L10 – Perpendicular Lines and Triangles Name _____________________________ 10.1 – Construction Warmup Per _____ Date _____________________

Geometry Q2: L10 - Perpendicular Lines & Triangles Handouts Page 1

1. Use a straightedge and compass to construct the perpendicular bisector for the line below. You may want to first review how we did this in L7 – Constructions.

2. If P is a point on the perpendicular bisector that does not lie on the line segment AB, what can you say about the lengths AP and BP?

A B

L10 – Perpendicular Lines and Triangles Name _____________________________ 10.2 – Grandma’s Roof Per _____ Date _____________________

Geometry Q2: L10 - Perpendicular Lines & Triangles Handouts Page 2

Grandma’s Roof

It’s time to design the roof to Grandma’s House. Many roofs are shaped like isosceles triangles.

1. List below all the things you know about isosceles triangles.

2. Can any of these help us build an isosceles shaped roof?

3. Sketch several examples of an isosceles triangle. What are some of the ways you can test to see if it’s really isosceles?

4. Think about the perpendicular bisector, as depicted below. This can help us with building an isosceles triangle. Draw an isosceles triangle on the diagram below. How do you KNOW what you’ve drawn is isosceles?

L10 – Perpendicular Lines and Triangles Name _____________________________ 10.2 – Grandma’s Roof Per _____ Date _____________________

Geometry Q2: L10 - Perpendicular Lines & Triangles Handouts Page 3

5. This construction can help us in building our roof. Imagine that this is the front view of Grandma’s house. What are the four steps you would follow to sketch the roof in such a manner that you could be assured your roof would be isosceles?

6. Choose how high you want Grandma’s roof, and make a sketch of the house roof below.

7. Grandma sees your plans and decides she wants a roof that is half as tall in the middle. Draw this shorter roof on the same sketch above. In both examples, which sides of the roof triangle appear to be congruent?

L10 – Perpendicular Lines and Triangles Name _____________________________ 10.2 – Grandma’s Roof Per _____ Date _____________________

Geometry Q2: L10 - Perpendicular Lines & Triangles Handouts Page 4

M BA

P

C

8. Let’s take a break from our building project to do a quick proof. Given: Point P is located on the perpendicular bisector of line segment AB. Prove: !" ≅ !"  

You just proved the Perpendicular Bisector Theorem: Any point on the perpendicular bisector of a line segment will be __________________ from the two endpoints of that line segment. (#THM).

Statement Reason

1. 1. Given

2. !"!!!!! ≅ ______ 2.

3. !"!!!!! ≅ ______ 3.

4. 4.

5. _________ ≅ ________ 5.

6. !"!!!! ≅ !"!!!! 6.

L10 – Perpendicular Lines and Triangles Name _____________________________ 10.2 – Grandma’s Roof Per _____ Date _____________________

Geometry Q2: L10 - Perpendicular Lines & Triangles Handouts Page 5

T

US

9. Let’s do another isosceles triangle proof. First, use patty paper to reproduce the triangle STU. Fold the patty paper in such a way that the fold contains the midpoint of SU and the point T. Which angles appear to be equal? Let’s prove it!

Given: In triangle STU, !" ≅ !" Prove: S U∠ ≅ ∠

Hint: First draw the perpendicular line from T to !".

You just proved the Isosceles Triangle Theorem: If two sides of a triangle are

_______________, then the angles opposite those sides are also ________________. (#THM)

10. Make a conjecture for the converse of this theorem.

If two angles of a triangle are ____________________, then the _________ opposite

____________ are also ____________________.

Note: The Isosceles Triangle Theorem is equivalent to the Converse of the Perpendicular Bisector Theorem: If !" ≅ !" then P must lie on the perpendicular bisector of line segment AB. (#THM) Thus, when you drop down a perpendicular line from the peak to the base, it must bisect the base (as long as you already know the two sides are equal, which is given).

Statement Reason

L10 – Perpendicular Lines and Triangles Name _____________________________ 10.2 – Grandma’s Roof Per _____ Date _____________________

Geometry Q2: L10 - Perpendicular Lines & Triangles Handouts Page 6

Back to Grandma’s roof…

11. Use what you just discovered to again construct an isosceles triangle roof for grandma’s house, using only a compass and straightedge.

What if Grandma decides she wants a scalene triangle for the roof instead of isosceles? Using the example on the right:

12. Construct a perpendicular line from the peak to the base of the roof (i.e. the ceiling).

Place a point where the perpendicular line intersects the base of the roof.

This is where the opening to the attic will be installed. We need to make sure that there is enough space to move around in and stand up when you climb through. To do this, we want to maximize the distance from the opening to BOTH sides of the roof.

How do you think we should define distance from a point to a line? Draw dotted line segments to represent the shortest distance from the attic access point to both roof sides. What do you notice about the angles the line segments form in relation to the roof sides? Move one of your line segments from side to side, keeping one end of your segment at the attic opening. What happens to the distance between the attic opening and the roof side as you move your line segment? What happens to the angle of intersection with the roof sides?

L10 – Perpendicular Lines and Triangles Name _____________________________ 10.2 – Grandma’s Roof Per _____ Date _____________________

Geometry Q2: L10 - Perpendicular Lines & Triangles Handouts Page 7

L

K

A

E

P

H

T

13. The distance from a point to a line can be defined as the length of the

__________________ line segment from the point to the line, which is also the line segment

with the __________________ length. (#VOC)

14. Looking at the two distances (dotted lines) you just drew, are they the same length?

15. If we want to make sure there’s enough space to move around on both sides of the attic

opening, it would be best if the distances from the attic opening to each roof side are

_____________________.

16. Using your answer to the previous question, roughly sketch where you think the opening

should be.

17. Let’s take a break again to prove where the opening should be located. We will prove a more general result. Use the diagram at the right.

Given: !" bisects angle HAT and Point P lies on !" Prove: !" ≅ !"

*Hint: use the triangles

Assuming this proof helps us locate the attic opening, draw where the ceiling would be located in

the proof sketch above (i.e. duplicate the scalene roof) and indicate the attic opening. Note: your

ceiling does not need to pass through points H and T, and you may wish to rotate the sketch.

Statement Reason _____

L10 – Perpendicular Lines and Triangles Name _____________________________ 10.2 – Grandma’s Roof Per _____ Date _____________________

Geometry Q2: L10 - Perpendicular Lines & Triangles Handouts Page 8

You just proved Theorem 10.1 (#THM): Given an angle bisector, all points on that bisector are

_________________ from the sides of the angle. Use this information to locate the attic opening

in the house diagram of problem 12 above.

18. Work with a neighbor to come up with a converse for Theorem 10.1. Do you think this will always be true? Be sure to include it in your Theorem Booklet.

19. One more proof.

Given: Triangle XYZ is equilateral. (I.e. all three sides are congruent.)

Prove: X Y Z∠ ≅ ∠ ≅ ∠

Theorem 10.2: An equilateral triangle is also ______________________, and each angle has a

measure of ______. (#THM)

Statement Reason ______

1. 1. Given

2. !"!!!! ≅ ______ ≅ ______ 2.

3. 3. Isosceles Triangle Theorem,

4. 4. Isosceles Triangle Theorem,

5. 5.

X

Y Z

L10 – Perpendicular Lines and Triangles Name _____________________________ 10.3 – Practice/Homework Per _____ Date _____________________

Geometry Q2: L10 - Perpendicular Lines & Triangles Handouts Page 9

41°VS

T68°

X

Y

R

M O

L

7 cm

U Y

W

12 in

D

K

J

9 mi7 mi

T

F

A

Practice

Find the measure of the indicated side or angle

1. !" 2. !" 3. !"

4. V 5. X 6. A

Use the diagram on the right for #7 - 10. Solve for x.

7. LM = 5, LO = x

8. LM = 2x + 4, LO = 18

9. LM = 3x - 6, LO = 2x + 21

10. 20 , 90m M x m O x∠ = + ∠ = −

 

L10 – Perpendicular Lines and Triangles Name _____________________________ 10.3 – Practice/Homework Per _____ Date _____________________

Geometry Q2: L10 - Perpendicular Lines & Triangles Handouts Page 10

B

A

E

K

P

Y

H

Use the diagram at the right for #11-14. !" bisects YEH∠ . Solve for x.

11. AP = ! − 5, PB = −2! + 25

12. AP = 4!! − 12, PB =2!! + 6

13. 82 , 4 9m YEH m BEP x∠ = ° ∠ = + 14. 82 , 7m YEH m APE x∠ = ° ∠ =

15. Prove the Converse of the Isosceles Triangle Theorem, stated below (10.2 #10). Given: A ≅ B in triangle ABC Prove: !" ≅ ______

Statement Reason _ _____

A B

C

L10 – Perpendicular Lines and Triangles Name _____________________________ 10.3 – Practice/Homework Per _____ Date _____________________

Geometry Q2: L10 - Perpendicular Lines & Triangles Handouts Page 11

S

M

A

P

16. Prove the Converse of Theorem 10.1, stated below (#18 in lesson 10.2). Given: !" ≅ !" Prove: !" is the angle bisector of MAS∠ .

*Hint: In right triangles, if you know 2 sides, how many possible measures are there for the third side?

17. Given: ,BC CD DE FDE F≅ ≅ ∠ ≅ ∠ Prove: ΔBCD ≅ ΔFED

Statement Reason _ _____

Statement Reason _ _____

B

D

F

C

E

L10 – Perpendicular Lines and Triangles Name _____________________________ 10.4 – Locating the Center of a Rotation Per _____ Date _____________________

Geometry Q2: L10 - Perpendicular Lines & Triangles Handouts Page 12

1. ABCΔ below was rotated 180° about point D. Find point D. Hint: when a single point P is rotated about point D, the distance from its image P’ and D remains the same as from P to D.

a. Verify that AD = A’D, BD = B’D, and CD = C’D.

b. Verify that the distance from the midpoint of BC to D is the same as the distance from the image of the midpoint of BC to D.

 

L10 – Perpendicular Lines and Triangles Name _____________________________ 10.4 – Locating the Center of a Rotation Per _____ Date _____________________

Geometry Q2: L10 - Perpendicular Lines & Triangles Handouts Page 13

Recall the Converse of the Perpendicular Bisector Theorem: If point P is equidistant from points A and B, then P must lie on the perpendicular bisector of line segment AB.

2. When a figure is rotated 180° about a point, it is relatively easy to find the central point of rotation. When the figure is rotated X°, where X is NOT 180°, the problem becomes more difficult. We can use what we’ve learned in this lesson to locate the center. Let’s begin by rotating a point P, CW X° about a given point D, as illustrated in the diagram below. (D has purposefully been hidden, and the value for X is irrelevant to this discussion.) Since P is rotated about D, its image P’ must lie on the circle centered at D with radius PD. How might you be able to find D? Assume the dotted line circle is hidden from your view. a. Draw the line segment 'PP .

b. The values PD and P’D must be _________________.

c. Using your answer to part b, along with what you’ve learned in this lesson, it must be the

case that D lies somewhere on the ___________________________ of 'PP .

d. Draw a line that must contain the point D.

Note: since in a rotation problem you are not given the dotted-line circle, it is not possible to locate D with such limited information; you can only limit its location to being some point on a line. However, if your figure being rotated contains more than a single point, such as with a triangle for example, then you can use this technique on multiple points (e.g. the vertices) to find D. We do that in the next problem.

 

L10 – Perpendicular Lines and Triangles Name _____________________________ 10.4 – Locating the Center of a Rotation Per _____ Date _____________________

Geometry Q2: L10 - Perpendicular Lines & Triangles Handouts Page 14

3. EFGΔ below was rotated clockwise X° about point D. Find point D. Hint: Follow the

technique used in the previous problem for the three vertices, along with the Converse of the Perpendicular Bisector Theorem.

Reflections:

a. Explain how to find the center of rotation when given a triangle and its rotated image.

b. Do you think this technique would work with other figures? Why?

c. How many pre-image/image pairs of points do you need to locate the center of rotation?

 

L10 – Perpendicular Lines and Triangles Name _____________________________ 10.5 – Sewing Time Per _____ Date _____________________

Geometry Q2: L10 - Perpendicular Lines & Triangles Handouts Page 15

Sewing Time!

While everyone else is working on her house, Grandma decides to make some tablecloths. She decides to make a pattern that includes different sized equilateral triangles (similar to the diagram on the right). What do you know about equilateral triangles?

Help her figure out how to construct an equilateral triangle that she can use as a template, given a specified side length. She can’t use a protractor so you have to teach her to do it with a compass and straightedge. (Note: she could also use a piece of string instead of a compass.) *Hint: Think about what information a compass can provide. Practice on a separate sheet of paper with the given lengths below.

Explain briefly your method and why it works.

Activity:

Make several different sized templates of your own (out of cardstock or index cards) and create a pattern on a separate page.