D AY 36 – C OPYING AND BISECTING A LINE SEGMENT AND AN ANGLE
I NTRODUCTION We are familiar with the terms line segment and angle, having defined and discussed their properties in our previous lessons. In the real world, angles and line segments are usually copied to achieve a number of desirable effects, for instance, in architecture, to come up with equal lengths and angles with the same measure. An angle and a line segment can also be divided into two equal parts. In this lesson, we are going to learn how to copy a line segment as well as an angle and later discuss how to divide them into two equal parts.
V OCABULARY Line segment A portion of a line between two endpoints. Line segment AB, written as 𝐵𝐶 is shown below. B A Angle When two rays diverge from a common point or meet at the same endpoint, the amount of turn between them is called an angle.
Ray A line with one endpoint and extending infinitely in the other direction. Vertex The common endpoint, intersection point or starting point of two or more rays or lines. Bisect To divide something, in this case, a line segment or an angle into two equivalent parts. Straightedge An uncalibrated ruler.
The midpoint of a line segment A point on a line segment that divides it into two equal parts. The perpendicular bisector of a line segment A line segment which passes through the midpoint of the line segment and it is also perpendicular to that segment.
We can copy and bisect both a line segment and an angle in geometry using various methods and a variety of tools such as a pair of compasses, a straight edge, a string, a reflective device, use of paper folding and a dynamic geometric software. We will mainly use a pair of compasses and a straight edge to carry out the activities.
C OPYING A LINE SEGMENT We are going to copy a given line segment using a compass and a straight edge. Consider the line segment KL below. K L M To copy KL : 1. We mark a point M below KL as shown above, this becomes one endpoint of the new line segment.
2.We draw another line segment below KL using a straightedge with point M as one of its endpoints as shown below. The segment should be longer than KL . K L M
3. We place the needle of the compass at point K and open the compass until the tip of the pencil coincides with point L. The length of KL is now equal to the width of the compass. 4. Without interfering with the width of the compass, we place the needle of the compass at point M and draw an arc on the new line segment as shown below. K L M
5. We label the point where the arc intersects the new segment as point N as shown below. K L M N 6. MN is now a copy of KL . We say that, MN is congruent to KL .
B ISECTING A LINE SEGMENT To bisect a line segment means to divide it into two equal parts. It is also referred to as constructing the perpendicular bisector of a line segment. We are going to bisect a given line segment using a compass and a straight edge. Consider the line segment KL below. K L
To bisect KL : 1. We place the needle of the compass at point K and open the compass so that its width is more than half of KL . The exact width is immaterial in this case. 2. We then draw an arc of appropriate length below and above KL without changing the compass width as shown below.
The two arcs appear as shown below. L K
3. Now, without changing the compass width we place the needle of the compass at point L and in the same way, we draw an arc below and above KL in such a way that the arcs intersect the first two arcs as shown below. L K
4. Using a straightedge we draw a straight line through the points where the arcs intersect as shown below. We then label the intersection point M. K M L
5. We will discover that the line we have just drawn is perpendicular to the first line and it bisects KL at point M, which is its midpoint. We should note that in this case, KM = ML .
C OPYING AN ANGLE We are going to copy a given angle using a compass and a straightedge, that is, we are to construct an angle which is congruent to a given angle. Consider ∠ABC shown below. A C B
To copy ∠ABC using a compass and a straightedge: 1. We draw a ray longer than either arm of ∠ABC using a straightedge and label its endpoint E as shown below. This ray forms one of the arms of the duplicate angle. A B C E
2. We use a compass to draw an arc to intersect both arms of ∠ABC , using point B as the center as shown below. A B C
3. Without changing the compass width, we draw another arc on the ray with endpoint E, now, using point E as the center. We label the point of intersection F as shown below. F E
4. We use the point of intersection of the arc and arm BC as the center and open the compass up to the exact point where the arc intersects arm BA, then we make an arc as shown below. A C B
5. Without changing the compass width, we use point F as the center and make another arc that intersects the first one as shown below. We label the point of intersection D. D E F
6. To complete the construction, we draw a ray from point E through point D as shown below. D F E We can measure and confirm that ∠𝐁𝐂𝐃 ≅ ∠𝐄𝐅𝐆
B ISECTING AN ANGLE Bisecting an angle means dividing the angle into two equal angles. An angle bisector is simply a ray which divides the angle into two equal parts. We use an angle bisector to bisect a given angle. Let us learn how to bisect an given angle using a straightedge and a compass. Consider ∠ABC shown below. A B C
To bisect ∠ABC : 1. We draw an arc to cross the two arms of ∠ABC using the vertex B as the center. We label the points of intersection as D and E as shown below. A D E C B
2. To construct the angle bisector, we use point D as the center and make a draw an arc. Then without changing the width of the compass, we use point E as the center and draw an arc to intersect the first one as shown below. A D E C B
3. We then use a straightedge to draw a ray from the vertex B through the point of intersection F of the two arcs as shown below. This ray is referred to as the angle bisector. A F D E C B In this case ∠ABC has been bisected into ∠ABF and ∠CBF such that ∠𝐁𝐂𝐆 ≅ ∠𝐃𝐂𝐆 .
Example Consider ∠PQR shown below. Given that ∠PQR = 90° , bisect it into two congruent angles using a straightedge and a compass only. P Q R
Solution To divide ∠PQR into two congruent angles. 1. We draw an arc to cross the two arms of ∠PQR using the vertex Q as the center. P Q R
2. We use the points of intersection of the arc and the arms of ∠PQR to draw two intersecting arcs using the same compass width. P Q R
3. We then use a straightedge to draw a ray from the vertex Q through the point of intersection of the two arcs. We then label the point of intersection S. S P Q R ∠PQR has been bisected into ∠PQS and ∠RQS such that ∠𝐐𝐑𝐓 ≅ ∠𝐒𝐑𝐓 .
HOMEWORK Consider AB shown below. Copy AB to another location using a straightedge and a compass only then label it AB . A B
A NSWERS TO HOMEWORK Accuracy should be ascertained. The length of AB should be equal to the length of CD . A B C D
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