G7-20 Constructing Angle Bisectors

G7-20 Constructing Angle Bisectors

Pages 125¨C126

Goals

Curriculum

Expectations

Students will construct angle bisectors using a compass and a

straightedge, explain why the construction works, and use the same

method to construct lines that intersect at 60¡ã and 30¡ã.

Ontario: 7m2, 7m48

WNCP: 7SS3, [CN, R, V]

PRIOR KNOWLEDGE REQUIRED

Vocabulary

Can draw line segments with a ruler

Can measure angles with a protractor

Can draw arcs of given radius with a compass

Can name angles and line segments

Is familiar with notation for equal sides, equal angles, and

congruent triangles

Can identify congruent triangles

Is familiar with SSS congruence rule

Knows what an angle bisector is

Can construct triangles with given sides using a compass and

a straightedge

straightedge

compass

angle bisector

triangle

corresponding angles

corresponding sides

congruence rule

SSS (side-side-side)

congruent triangles

equilateral, isosceles

(triangle)

Materials

compasses

straightedges

protractors

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Remind students that a straightedge is any tool with a straight side. When

they are asked to use a straightedge, they can use a ruler, but they cannot

use the markings on the ruler. Instead, they can copy line segments with

a compass.

Constructing angle bisectors. Model the construction of an angle

bisector using a compass and a straightedge. Have students practise the

construction using Questions 1 through 4 on Workbook pages 125 and 126.

Then ask students to draw a large obtuse scalene triangle and to construct

angle bisectors to all three angles of the triangle. Assessment tip: If the

construction is performed correctly, the bisectors will intersect at the same

point. When you have checked students¡¯ work, point out to the students

that the bisectors should intersect at the same point, and they can now use

it as a self-checking mechanism.

P

T

Q

Geometry 7-20

U

S

R

Why does the construction work? Do Question 6 on Workbook page 126

together as a class. Have students articulate how the fact that ?QTU and

?QSU are congruent helps them to explain why the construction works.

(Since the triangles are congruent, the corresponding angles ¡ÏTQU and

¡ÏSQU are equal too. This means QU splits ¡ÏTQS into two equal angles,

so QU is the angle bisector of ¡ÏTQS.)

P-25

Review constructing triangles with a compass and a straightedge.

Review the method briefly, then discuss with students how the diagram for

an equilateral triangle is different from the diagrams for constructing other

triangles with a compass and a straightedge: since all the sides are equal,

the circles have equal radii, and they also pass through the centres of each

other. Model the construction of an equilateral triangle on the board or invite

volunteers to do so. Have students individually construct an equilateral

triangle with a compass and a straightedge.

Process assessment

7m7, [C]

Review with students the fact that all angles of equilateral triangles are

equal. ASK: What is the measure of the angles in an equilateral triangle?

(60¡ã) How do you know? (The sum of the angles in a triangle is 180¡ã, and

the angles are equal, so they are 180¡ã ¡Â 3 = 60¡ã.)

Constructing lines that intersect at an angle of 60¡ã and 30¡ã. Ask

students to use the construction of an equilateral triangle to draw two lines

that intersect at an angle of 60¡ã. How could you use this construction to

construct a pair of lines that intersect at an angle of 30¡ã? (bisect a 60¡ã angle)

Have students perform the construction and check its accuracy using

a protractor.

ACTIVITY

a) Draw an acute angle A on a blank sheet of paper (not in your

notebook as you will need to fold paper).

b) Draw an angle bisector using a set square.

Step 1: Place the set square

as shown. Make sure the

vertex of the set square is at

the vertex of your angle. Draw

a line from B as shown.

Step 2: Draw a line from C as shown.

Make sure that the same side of the set

square is placed along the arm of your

angle and that the vertex of the set square

is at the vertex of your angle. Mark D.

C

D

B

A

B

Step 3: Draw a line through A and D as shown

C

D

A

B

c) Check your answer by folding the paper along AD. Does line AB

meet line AC?

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Teacher¡¯s Guide for Workbook 7.2

COPYRIGHT ? 2010 JUMP MATH: NOT TO BE COPIED

A

d) Draw an obtuse angle and repeat steps b) and c).

e) Right triangles are special: SSA is a congruence rule for them if the

angle used is the right angle. Find two right triangles in the picture

in Step 3 and formulate the SSA congruence rule for them. (If in

triangles ABD and ACD AB = AC and BD is the same in both triangles,

and angles ¡ÏB = ¡ÏD = 90¡ã, then these triangles are congruent.)

f) How is the construction in b) the same as the construction of an

angle bisector using a compass and a straightedge? Why does this

construction work? (A set square is used to create points B and C

that are at the same distance from A. In both constructions you

are producing two congruent triangles (?QTU and ?QSU in

the construction in the workbook and ?ABD and ?ACD in this

construction) with a common side that is the bisector of the angle.

In this construction the triangles are congruent because they are

both right triangles, and AB = AC and AD is the common side, so

the triangles are congruent by SSA. (NOTE: SSA is a congruence

rule only in the case when the equal angles are the largest angles

in the triangle. For example, if both triangles are right triangles, the

rule works.)

Extensions

1. If you bisect an angle, and then bisect each half again, you get 4 equal

angles. Trisecting an angle (splitting it into 3 equal parts) is impossible

using a compass and a straightedge. However, trisecting a line segment

is possible¡ªsee the Extension in G7-36.

E

2. a) Draw a line AC and mark a point B on it. Draw a line segment BD

intersecting AC.

D

F

A

B

C

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c) Find ¡ÏEBF without using a protractor and then verify your answer

using a protractor.

3. a) Draw an acute angle ABD. Extend the arm BA beyond B and mark

a point C on the extension. You should get two angles, ¡ÏDBC and

¡ÏABD, that add to 180¡ã. (See sample in margin.)

A

D

b) Using a protractor, draw lines EB and FB so that line EB bisects

¡ÏABD and line FB bisects ¡ÏDBC. (See sample in margin.)

B

C

b) Construct the angle bisector EB of ¡ÏABD and the angle bisector BF

of ¡ÏDBC by using a compass and a straightedge.

c) What is the measure of ¡ÏEBF? Predict, then verify with a protractor.

d) Repeat a), b), and c) for a right angle and an obtuse angle.

e) Does the measure of ¡ÏEBF depend on the size of ¡ÏABD?

Geometry 7-20

P-27

f) To explain your answer in e), let x be the measure of ¡ÏABE and y be

the measure of ¡ÏCBF. Which other angles also have the measures

x and y? Mark them on the diagram.

g) x + x + y + y =

¡ã

2x + 2y =

¡ã

2(x + y) =

¡ã

x+y=

¡ã

h) What is the measure of ¡ÏEBF in terms of x and y?

How many degrees is that?

Find the measure of ¡ÏEBF by using a protractor.

COPYRIGHT ? 2010 JUMP MATH: NOT TO BE COPIED

i)

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Teacher¡¯s Guide for Workbook 7.2

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