3.1 Duplicating Segments and ngles [and riangles] hese notes replace pages 144 146 in the book. You can read these pages for extra clarifications. Instructions for making geometric figures: You can sketch a figure without using geometry tools. You need to mark the diagram with equal segments, equal angles and parallel lines (or parts of lines) or label the measures of the parts to indicate more accurate measures. You can draw a figure using measuring tools, such as a protractor and a ruler. Make a drawing when it is important for lengths and angle measures to be fairly precise. Mark the measures in the diagrams. You can construct a figure using only a compass and straightedge. When you make a construction, do not use your measuring tools. You MUS show your arc marks to show your work!! Duplicate will mean to make an exact copy. onstruct a duplicate line segment. age 145 Investigation 1: onstruct D. Stage 1: Draw a ray longer than label endpoint. Stage 2: With compass, measure and make an arc. Stage 3: ut point of compass on point and make an arc. Label the intersection D. onstruct an Equilateral raingle. age 147 #8. onstruct Equilateral triangle Δ EQU, with sides all equal to. Stage 1: Make EQ. Stage 2: With compass equal to. Swing an arc with center E. Stage 3: With compass equal to. Swing an arc with center Q. Mark the intersection of the two arcs U. Stage 4: onstruct sides EU and QU. D Duplicate a triangle. SSS Method. age 147 #7. onstruct ΔI Δ. Stage 1: Make. Stage 2: Use compass to measure I. Swing an arc with center. Stage 3: Use compass to measure I. Swing an arc with center. Mark the intersection of the two arcs. Stage 4: onstruct sides and. U I E Q an also be used to construct an equiangular triangle, and to construct a 60 angle. S. Stirling age 1 of 14
EXEISES Lesson 3.1 elow is age 147-148 #1 3, 7, 8, 17 Use only a compass and a straight edge unless the instructions say to draw or measure! 1. Duplicate the line segments below them. D E F D E F 2. onstruct line segment XY with length + D. 3. onstruct line segment XY with length + 2 EF D. 7. Duplicate triangle Δ by copying the three sides, SSS method. 8. onstruct an equilateral triangle. Each side should be the length of this segment. 17. Use your ruler to draw a triangle with side lengths 8 cm, 10 cm, and 11 cm. Explain your method! S. Stirling age 2 of 14
3.2 onstructing erpendicular isectors segment bisector is a line, ray, or segment that passes through the midpoint of the segment. perpendicular bisector of a segment is a line (or part of a line) that passes through the midpoint of a segment and is perpendicular to the segment. onstruct a erpendicular isector age 150 Investigation 2 onstruct D, the perpendicular bisector of. Stage 1: Set compass to a radius longer than 1. 2 Stage 2: With as center, make an arc above and below. Stage 3: With as center, make an arc above and below. Label the intersections and D. Stage 4: onstruct D. Label intersection M. segment has an infinite number of bisectors, but in a plane it has only one perpendicular bisector. omplete Investigation 1 page 149-150. erpendicular isector onjecture ( 149 Inv 1): If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints. onverse of the erpendicular isector onjecture: If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment. E M Given that M is the midpoint of. Name segment bisectors: D, ME Name perpendicular bisectors: D, D, ME, etc Since point D is on the perpendicular bisector of, D = D. D Make a point G that is equidistant from and. Where is it? On the perpendicular bisector. Is a bisector? perpendicular bisector? Neither S. Stirling age 3 of 14
EXEISES Lesson 3.2 elow is age 151-153 #1 3 and Exercise #12. Do page 153 #15 20 on a separate sheet of paper. Use only a compass and a straight edge on 1 3! 1 & 3. onstruct the perpendicular bisector of then construct the perpendicular bisector of EF at the right. E 2. onstruct perpendicular bisectors to divideqd into four congruent segments. F Q D 3.1 age 148 Exercises #12 a = 50, b = 130, c = 50, d = 130, e = 50, f = 50, g = 130, h = 130, k = 155, m = 115, n = 65 S. Stirling age 4 of 14
3.3 onstructing erpendiculars to a Line ead the top of page 154 in the book. Shortest Distance onjecture he shortest distance from a point to a line is measured along the perpendicular segment from the point to the line. Draw a erpendicular Line to a line from a point NO on the line. Stage 1: lace your compass on line j and slide it until the perpendicular ray goes through. Stage 2: Mark the vertex of the 90 degree angle. Stage 3: Draw the perpendicular. Label intersection Q. he distance from a point to a line is the length of the perpendicular segment from the point to the line. j M Q S What is the shortest distance from to? M Is Q the shortest distance from to line j? Yes ead the rest of page 156 top. he altitude of a triangle is a perpendicular segment from a vertex of a triangle to the line containing the opposite side. he length of this segment is the height of the triangle. n altitude can be inside or outside the triangle, or it can be one of the triangle s sides. cute L Δ with altitude L. ltitude is inside the triangle. L is the height from the base. ight Δ with altitudes (or ). ltitude is on the triangle. is the height from base and is the height from base L Δ Obtuse with altitude L ltitude is outside the triangle. L is the height from base S. Stirling age 5 of 14
triangle has three different altitudes, so it has three different heights! In the figure below, use a colored pencil to indicate the heights from each side of the triangles. Use a note card to help you! Draw all ltitudes [in an acute triangle]. Draw all ltitudes [in an obtuse triangle]. O Where are all of the altitudes located? ll inside the triangle. Where are all of the altitudes located? One inside and 2 outside the triangle. EXEISES Lesson 3.3 elow is age 156-158 #1 3, 8, 10, 12, 18, 20. Do page 158 #13, 16 on a separate sheet of paper. 1. onstruct perpendiculars from the point to both sides of IG. Which side is closer to point? I G 2 & 3. Draw altitudes from all three vertices of each triangle below. Note where the altitudes are located. lso identify the type of triangle (acute, right or obtuse). G S. Stirling age 6 of 14
O 8. Draw an altitude M from the vertex angle of the isosceles right triangle. What do you notice about this segment? 10. Draw and/or construct a square LE given L as a diagonal. Explain how did it and support your reasoning with properties we ve learned. L 12. Draw the complement of. Explain how did it and support your reasoning with properties we ve learned. S. Stirling age 7 of 14
18. Draw a triangle with a 6 cm side and an 8 cm side and the angle between them measuring 40º. Draw a second triangle with a 6 cm side and an 8 cm side and exactly one 40º angle that is not between the two given sides. re the two triangles congruent? 20. Sketch two triangles. Each should have one side measuring 5 cm and one side measuring 7 cm, but they should not be congruent. S. Stirling age 8 of 14
3.4 onstructing ngle isectors ead the top of page 159 in the book. omplete Investigation 1 page 159, need tracing paper. ngle isector onjecture If a point is on the bisector of an angle, then it is equidistant from the sides of the angle. If is on angle bisector Q, then X = Y. or is equidistant from Q and Q. Q X Y onverse of ngle isector onjecture If a point is equidistant from the sides of the angle, then it is on the bisector of an angle. Does every angle have a bisector? Yes Is it possible for an angle to have more than one bisector? No, only one. EXEISES Lesson 3.4 elow is age 161-162 #6 8, 12. On a separate sheet of paper: Do page 162 #14 16 Must use algebra to solve!.; Do #19, 20. 6. Draw and/or construct an isosceles right triangle with z the length of each of the two legs. z S. Stirling age 9 of 14
7. Draw and/or construct Δ with angle bisector and the perpendicular bisector of. 8. Draw and/or construct Δ MSE with angle bisector S and altitude S. M E M S M 12. onstruct a linear pair of angles. arefully bisect each angle in the linear pair. What do you notice about the two angle bisectors? an you make a conjecture? an you prove that it is always true? S. Stirling age 10 of 14
EXEISES Lesson 3.5 & 3.6 age 164-166 #1, 2, 4, 5, 17; age 172 #6 On a separate sheet of paper: age 165 #14, 15; age 173 eview #15, 16 1. Draw a line parallel to n through using alternate interior angles. Label what you measured and state the property you used. 2. Draw a line parallel to n through using corresponding angles. Label what you measured and state the property you used. n n 4. Draw and/or construct a rhombus with x as the length of each side and as one of the acute angles. x 5. Draw and/or construct trapezoid with and as the two parallel sides and with as the distance between them. (here are many solutions.) S. Stirling age 11 of 14
3.5 age 165 eview Exercise #17 a = 72, b = 108, c = 108, d = 108, e = 72, f = 108, g = 108, h = 72, j = 90, k = 18, l = 90, m = 54, n = 62, p = 62, q = 59, r = 118 age 172 # 6. Draw and/or construct isosceles triangle with perimeter y and length of the base equal to x. x y S. Stirling age 12 of 14
Misc. EXEISES roblem Solving:. hree towns fire stations are shown on the map below. hey are planning to join each with straight access roads and then need to locate a central communication tower that is equidistant from the three roads. Find the location of the communication tower and explain to the planners why you know that it is the correct location. hree roads form a triangle. reate the angle bisectors to find a point equidistant from the sides of the triangle, so the tower will be equidistant from the 3 roads.. hree towns fire stations are shown on the map below. hey are planning to consolidate their resources and need to locate a central communication center that is equidistant from the three fire stations. Find the location of the communication center and explain to the planners why you know that it is the correct location. hree roads form a triangle. reate the perpendicular bisectors to find points equidistant from the endpoints. he point is equidistant from the vertices of the triangle, so the tower will be equidistant from the 3 fire stations. S. Stirling age 13 of 14
3.8 age 190 eview Exercise #14 a = 128, b = 52, c = 128, d = 128, e = 52, f = 128, g = 52, h = 38, k = 52, m = 38, n = 71, p = 38 3. age 197-198 eview Exercise #62 & 64 a = 38, b = 38, c = 142, d = 38, e = 50, f = 65, g = 106, h = 74 m FD = 30 SO m D = 30 but its vertical angle has a measure of 26. his is a contradiction. S. Stirling age 14 of 14