Slide 1-1 Copyright © 2014 Pearson Education, Inc. 1.6 Constructions Involving Lines and Angles.

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Presentation transcript:

Slide 1-1 Copyright © 2014 Pearson Education, Inc. 1.6 Constructions Involving Lines and Angles

Slide 1-2 Congruent Segments Two segments that have the same length are called congruent segments. The symbol ≅ means congruent. We mark congruent segments in a figure with exactly the same number of tick marks. Copyright © 2014 Pearson Education, Inc.

Slide 1-3 Example Measuring Congruent Segments a. Use the figure to write the congruent segments and the equal distances. are marked the same, so Copyright © 2014 Pearson Education, Inc.

Slide 1-4 Example Measuring Congruent Segments b. If BC = 2 feet, find AD. BC = AD, so if BC = 2 feet, then AD = 2 feet. c. If DC = 6 feet, find AB. DC = AB, so if DC = 6 feet, then AB = 6 feet. Copyright © 2014 Pearson Education, Inc.

Slide 1-5 Definitions The midpoint of a segment is a point that divides, or bisects, a segment into two congruent segments. True statements: Copyright © 2014 Pearson Education, Inc. A line, ray, segment, or plane that intersects a segment at its midpoint is called a segment bisector.

Slide 1-6 Definitions A straight edge is a ruler with no markings on it. A compass is a geometric tool used to draw circles and parts of circles called arcs. A construction is a geometric figure drawn using a straight edge and a compass. Copyright © 2014 Pearson Education, Inc.

Slide 1-7 Example Constructing Congruent Segments Construct a segment congruent to a given segment. Given: segment AB Construct: segment CD so that segment CD ≅ segment AB Solution Step 1. Draw a ray with endpoint C. Copyright © 2014 Pearson Education, Inc.

Slide 1-8 Example Constructing Congruent Segments Step 2. Open the compass to the length of segment AB. Step 3. With the same compass setting, put the compass point on point C. Draw an arc that intersects the ray. Label the point of intersection D. Copyright © 2014 Pearson Education, Inc.

Slide 1-9 Example Constructing Congruent Angles Construct an angle congruent to a given angle. Given: ∠ A Construct: ∠ S so that ∠ S ≅ ∠ A Solution Step 1. Draw a ray with endpoint S. Copyright © 2014 Pearson Education, Inc.

Slide 1-10 Example Constructing Congruent Angles Step 2. With the compass point on vertex A, draw an arc that intersects the sides of ∠ A. Label the points of intersection B and C. Copyright © 2014 Pearson Education, Inc.

Slide 1-11 Example Constructing Congruent Angles Step 3. With the same compass setting, put the compass point on point S. Draw an arc and label its point of intersection with the ray as R. Copyright © 2014 Pearson Education, Inc.

Slide 1-12 Example Constructing Congruent Angles Step 4. Open the compass to the length BC. Keeping the same compass setting, put the compass point on R. Draw an arc to locate point T. Step 5. Draw ray ST. The angles are congruent, or ∠ S ≅ ∠ A. Copyright © 2014 Pearson Education, Inc.

Slide 1-13 Definitions Copyright © 2014 Pearson Education, Inc.

Slide 1-14 Example Constructing the Perpendicular Bisector Copyright © 2014 Pearson Education, Inc.

Slide 1-15 Example Constructing the Perpendicular Bisector Step 2. With the same compass setting, put the compass point on point B and draw another long arc. Label the points where the two arcs intersect as X and Y. Copyright © 2014 Pearson Education, Inc.

Slide 1-16 Example Constructing the Perpendicular Bisector Step 3. Draw line XY. Label the point of intersection of segment AB and line XY as M, the midpoint of AB. Copyright © 2014 Pearson Education, Inc. at midpoint M, so line XY is the perpendicular bisector of segment AB.

Slide 1-17 Example Constructing the Angle Bisector Construct the bisector of an angle. Given: ∠ A Construct: ray AD, the bisector of ∠ A Solution Step 1. Put the compass point on vertex A. Draw an arc that intersects the sides of ∠ A. Label the points of intersection B and C. Copyright © 2014 Pearson Education, Inc.

Slide 1-18 Example Constructing the Angle Bisector Step 2. Put the compass point on point C and draw an arc. With the same compass setting, draw an arc using point B. Be sure the arcs intersect. Label the point where the two arcs intersect as D. Copyright © 2014 Pearson Education, Inc.

Slide 1-19 Example Constructing the Angle Bisector Step 3. Draw ray AD. ray AD is the angle bisector of ∠ CAB. Copyright © 2014 Pearson Education, Inc.