. B C 3) Using R as center and the SAME RADIUS as in Step 2, draw an arc intersecting RY. Label point E2 the point where the arc intersects RY R Y Construct an angle congruent to a given angle Given: Procedure: D E Construct: D2 4) Measure the arc from D to E. E2 5) Move the pointer toE2 and make an arc that that intersects the blue arc to get point D2 6) Draw a ray from R through D2

6. definitions • Perpendicular lines • Two lines that intersect to form right angles. • The symbol • Perpendicular bisector • Is a line, segment, or ray that is perpendicular to the segment at its midpoint, thereby bisecting the segment into two congruent segments.

7. definitions • Angle bisector • Ray that divides an angle into two congruent coplanar angles. • Its endpoint is a the angle vertex

8. Construction #3 A B C Using B as center and any radius, draw and arc that intersects BA at X and BC at point Y. 3. Draw BZ. How do I construct a Bisector of a given angle? Z Given: X Y Procedure: 2. Using X as center and a suitable radius, draw and arc. Using Y as center and the same radius, draw an arc that intersects the arc with center X at point Z.

9. Construction #4 B A 2. Draw XY How do I construct a perpendicular bisector to a given segment? X Given: Y Procedure: Using any radius greater than 1/2 AB, draw four arcs of equal radii, two with center A and two with center B. Label the points of intersection X and Y.

10. Construction #5 C k 3. Draw CZ. How do I construct a perpendicular bisector to a given segment at a given point? Z Given: X Y Procedure: Using C as center and any radius, draw arcs intersecting k at X and Y. Using X as center and any radius greater than CX,draw an arc. Using Y as center and the same radius, draw and arc intersecting the arc with center X at Z.

11. Construction #6 P k 3. Draw PZ. How do I construct a perpendicular bisector to a given segment at a given point outside the line? Given: X Y Z Procedure: Using P as center, draw two arcs of equal radii that intersect k at points X and Y. Using X and Y as centers and a suitable radius, draw arcs that intersect at a point Z.

12. Construction #7 P k Let A and B be two points on line k. Draw PA. How do I construct a line parallel to a given line through a given point? 1 l Given: A B Procedure: At P, construct <1 so that <1 and <PAB are congruent corresponding angles. Let l be the line containing the ray you just constructed.

13. Step 1: Draw a ray with endpoint T. Step 2: Open the compass to the length of KM. Step 3: With the same compass setting, put the compass point on point T. Draw an arc that intersects the ray. Label the point of intersection W. TWKM Basic Constructions LESSON 1-7 Additional Examples Construct TW congruent to KM. Quick Check

14. Step 1: Draw a ray with endpoint Y. Step 2: With the compass point on point G, draw an arc that intersects both sides of G. Label the points of intersection E and F. 75° Step 3: With the same compass setting, put the compass point on point Y. Draw an arc that intersects the ray. Label the point of intersection Z. Basic Constructions LESSON 1-7 Additional Examples Construct Y so that YG.

15. Step 4: Open the compass to the length EF. Keeping the same compass setting, put the compass point on Z. Draw an arc that intersects the arc you drew in Step 3. Label the point of intersection X. Step 5: Draw YX to complete Y. Y G Basic Constructions LESSON 1-7 Additional Examples (continued) Quick Check

16. Start with AB. Step 2: With the same compass setting, put the compass point on point B and draw a short arc. Step 1: Put the compass point on point A and draw a short arc. Make sure that the opening is less than AB. 1 2 Basic Constructions LESSON 1-7 Additional Examples Quick Check 1 2 Use a compass opening less than AB. Explain why the construction of the perpendicular bisector of AB shown in the text is not possible. Without two points of intersection, no line can be drawn, so the perpendicular bisector cannot be drawn.

17. Draw and label a figure to illustrate the problem mAWR = mBWRDefinition of angle bisector x = 4x – 48Substitute x for mAWR and 4x – 48 for mBWR. m AWR = 16m BWR = 4(16) – 48 = 16 Substitute 16 for x. mAWB = m AWR + mBWRAngle Addition Postulate mAWB = 16 + 16 = 32 Substitute 16 for m AWR and for m BWR. Basic Constructions LESSON 1-7 Additional Examples Quick Check WR bisects AWB. mAWR = x and m BWR = 4x – 48. Find mAWB. –3x = –48 Subtract 4x from each side. x = 16 Divide each side by –3.

18. Step 1: Put the compass point on vertex M. Draw an arc that intersects both sides of M. Label the points of intersection B and C. Step 2: Put the compass point on point B. Draw an arc in the interior of M. Basic Constructions LESSON 1-7 Additional Examples Construct MX, the bisector of M.

19. Step 3: Put the compass point on point C. Using the same compass setting, draw an arc in the interior of M. Make sure that the arcs intersect. Label the point where the two arcs intersect X. Step 4: Draw MX. MX is the angle bisector of M. Basic Constructions LESSON 1-7 Additional Examples (continued) Quick Check

20. No assignment today! Today is a hands-on activity day.