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Mastering Geometric Constructions: Duplicating Segments, Angles, and Perpendicular Bisectors

In this lesson, students will learn how to duplicate line segments and angles, and construct perpendicular bisectors. Through hands-on practice using a compass and straightedge, students will master essential geometric constructions. The objectives include solving for angle measures, applying problem-solving skills, and making conjectures about perpendicular bisectors. Key assessments include a test on angle relationships, vocabulary review, and weekly tutoring sessions. Engage with interactive practice exercises to reinforce understanding of geometric concepts.

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Mastering Geometric Constructions: Duplicating Segments, Angles, and Perpendicular Bisectors

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  1. DO NOW • Take out a protractor • Take out your compass • Put your 2.6 Worksheet on your desk ready to be stamped. • In the picture to the left, solve for x. Then find the angle measures.

  2. Announcements • Test next Wednesday over angle relationships and constructions. • Vocab and review (on website) are due Wednesday. • Tutoring Tuesdays/Thursdays • Test corrections from Unit 1 are due next Friday.

  3. Constructions Duplicating Segments and Angles, Constructing Perpendicular Bisectors

  4. Today’s Objectives • Duplicate a line segment, an angle and a polygon • Construct perpendicular bisectors and midpoints • Make conjectures about perpendicular bisectors • Use Problem Solving skills

  5. Creating Geometric Figures • Sketch—Freehand, “looks like,” not accurate • Draw—Carefully use a straightedge and use measurements from protractor, fairly accurate. • Construct—Don’t rely on protractor measurements. Use compass and straightedge. Guaranteed to be accurate.

  6. Duplicating a Segment • Step 1: Draw segment AB. • Step 2: Draw a ray starting at C. • Step 3: Open your compass to match the length of AB. • Step 4: Put the point of your compass on point C and drag the pencil to make an arc that intersects the ray. • Step 5: Mark the intersection D. • AB ≅ CD

  7. Duplicating an Angle • Step 1: Draw ∠DEF. • Step 2: Draw a ray from point G. • Step 3: With your compass, put an arc around E that intersects the rays ED and EF. • Step 4: Without changing the width of your compass, put a matching arc around G. • Step 5: Open your compass to measure the distance between intersection points in ∠DEF. • Step 6: Mark the same distance from the intersection point in G through the arc. • Step 7: Draw a ray from G to the new intersection point. • ∠E ≅∠G

  8. Practice • Draw a segment that is 5cm (with ruler measurements). • Construct a duplicate segment (without ruler measurements, only using a compass and straightedge).

  9. Practice • Draw an angle that is 70o(using a protractor) • Construct a duplicate angle (not using a protractor).

  10. Perpendicular Bisectors—Terms • A segment bisector—a line, ray, or segment that passes through the midpoint of a segment. • Cuts the line segment in half • Perpendicular lines—intersect at a right angle. • Perpendicular bisector—passes through the midpoint of a segment at a right angle. • Equidistant—the same distance

  11. Constructing Perpendicular Bisectors • Step 1: Draw a line segment. Set your compass to more than half the distance between the two endpoints. • Step 2: Using one endpoint as center, swing an arc on both sides of the segment. • Step 3: Using the same compass setting, swing an arc from the other endpoint to intersect each arc. • Step 4: Mark your two intersection points and connect them.

  12. Perpendicular Bisector Conjecture • If a point is on the perpendicular bisector of a segment, then it is _________ from the endpoints. • equidistant

  13. Converse of Perpendicular Bisector Conjecture • If a point is equidistant from the endpoints of a segment, then it is on the _______________of the segment. • perpendicular bisector • Also true!

  14. Practice • Draw and label AB. Construct the perpendicular bisector of AB.

  15. Practice • Draw and label QD. Construct perpendicular bisectors to divide QD into four congruent segments.

  16. Stations! • Direct: Practice Constructions • Collaborative: Without writing on the worksheets, complete 3.1 and 3.2 worksheets on a separate sheet of paper as a group. (Each person turn in your own paper.) • Independent: Take your test and your notebook and begin test corrections. If you are satisfied with your test score, begin vocabulary that is due on Wednesday.

  17. Practice

  18. Today’s Objectives • Duplicate a line segment, an angle and a polygon • Construct perpendicular bisectors and midpoints • Make conjectures about perpendicular bisectors • Use Problem Solving skills

  19. Exit Slip For all exercises, do not erase your construction marks. • Draw an obtuse angle. Label it ∠LGE, then duplicate it. • Draw a line segment. Label it RS, then duplicate it. • Draw a line segment. Label it PQ, then construct its perpendicular bisector. • Draw two acute angles. Construct a third angle with measure equal to the sum of the measures of the first two angles. You may not use a protractor.

  20. Exit Slip Honors For all exercises, do not erase your construction marks. • Draw an obtuse angle. Label it ∠LGE, then duplicate it. • Draw a line segment. Label it PQ, then construct its perpendicular bisector. • Draw two acute angles. Construct a third angle with measure equal to the sum of the measures of the first two angles. You may not use a protractor. • Draw a quadrilateral QUAD. Duplicate it, matching all the angles AND segments. Label the construction COPY so that QUAD ≅COPY.

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