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6.1 Justifying Constructions

Essential Question: How can you be sure that the result of a construction is valid?

Resource
Locker

Explore 1 Using a Reflective Device to Construct

a Perpendicular Line
You have constructed a line perpendicular to a given line through a point not on the line using
a compass and straightedge. You can also use a reflective device to construct perpendicular lines.

A Step 1 Place the reflective device along line ℓ. Look through the device to locate the
image of point P on the opposite side of line ℓ. Draw the image of point P and
label it P '.
‹ ›
−
Step 2 Use a straightedge to draw PP '.

ℓ P ℓ P

‹ ›
−
Explain why PP ' is perpendicular to line ℓ.

B Place the reflective device so that it passes through point Q and is approximately
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perpendicular to line m. Adjust the angle of the device until the image of line m
coincides with line m. Draw a line along the reflective device and label it line n.
Explain why line n is perpendicular to line m.

Q
Q
m m

Module 6 273 Lesson 1

 Reflect

1. How can you check that the lines you drew are perpendicular to lines ℓ and m?

2. Use the reflective device to draw two points on line ℓ that are reflections of each other. Label the
points X and X'. What is true about PX and PX'? Why? Use a ruler to check your prediction.

3. Describe how to construct a perpendicular bisector of a line segment using paper folding.
Use a rigid motion to explain why the result is a perpendicular bisector.

Explore 2   Justifying the Copy of an Angle Construction

You have seen how to construct a copy of an angle, but how do you know that the copy must be
congruent to the original? Recall that to construct a copy of an angle A, you use these steps.
Step 1 Draw a ray with endpoint D.

Step 2 Draw an arc that intersects both rays of ∠A. Label the intersections B and C.

Step 3 Draw the same arc on the ray. Label the point of intersection E.

Step 4 Set the compass to the length BC.

​→ compass at E and draw a new arc. Label the intersection of the new arc F.
Step 5 Place the
​‾ . ∠D is congruent to ∠A.
Draw DF ​

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C F

A B E
D

A Sketch and name the two triangles that are created when you construct a copy of an angle.

A D

Module 6 274 Lesson 1

 B What segments do you know are congruent? Explain how you know.

 Are the triangles congruent? How do you know?

Reflect
_
4. Discussion Suppose you used a larger compass setting to create AB than another
student when copying the same angle. Will your copied angles be congruent?

5. Does the justification above for constructing a copy of an angle work for obtuse angles?

Explain 1 Proving the Angle Bisector and Perpendicular

Bisector Constructions
You have constructed angle bisectors and perpendicular bisectors. You now have the tools you
need to prove that these compass and straightedge constructions result in the intended figures.

Example 1 Prove two bisector constructions.

A You have used the following steps to construct an angle bisector.

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Step 1 Draw an arc intersecting the sides of the angle.

Label the intersections B and C. B
Step 2 Draw intersecting arcs from B and C. D
Label the intersection of the arcs as D.
_ A
C
Step 3 Use a straightedge to draw AD.

Prove that the construction results in the angle bisector.

The construction results in the triangles ABD and ACD. B

Because _the same
_ compass
_ setting
_ was used to_ create D
them, AB ≅ AC and BD ≅ CD. The segment AD is congruent
to itself by the Reflexive Property of Congruence. So, by the SSS A
C
Triangle Congruence Theorem, △ABD ≅ △ACD.

Corresponding parts of congruent figures are congruent, so ∠BAD ≅ ∠DAC.

→
‾ is the angle bisector of ∠A.
By the definition of angle bisector, AD
Module 6 275 Lesson 1
B You have used the following steps to construct a perpendicular bisector.
C
Step 1 Draw an arc centered at A.

Step 2 Draw an arc with the same diameter A B

centered at B. Label the intersections C and D.
_ D
​  .
Step 3 Draw CD​

Prove that the construction results in the perpendicular bisector.

The point C is equidistant from the endpoints of    , so by the               

_
      Theorem, it lies on the             of AB​ ​  . The point D is also equidistant
_
from the endpoints of   , so it also lies on the             of AB​ ​  . Two points

determine a line, so                    

Reflect

6. In Part B, what _ can you_ conclude about the measures of the angles made by the
intersection of AB​
​  and CD​
​  ?

7. Discussion
_ A classmate claims that_ in the construction shown in Part B,
​ AB​is the perpendicular bisector of CD​
​  . Is this true? Justify your answer.

Your Turn
_
8. The construction in Part B is also used to construct the midpoint R of MN​
​  .
How is the proof of this construction different from the proof of the M
perpendicular bisector construction in Part B?

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P R Q

9. How could you combine the constructions in Example 1 to construct a 45° angle?

Module 6 276 Lesson 1

 Elaborate
10. Describe how you can construct a line that is parallel to a given line using the
construction of a perpendicular to a line.

11. Use a straightedge and a piece of string to construct an equilateral triangle

that has AB as one of its sides. Then explain how you know your construction
works. (Hint: Consider an arc centered at A with radius AB and an arc centered
at B with radius AB.)

A B

12. Essential Question Check-In Is a construction something that must be proven? Explain.

Evaluate: Homework and Practice

• Online Homework
1. Julia is given a line ℓ and a point P not on line ℓ. She is asked to use a reflective device • Hints and Help
to construct a line through P that is perpendicular to line ℓ. She places the device as • Extra Practice
shown in the figure. What should she do next to draw the required line?
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P ℓ

2. Describe how to construct a copy of a segment. Explain how you know that the segments are congruent.

Module 6 277 Lesson 1

Complete the proof of the construction of a segment bisector.
_
3. Given: the construction of the segment bisector of AB​
​ 
_
​  ¯ bisects AB​
Prove: CD​ ​ 
C

A B

Statements Reasons
1. AC =    and AD =    . 1. Same compass setting used
_
2. C is on the perpendicular bisector of AB​
​  . 2.               
              
_
3. D is on the perpendicular bisector of AB​
​  . 3.               
              
_
4.    is the perpendicular bisector of AB​
​  . 4. Through any two points, there is exactly
one line.

5.          5. Definition of             

4. Complete the proof of the construction of a congruent angle.

C
Given: the construction of ∠CAB given ∠HFG
Prove: ∠CAB ≅ ∠HFG

A B n

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F G m

Statements Reasons
1. FG = FH =    = AC 1. same compass setting

2. GH = CB 2.                 

3. △FGH ≅ △ABC 3.                 

4. ∠CAB ≅ ∠HFG 4.                 

Module 6 278 Lesson 1

 To construct a line through the given point P,
parallel to line ℓ, you use the following steps.
T
Step 1 Choose _a point Q on line ℓ and
draw QP​
​  .

Step 2 Construct an angle congruent to V P S m

∠l at P.

Step 3 Construct the line through the U

given point, parallel to the line shown.

1
W Q R ℓ

Describe the relationship between the given

angles or segments. Justify your answer.
5. ∠TPS and ∠UQR 6. ∠SPU and ∠RQU

7. ∠VPU and ∠UQR 8. ∠TPS and ∠WQU

_ _ _ _
9. ​ QU​and PS​
​  10. QU​
​  and PT​
​ 

11. To construct a line through the given point P, parallel to line ℓ, you use ℓ n
the
following steps.
Step 1 Draw line m through P and intersecting line ℓ.
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Step 2 Construct an angle congruent to ∠l at P. 1 2 m

F P
Step 3 Construct the line through the given point, parallel to
the line shown.

How do you know that lines ℓ and n are parallel? Explain.

12. Construct an angle whose measure is __

​ 14 ​the measure of ∠Z. Justify the
construction.

Module 6 279 Lesson 1

In Exercises 13 and 14, use the diagram shown. The diagram shows
the result of constructing a copy of an angle adjacent to one of the
rays of the original angle. Assume the pattern continues.

13. If it takes 10 more copies of the angle for the last

angle to overlap the first ray (the horizontal ray),
what is the measure of each angle?

14. If it takes 8 more copies of the angle for the last

angle to overlap the first ray (the horizontal ray),
what is the measure of each angle?

15. Sonia draws a segment on a piece of paper. She wants to find three points that are
equidistant from the endpoints of the segment. Explain how she can use paper folding
to help her locate the three points.

In Exercises 16–18, a polygon is inscribed in a circle if all of the

polygon’s vertices lie on the circle.

16. Follow the given steps to construct a square inscribed in a circle.

Use your compass to draw a circle. Mark the center.
_

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Draw a diameter, AB​
​  , using a straightedge.
_
Construct the perpendicular bisector of AB​ ​  . Label the
points where the perpendicular bisector intersects the
circle as C and D.
_ _ _ _
Use the straightedge to draw AC​
​  , CB​
​  , BD​
​  , and DA​
​  .

17. Suppose you are given a piece of tracing paper with a circle on it and you do not have
a compass. How can you use paper folding to inscribe a square in the circle?

Module 6 280 Lesson 1

 18. Follow the given steps to construct a regular hexagon inscribed in a circle.
Tie a pencil to one end of the string.

Mark a point O on your paper. Place the string on point O

and hold it down with your finger. Pull the string taut and
draw a circle. Mark and label a point A.

Hold the point on the string that you placed on point O,

and move it to point A. Pull the string taut and draw an arc
that intersects the circle. Label the point as B.

Hold the point on the string that you placed on point A,

and move it to point B. Draw an arc to locate point C on
the circle. Repeat to locate points D, E, and F. Use your
straightedge to draw ABCDEF.
A B

H.O.T. Focus on Higher Order Thinking

19. Your teacher constructed the figure shown. It shows the construction of line PT
through point P and parallel to line AB.
a. Compass settings of length AB and AP were P T
used in the construction. Complete the statements:
With the compass set to length AP, an arc
was drawn with the compass point at point   .

 ith the compass set to length   

W , an arc was
drawn with the compass point at point   .
A B
The two arcs intersect at point   
.

b. Write two congruence statements involving

segments in the construction.

c. Write a_ proof P
_that the construction is true. That is, given the construction, T
​  ||AB​
prove PT​  ​  . (Hint: Draw segments to create two congruent triangles.)
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_
20. Use the segments shown. Construct and label_ _ XY​
a segment, ​  , A B
whose length is the average of the lengths of AB​
​  and CD​
​  . C D
Justify the method you used.

Module 6 281 Lesson 1

Lesson Performance Task
A plastic “mold” for copying a 30° angle is shown here.

a. If you drew a 30°-60°-90° triangle using the mold, how

would you know that your triangle and the mold were
congruent?

b. Explain how you know that any angle you would draw using
the lower right corner of the mold would measure 30°.

c. Explain the meaning of “tolerance” in the context of drawing

an angle using the mold.

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Module 6 282 Lesson 1