Unit 10-Aim 5: What are constructions?
Motivation:
1.
A construction is the drawing of various shapes using only a compass and straightedge or ruler. No measurement of lengths or angles is allowed. Very importantly, you are not allowed to measure angles with a protractor or measure lengths with a ruler.
Challenge: Aug 2009: 32 Using a compass and straightedge, construct the angle bisector of shown below
Step I: Construct an arc from the vertex Step III: Construct an arc from the bottom intersection
Put your compass at point A and draw a large arc that intersects both sides of

A .
Step II: Construct an arc from the top intersection
Keeping the same compass setting, place the compass at point C and draw an arc that intersects the arc drawn in Step II.
Step IV: Construct a line through the vertex and the intersections of the arcs
With the compass at point B , draw an arc in the interior of the angle
Draw

AD http://www.mathopenref.com/constbisectangle.html
http://www.mathsisfun.com/geometry/construct-anglebisect.html

2. Course II Jan 87: 35 Construct the angle bisector of angle C in parallelogram ABCD

3. June 2009: 25 Which illustration shows the correct construction of an angle bisector?
1) 2)
3) 4)
Step I: Construct an arc from the vertex Step III: Construct an arc from the bottom intersection
Put your compass at point A and draw a large arc that intersects both sides of

A .
Step II: Construct an arc from the top intersection
Keeping the same compass setting, place the compass at point C and draw an arc that intersects the arc drawn in Step II.
Step IV: Construct a line through the vertex and the intersections of the arcs
With the compass at point B , draw an arc in the interior of the angle
Draw

AD
ANSWER: 3)

4. Aug 2010: 28. Which geometric principle is used in the construction shown below?
1) The intersection of the angle bisectors of a triangle is the center of the inscribed circle.
2) The intersection of the angle bisectors of a triangle is the center of the circumscribed circle.
3) The intersection of the perpendicular bisectors of the sides of a triangle is the center of the inscribed circle.
4) The intersection of the perpendicular bisectors of the sides of a triangle is the center of the circumscribed circle.
Step I: Name the Construction ( Angle Bisector ) Step II: Definition
The largest circle contained in a triangle is inscribed in the triangle. (remember “in” scribed)
ANSWER: 1) The intersection of the angle bisectors of a triangle is the center of the inscribed circle.
Eliminate choices 3) and 4) because they include perpendicular bisectors

5. Jan 2010: 4. Based on the construction below, which statement must be true?
1) m

ABD

1
2 m

CBD 2) m

ABD
 m

CBD
3) m

ABD
 m

ABC 4) m

CBD

1
2 m

ABD
Step I: Construction angle bisector
Step II: Definition
An angle bisector is a ray that divides an angle into two congruent angles.
ANSWER: 2) m

ABD
 m

CBD

6. Why we learn about constructions
The ancient Greek mathematician Euclid is the acknowledged inventor of geometry. He did this over 2000 years ago, and his book "Elements" is still regarded as the ultimate geometry reference. In that work, he uses these construction techniques extensively, and so they have become a part of the geometry field of study. They also provide a greater insight into geometric concepts and give us tools to draw things when direct measurement is not appropriate.
Why did Euclid do it this way?
Why didn't Euclid just measure angles with a compass and calculate angle measures? For example, one of the basic constructions is bisecting an angle (dividing it into two equal angles). Why not just measure it with a protractor and divide by two?
The answer is surprising. The Greeks could not do arithmetic. They had only whole numbers, no zero, and no negative numbers. This meant they could not for example divide 5 by 2 and get 2.5, because 2.5 is not a whole number - the only kind they had. Also, their numbers did not use a positional system like ours, with units, tens , hundreds etc, but more like the Roman numerals. In short, they could perform very little useful arithmetic.
So, faced with the problem of finding the position of an angle bisector, they could not do the obvious - measure it and divide by two. They had to have other ways, and this lead to the constructions using compass and straightedge or ruler. It is also why the straightedge has no markings. It is definitely not a graduated ruler, but simply a pencil guide for making straight lines. Euclid and the Greeks solved problems graphically, by drawing shapes, as a substitute for using arithmetic.
Why didn't Euclid just measure and use division to bisect angles?
Answer: The Greeks could not do arithmetic. They had only whole numbers, no zero, and no negative numbers. This meant they could not for example divide 5 by 2 and get 2.5, because 2.5 is not a whole number - the only kind they had. As a result, they could not divide an odd angle measure by 2 to find the measure to draw an angle bisector.

7. Fill in the proof of the angle construction:
This construction works by effectively building two congruent triangles. The image below is the final drawing above with the line segments AB and BC added and points A,B,C labeled.
Statement
1. QA

QB
2. AC

BC
3. QC

QC
4.

QAC
 
QBC
5.

AQC
 
BQC
6. The line QC bisects the angle PQR
Reason
1. They were both drawn with the same compass width.
2. They were both drawn with the same compass width
3. Reflexive Property
4. Three sides congruent (SSS).
5. CPCTC Corresponding parts of congruent triangles are congruent
6.

AQC and

BQC are adjacent and congruent
8. Course II Jan 96: 35 In the accompanying diagram, the bisector of an angle has been constructed.
In proving this construction, which reason is used for the congruence involved?
(1) ASA (2) SSS
PI GG17 GG18 Lesson #11
(3) AAS (4) SAS
AIM: How do we use a compass and straight edge to bisect an angle or a line segment?
Students will be able to:
1. construct a bisector of a given angle
2. justify the construction for bisecting a given angle
3. construct a perpendicular bisector of a given line segment using a straight edge and compass
4. justify the construction for the perpendicular bisector of an angle
5. apply the above to constructing an angle bisector or a median of a triangle
Writing Exercise:
1. Explain why joining the vertex of an angle to the point found in the angle bisector construction guarantees that the angle is bisected.
2. In constructing the perpendicular bisector of a line segment, why must the two arcs have the same radius?

Student Name ____________________________
Unit 10-Aim 5: What are constructions?
1.
A construction is the drawing of various shapes using only a compass and straightedge or ruler. No measurement of lengths or angles is allowed. Very importantly, you are not allowed to measure angles with a protractor or measure lengths with a ruler.
Challenge: Using a compass and straightedge, construct the angle bisector of shown below

Step I: Construct an arc from the vertex Step III: Construct an arc from the bottom intersection
Put your compass at point A and draw a large arc that intersects both sides of

A .
Step II: Construct an arc from the top intersection
Keeping the same compass setting, place the compass at point C and draw an arc that intersects the arc drawn in Step II.
Step IV: Construct a line through the vertex and the intersections of the arcs
With the compass at point B , draw an arc in the interior of the angle
Draw

AD http://www.mathopenref.com/constbisectangle.html
http://www.mathsisfun.com/geometry/construct-anglebisect.html

2. Construct the angle bisector of angle C in parallelogram ABCD
3. Which illustration shows the correct construction of an angle bisector?
1) 2)
3) 4)

4. Which geometric principle is used in the construction shown below?
1) The intersection of the angle bisectors of a triangle is the center of the inscribed circle.
2) The intersection of the angle bisectors of a triangle is the center of the circumscribed circle.
3) The intersection of the perpendicular bisectors of the sides of a triangle is the center of the inscribed circle.
4) The intersection of the perpendicular bisectors of the sides of a triangle is the center of the circumscribed circle.
5. Based on the construction below, which statement must be true?
1) m

ABD

1
2 m

CBD
3) m

ABD
 m

ABC
2) m

ABD
 m

CBD
4) m

CBD

1
2 m

ABD

6. Why we learn about constructions
The ancient Greek mathematician Euclid is the acknowledged inventor of geometry. He did this over 2000 years ago, and his book "Elements" is still regarded as the ultimate geometry reference. In that work, he uses these construction techniques extensively, and so they have become a part of the geometry field of study. They also provide a greater insight into geometric concepts and give us tools to draw things when direct measurement is not appropriate.
Why did Euclid do it this way?
Why didn't Euclid just measure angles with a compass and calculate angle measures? For example, one of the basic constructions is bisecting an angle (dividing it into two equal angles). Why not just measure it with a protractor and divide by two?
The answer is surprising. The Greeks could not do arithmetic. They had only whole numbers, no zero, and no negative numbers. This meant they could not for example divide 5 by 2 and get 2.5, because 2.5 is not a whole number - the only kind they had. Also, their numbers did not use a positional system like ours, with units, tens , hundreds etc, but more like the Roman numerals. In short, they could perform very little useful arithmetic.
So, faced with the problem of finding the position of an angle bisector, they could not do the obvious - measure it and divide by two. They had to have other ways, and this lead to the constructions using compass and straightedge or ruler. It is also why the straightedge has no markings. It is definitely not a graduated ruler, but simply a pencil guide for making straight lines. Euclid and the Greeks solved problems graphically, by drawing shapes, as a substitute for using arithmetic.
Why didn't Euclid just measure and use division to bisect angles?
Answer: The Greeks could not do arithmetic. They had only whole numbers, no zero, and no negative numbers. This meant they could not for example divide 5 by 2 and get 2.5, because 2.5 is not a whole number - the only kind they had. As a result, they could not divide an odd angle measure by 2 to find the measure to draw an angle bisector.

7. Fill in the proof of the angle construction:
This construction works by effectively building two congruent triangles. The image below is the final drawing above with the line segments AB and BC added and points A,B,C labeled.
Statement
1. QA

QB
2. AC

BC
3. QC

QC
4.

QAC
 
QBC
5.

AQC
 
BQC
6. The line QC bisects the angle PQR
Reason
1. They were both drawn with the same compass width.
2.
3.
4.
5.
6.

AQC and

BQC are _______________and
___________________.
8. In the accompanying diagram, the bisector of an angle has been constructed.
In proving this construction, which reason is used for the congruence involved?
(1) ASA (2) SSS (3) AAS (4) SAS

Extra:
HW5 Jan 2011: 33. On the diagram below, use a compass and straightedge to construct the bisector of

ABC .
[Leave construction marks.]
Step I: Construct an arc from the vertex Step III: Construct an arc from the bottom intersection
Put your compass at point B and draw a large arc that intersects both sides of

B .
Step II: Construct an arc from the top intersection Keeping the same compass setting, place the compass at point E and draw an arc that intersects the arc drawn in Step II.
Step IV: Construct a line through the vertex and the intersections of the arcs
With the compass at point D , draw an arc in the interior of the angle
Draw

AF

HW 6 Course II Jan 03: 35 Which diagram shows a correct mathematical construction using only a compass and a straight edge to bisect an angle?
Course II Jan 99: 35 On the diagram below, use a compass and straightedge to construct the bisector of the obtuse angle MAE .
[Leave construction marks.]
Jan 2008: 32. Using a compass and straightedge, construct the bisector of the angle shown below.
[ Leave all construction marks .]

Step I: Construct an arc from the vertex Step III: Construct an arc from the bottom intersection
Put your compass at point A and draw a large arc that intersects both sides of

A .
Step II: Construct an arc from the top intersection
Keeping the same compass setting, place the compass at point C and draw an arc that intersects the arc drawn in Step II.
Step IV: Construct a line through the vertex and the intersections of the arcs
With the compass at point B , draw an arc in the interior of the angle
Draw

AD
Course II Jan 86: 35 On the diagram below, use a compass and straightedge to construct the bisector of

ABC .
[Leave construction marks.]

Course II June 2000: 35 On the diagram below, use a compass and straightedge to construct the bisector of

ABC . [Leave construction marks.]
Course II Jan 85: 35 Construct the angle bisector of angle B in triangle ABC .

Course II June 99: 35 Construct the angle bisector of angle C in triangle ABC .
HW8 Course II June 2002: 35 Construct the locus of points equidistant from the sides of

ABC.
Given

ABC , construct and label

BD such that

ABD
 
CBD .
Course II June 1994: 35 Construct the ray that bisects

B .

HW5 Course II June 1991: 35 The diagram below shows the construction of the bisector of

ABC .
Which reason for triangle congruence is used in proving that BF bisects

ABC .
(1) ASA (2) SAS (3) SSS (4) SAA

Aug 2009: 2. The diagram below shows the construction of the bisector of

ABC .
Which statement is not true?
1)
3) m

EBF m

EBF


1 m

ABC
2 m

ABC
Step I: Definition two congruent angles.
Step II: Substitution
2)
4) m

DBF m

DBF
An angle bisector is a ray that divides an angle into


1 m

ABC
2 m

EBF
ANSWER: 3) m

EBF check : 35


 m

ABC
70
 is not true
Example:
Let

ABC

70

Then, m

EBF

1
2 m

ABC

1
2
 

35

What is the difference between sketching and constructing?
Course II June 00: 35 Construct the angle bisector of

ABC . Leave all construction lines.

Aug 2011: 6. A straightedge and compass were used to create the construction below. Arc EF was drawn from point B , and arcs with equal radii were drawn from E and F .
Which statement is false ?
(1) m
Ð
ABD
= m
Ð
DBC (2)
1
2
( m
Ð
ABC
) = m
Ð
ABD
(3) 2
( m
Ð
DBC
) = m
Ð
ABC (4) 2
( m
Ð
ABC
) = m
Ð
CBD
Jan 2012: 33. On the diagram below, use a compass and a straightedge to construct the bisector of

XYZ . {Leave all construction marks.]