Name _________________________________________
Geometry – Pd __
Date ____________
Constructions
Constructions
1. Congruent Line Segments
Given: (Line segment) AB
Task: To construct a line segment congruent to (line segment) AB
1. If a reference line does not already exist, draw a reference line with your
straightedge. Place a starting point on the reference line.
2. Place the point of the compass on point A.
3. Stretch the compass so that the pencil is exactly on B.
4. Without changing the span of the compass, place the compass point on the starting point on the
reference line and swing the pencil so that it crosses the reference line. Label your copy.
Your copy and (line segment) AB are congruent. Congruent means equal in length.
Example 1)
Example 2)
2. Congruent Angles
Given:
Regents
Common Core
BAC
Task: To construct an angle congruent to
BAC
1. If a reference line does not already exist, draw
a reference line with your straightedge. Place
a starting point on the reference line.
2. Place the point of the compass on the vertex
of BAC (point A).
3. Stretch the compass to any length so long as it stays ON the angle.
4. Swing an arc with the pencil that crosses both sides of BAC .
5. Without changing the span of the compass, place the compass point on the starting point of the
reference line and swing an arc that will intersect the reference line and go above the reference line.
6. Go back to BAC and measure the width (span) of the arc from where it crosses one side of the
angle to where it crosses the other side of the angle.
7. With this width, place the compass point on the reference line where your new arc crosses the
reference line and mark off this width on your new arc.
8. Connect this new intersection point to the starting point on the reference line.
Example 1)
Example 2)
3. Construct an Angle Bisector
Given:
Regents
Common Core
BAC
Task: Bisect
BAC
1. Place the point of the compass on the vertex
of BAC (point A).
2. Stretch the compass to any length so long as
it stays ON the angle.
3. Swing an arc so the pencil crosses both sides of BAC . This will create two intersection points
with the sides (rays) of the angle.
4. Place the compass point on one of these new intersection points on the sides of BAC . If needed,
stretch your compass to a sufficient length to place your pencil more than halfway into the interior of
the angle.
Stay between the sides (rays) of the angle. Place an arc in this interior - you do not need to cross the
sides of the angle.
5. Without changing the width of the compass, place the point of the compass on the other intersection
point on the side of the angle and make the same arc. Your two small arcs in the interior of the angle
should be crossing.
6. Connect the point where the two small arcs cross to the vertex A of the angle.
Example 1)
Example 2)
Name _________________________________________
Geometry – Pd __
Date ____________
Constructions
Constructions HW
̅̅̅̅, construct a congruent segment.
1) Using 𝐴𝐵
[Leave all construction marks]
2) Construct an angle congruent to < 𝐴𝐵𝐶
[Leave all construction marks]
3) Create an angle bisector for < 𝑀𝐽𝐻.
[Leave all construction marks]
4) Using a compass, construct the bisector
of < ABC. [Leave all construction marks]
_____5) When given a square, the construction of an angle bisector at any vertex will create the
diagonal of the square.
(1) True
(2) False
_____ 6) Which illustration shows the correct construction of an angle bisector?
(1)
(2)
(3)
(4)
_____ 7) Based on the construction to the right, which statement must be true?
1
(1) 𝑚 < 𝐴𝐵𝐷 = 𝑚 < 𝐶𝐵𝐷
2
(2) 𝑚 < 𝐴𝐵𝐷 = 𝑚 < 𝐶𝐵𝐷
(3) 𝑚 < 𝐴𝐵𝐷 = 𝑚 < 𝐴𝐵𝐶
1
(4) 𝑚 < 𝐶𝐵𝐷 = 2 𝑚 < 𝐴𝐵𝐷
_____ 8) As shown in the diagram, a compass is used to find points D and E,
equidistant from point A. Next the compass is used to find point F,
equidistant from point D and E. Finally, a straight edge is used to
draw AF. Which statement must be true?
̅̅̅̅ bisects side 𝐵𝐶
̅̅̅̅
(1) 𝐴𝐹
̅̅̅̅
(2) 𝐴𝐹 bisects < 𝐵𝐴𝐶
̅̅̅̅ ⊥ 𝐵𝐶
̅̅̅̅
(3) 𝐴𝐹
(4) ∆ ABG ~∆ ACG
_____ 9) The diagram shows the construction of the bisector of < 𝐴𝐵𝐶.
Which statement is not true?
1
(1) 𝑚 < 𝐸𝐵𝐹 = 𝑚 < 𝐴𝐵𝐶
2
1
(2) 𝑚 < 𝐷𝐵𝐹 = 𝑚 < 𝐴𝐵𝐶
2
(3) 𝑚 < 𝐸𝐵𝐹 = 𝑚 < 𝐴𝐵𝐶
(4) 𝑚 < 𝐷𝐵𝐹 = 𝑚 < 𝐸𝐵𝐹
_____ 10) A straight edge and compass were used to create the following construction.
Which statement is false?
(1) 𝑚 < 𝐴𝐵𝐷 = 𝑚 < 𝐷𝐵𝐶
1
(2) (𝑚 < 𝐴𝐵𝐶) = 𝑚 < 𝐴𝐵𝐷
2
(3) 2(𝑚 < 𝐷𝐵𝐶) = 𝑚 < 𝐴𝐵𝐶
(4) 2(𝑚 < 𝐴𝐵𝐶) = 𝑚 < 𝐶𝐵𝐷
Review Section!!
_____ 11) In the diagram, transversal TU intersects PQ and RS at V and W respectively.
If m<TVQ = 5x – 22 and m<VWS = 3x + 10, for which value of x is PQ//RS?
(1) 6
(2) 16
(3) 24
(4) 28
_____ 12) When the system of equations 𝑦 + 2𝑥 = 𝑥 2 and 𝑦 = 𝑥 is graphed on a
set of axes, what is the total number of points of intersection?
(1) 0
(2) 1
(3) 2
(4) 3
_____ 13) The equation of a line is 3𝑦 + 2𝑥 = 12. What is the slope of the line
perpendicular to the given line?
3
2
(1)
(2)
2
(3)
3
−2
3
2
(4) − 3
Constructions HW Answers
1)
2)
3)
4)
5) (1)
6) (3)
7) (2)
8) (2)
9) (3)
10) (4)
11) (2)
12) (3)
13) (1)
Review Section!!
11) In the diagram, transversal TU intersects PQ and RS at V and W respectively.
If m<TVQ = 5x – 22 and m<VWS = 3x + 10, for which value of x is PQ//RS?
12) When the system of equations 𝑦 + 2𝑥 = 𝑥 2 and 𝑦 = 𝑥 is graphed on a
set of axes, what is the total number of points of intersection?
13) The equation of a line is 3𝑦 + 2𝑥 = 12. What is the slope of the line
perpendicular to the given line?
Name _________________________________________
Geometry – Pd __
Date ____________
Constructions
Constructions Day 2
1. Construct an Equilateral Triangle
Regents
Common Core
Given: (Line segment) AB
Task: To construct an equilateral triangle
1. Place the point of the compass on point A and
measure (place the pencil) to point B. Swing a
large arc up above the given line.
2. Place the point of the compass on point B and
measure (place the pencil) to point A. Swing a large arc up above the given line.
3. Create the missing two sides of the equilateral triangle by connecting point A to the point of
intersection, and then point B to the point of intersection.
Example 1)
Example 2)
Common Core Application:
1) Margie has three cats. She has heard that cats in a room position themselves at equal distances from
one another and wants to test that theory. Margie notices that Simon, her tabby cat, is in the center of
her bed (at S), while JoJo, her Siamese, is laying on her desk chair (at J). If the theory is true, where will
she find Mack, her calico cat? Use the scale drawing of Margie’s room shown below, together with
(only) a compass and straightedge. Place an M where Mack will be if the theory is true.
2. Construct a 𝟑𝟎° Angle
Regents
Common Core
Given: (Line segment) AB
Task: To construct a 30° Angle
1. First, construct an equilateral triangle
based on segment ̅̅̅̅
𝐴𝐵.
a) Place the point of the compass on point A and measure
to point B. Swing a large arc up above the given line.
b) Place the point of the compass on point B and measure to point A. Swing a large arc up
above the given line.
c) Create the missing two sides of the equilateral triangle by connecting both point A and point B
to the point of intersection
2. Second, bisect any angle of the equilateral triangle.
a) Place the point of the compass on the vertex
b) Stretch the compass to any length so long as it stays ON the angle
c) Swing an arc so the pencil crosses both sides of the angle
d) Place the compass point on one of these new intersection points on the sides of the angle and
swing an arc (using the pencil) on the interior of the angle.
e) Without changing the width of the compass, complete the same for the other point of intersection.
f) Connect the point where the two small arcs cross to the vertex of the angle.
3. You have now created the construction of a 30° angle.
Name _________________________________________
Geometry – Pd __
Date ____________
Constructions
Constructions Day 2 HW
̅̅̅̅, construct an equilateral triangle.
1) Using 𝐴𝐵
[Leave all construction marks]
2) Construct an angle bisector to < 𝑀𝐽𝐻
[Leave all construction marks]
3) ∆ 𝐴𝐵𝐶 is shown below. Is it an equilateral
triangle? Justify your response.
4) Using equilateral ∆ 𝐴𝐵𝐶 to construct a 30°
angle. [Leave all construction marks]
_____ 5) One method of constructing an equilateral triangle is to simply construct a triangle using the
same segment for each side.
(1) True
(2) False
_____ 6) Which diagram shows the construction of a 30° angle?
_____ 7) The diagram shows the construction of an equilateral triangle.
Which statement justifies this construction?
(1) < 𝐴 + < 𝐵 + < 𝐶 = 180
(2) < 𝐴 = < 𝐵 = < 𝐶
(3) 𝐴𝐵 = 𝐵𝐶 = 𝐴𝐶
(4) 𝐴𝐵 + 𝐵𝐶 > 𝐴𝐶
_____ 8) The diagram below shows the construction of the bisector of ∡ 𝐴𝐵𝐶.
Which statement is not true?
1
1) 𝑚∡ 𝐸𝐵𝐹 = 2 𝑚∡ 𝐴𝐵𝐶
1
2) 𝑚∡ 𝐷𝐵𝐹 = 2 𝑚∡ 𝐴𝐵𝐶
3) 𝑚∡ 𝐸𝐵𝐹 = 𝑚∡ 𝐴𝐵𝐶
4) 𝑚∡ 𝐷𝐵𝐹 = 𝑚∡ 𝐸𝐵𝐹
Review Section!!
_____ 9) What is the equation of a line passing through the point (4, −1) and parallel to the line whose
equation is 2𝑦 − 𝑥 = 8?
1
1
(1) 𝑦 = 2 𝑥 − 3
(2) 𝑦 = 2 𝑥 − 1
(3) 𝑦 = −2𝑥 + 7
(4) 𝑦 = −2𝑥 + 2
_____ 10) In the diagram of ∆𝑃𝑄𝑅 shown below, PR is extended to S,
m<P = 110, m<Q = 4x, and m<QRS = 𝑥 2 + 5𝑥. What is the m<Q?
(1) 44
(2) 40
(3) 11
(4) 10
_____ 11) The solution of the system of equations 𝑦 = 𝑥 2 − 2 and 𝑦 = 𝑥 is:
(1) (1,1) and (−2, −2)
(2) (2,2) and (−1, −1)
(3) (1,1) and (2,2)
(4) (−2, −2) and (−1, −1)
_____ 12) Lines p and q are intersected by line r, as shown. If m<1 = 7x – 36 and
m<2 = 5x + 12, for which value of x would p//q?
(1) 17
(2) 24
(3) 83
(4) 97
Constructions Day 2 HW
1)
2)
3)
4)
5) (1)
6) (1)
7) (3)
8) (3)
9) (1)
10) (2)
11) (2)
12) (1)
9) What is the equation of a line passing through the point (4, −1) and parallel to the line whose
equation is 2𝑦 − 𝑥 = 8?
10) In the diagram of ∆𝑃𝑄𝑅 shown below, PR is extended to S,
m<P = 110, m<Q = 4x, and m<QRS = 𝑥 2 + 5𝑥. What is the m<Q?
11) The solution of the system of equations 𝑦 = 𝑥 2 − 2 and 𝑦 = 𝑥 is:
12) Lines p and q are intersected by line r, as shown. If m<1 = 7x – 36 and
m<2 = 5x + 12, for which value of x would p//q?
Name _________________________________________
Geometry – Pd __
Date ____________
Constructions
Constructions Day 3
Recall: An ______________________ is a line segment with one endpoint on any vertex of a
triangle that extends to the opposite side of the triangle and bisects the angle. Since there are
three vertices in every triangle, there are ___________ angle bisectors of a triangle. The point
of concurrency of the angle bisectors of a triangle is known as the ______________ of a
triangle. To construct the incenter of a given triangle construct the angle bisector on ________
_______________________ vertices. The incenter will always be located ______________ a
given triangle. The point of concurrency (the incenter) is the center of the circle that is
inscribed within a given triangle.
The Incenter
Regents
1) Using a compass and a straightedge, construct the incenter of ∆𝐴𝐵𝐶.
Common Core
The Incenter:
- The incenter is formed by connecting the three angle bisectors
- The three angle bisectors of a triangle are concurrent at a point
equidistant from the sides of a triangle.
Directions: Using the above information, complete the following questions. Don’t forget justifications.
1) The incenter of ∆𝑇𝑈𝑆 is located at point P. If CP = 4x + 9 and PB = 6x – 11, find the value of x and
the length of CP and PD. Justify all calculations.
2) Point P is the incenter of ∆𝐹𝐺𝐻. If m<TFP = 3x + 15, and m<UFP = 5x – 13, find the value of x.
Justify all calculations.
3) The incenter of ∆𝐶𝐷𝐸 is point P. If m<SDP = 7x + 5 and m<UDP = 9x – 5, find the value of x
and m<SDP. Justify all calculations.
4) P is the incenter of ∆𝑋𝑌𝑍. If m<SZP = 7x + 7, and m<SZT = 16x + 4, find the value of x and m<SZT.
Justify all calculations.
Name _________________________________________
Geometry – Pd __
Date ____________
Constructions
Constructions Day 3 HW
1) Construct the incenter of ∆𝑇𝑂𝑀.
2) Construct an equilateral triangle to DE.
_____ 3) P is the incenter of ∆𝑋𝑌𝑍. If m<RYP = 2x + 20, and m<TYP = x + 40,
what is the m<RYT?
(1) 20
(2) 40
(3) 60
(4) 120
_____ 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) The incenter of ∆𝐴𝐵𝐶 is located at point G. If EG = 3x + 14 and
DG = 5x – 8, what is the length of GF?
(1) 5
(2) 11
(3) 22
(4) 47
_____ 6) A straight edge and compass were used to create the following construction.
Which statement is false?
(1) 𝑚 < 𝐴𝐵𝐷 = 𝑚 < 𝐷𝐵𝐶
1
(2) 2 (𝑚 < 𝐴𝐵𝐶) = 𝑚 < 𝐴𝐵𝐷
(3) 2(𝑚 < 𝐷𝐵𝐶) = 𝑚 < 𝐴𝐵𝐶
(4) 2(𝑚 < 𝐴𝐵𝐶) = 𝑚 < 𝐶𝐵𝐷
_____ 7) The diagram shows the construction of an equilateral triangle.
Which statement justifies this construction?
(1) < 𝐴 + < 𝐵 + < 𝐶 = 180
(2) < 𝐴 = < 𝐵 = < 𝐶
(3) 𝐴𝐵 = 𝐵𝐶 = 𝐴𝐶
(4) 𝐴𝐵 + 𝐵𝐶 > 𝐴𝐶
Review Section!!
_____ 8) What is the slope of a line perpendicular to the line whose equation is 3𝑥 − 7𝑦 + 14 = 0?
3
7
(1) 7
(2) − 3
1
(4) − 3
(3) 3
_____ 9) Line m and point P are shown in the graph. Which equation represents
the line passing through P and parallel to line m?
(1) 𝑦 − 3 = 2(𝑥 + 2)
(2) 𝑦 + 2 = 2(𝑥 − 3)
1
(3) 𝑦 − 3 = − 2 (𝑥 + 2)
1
(4) 𝑦 + 2 = − 2 (𝑥 − 3)
_____ 10) In ∆𝐴𝐵𝐶, m<A = 3x + 1, m<B = 4x – 17, and m<C = 5x – 20. Which type of triangle is ∆𝐴𝐵𝐶?
(1) right
(2) scalene
(3) isosceles
(4) equilateral
_____ 11) Transversal EF intersects AB and CD as shown. Which statement
could always be used to prove AB // CD?
(1) < 2 ≅< 4
(2) < 3 and < 6 are supplementary
(3) < 7 ≅< 8
(4) < 1 and < 5 are supplementary
Constructions Day 3 HW
1)
2)
3) (4)
4) (1)
5) (4)
6) (4)
8) (2)
9) (2)
10) (3)
11) (2)
8) What is the slope of a line perpendicular to the line whose equation is 3𝑥 − 7𝑦 + 14 = 0?
9) Line m and point P are shown in the graph. Which equation represents
the line passing through P and parallel to line m?
10) In ∆𝐴𝐵𝐶, m<A = 3x + 1, m<B = 4x – 17, and m<C = 5x – 20. Which type of triangle is ∆𝐴𝐵𝐶?
11) Transversal EF intersects AB and CD as shown. Which statement
could always be used to prove AB // CD?
(1) < 2 ≅< 4
(2) < 3 and < 6 are supplementary
(3) < 7 ≅< 8
(4) < 1 and < 5 are supplementary
7) (3)