Download Name: Period: Geometry Unit 8: Similar Triangles Day 1 Guided

Name: _______________________________
Geometry Unit 8: Similar Triangles
Period: ___________________
Day 1 Guided Notes: Ratios, Proportions and Geometric Mean & Similar Triangles
HW: Pg 436 #14-26 Even, Pg 444 #9-14
Warm Up
1. State the largest side of the triangle.
2. In triangle ABC, the ratio of  A:  B:  C = 8:7:3. State the largest side of the
triangle.
3. State the largest side of the triangle.
4. State the smallest angle of the triangle.
5. State the largest side of the triangle.
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Ratio: A quotient of two numbers.
Can be written in 3 ways: 1/2
___________
____________
Proportions: When two ratios are set equal!
To solve a proportion: Cross multiply and set cross products equal.
Examples: Solve for the variable (cross multiply!!!)
4 x

5 15
1.
2.
2
3
=
m  5 m 1
3.
8
z2
=
4
z2
4. The lengths of the sides of a triangle are in the extended ratio 4 : 7 :9. The perimeter
is 60 cm. What are the lengths of the sides?
The mean proportional/geometric mean of two positive numbers a and b is the positive
number x such that
When solving,

.
.
It is always positive because it represents the length! This is the formula for
finding the mean proportional!
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Examples: Find the geometric mean/mean proportional between the two numbers. Put all
answers in simplest radical form.
1. 2 and 8
2. 3 and 9
3. 7 and 14
4. 8 and 16
5. 10 and 12
6. 9 and 13
Similar Triangles: Triangles are similar if their corresponding (matching) angles are
_____________ and the ratio of their corresponding sides are _______________.
Note: It is sufficient to show that if two angles of one triangle are congruent to two
angles of another triangle, the triangles are similar.
Recall, when you dilate a figure it creates similar figures. The scale factor is ALWAYS the
image
. If it is not a dilation question, the scale factor can be written as a ratio with
pre  image
the number before ‘to’ as the numerator and the number after ‘to’ as the denominator.
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Examples:
1.
 ABC is similar to  DEF. Find the scale factor of  ABC to  DEF.
2.
In the figure to the right:
a)  DEF is the image of  ABC after a dilation by a scale factor of __________.
b)  ABC is the image of  DEF after a dilation by a scale factor of __________.
c) What do you notice about how the scale factor relates to the sizes of the two
triangles?
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Similar Triangle Problems:
1.
The sides of a triangle are 5, 6 and 10. Find the length of the longest side of a
similar triangle whose shortest side is 15.
2.
Two triangles are similar. The sides of the first triangle are 7, 9, and 11. The
smallest side of the second triangle is 21. Find the perimeter of the second
triangle.
3. Two triangular roofs are similar. The corresponding sides of these roofs are 16
feet and 24 feet. If the altitude of the smaller roof is 6 feet, find the
corresponding altitude of the larger roof.
4.
Triangles KLM and STU are similar. The length of the sides of KLM are x + 82,
4x - 26, and 5x - 90. The perimeter of KLM is 306. The perimeter of STU is 459,
what is the length of the longest side of STU?
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Day 2 Guided Notes: Using Proportions To Solve Similar Triangle Problems
HW: Day 2 HW WS
Warm Up
1. AB and BC are two sides of a triangle. If the ratio of AB:BC is 3:4, AB = 60 and
BC = 10q + 15,find q.
2. Given angle A and angle A' are each 59º, find AC.
Setting up proportions when you have two overlapping triangles…
and two sides are parallel!
Theorem: If a line parallel to one side of a triangle intersects the other two sides, then
it divides the two sides proportionally. This is called the “SIDE SPLITTER THEOREM”.
Converse of the Side Splitter Theorem: If a line intersects two sides of a triangle and
divides the sides proportionally, the line is parallel to the third side of the triangle.
Note: If you dilate one triangle (centered at the shared angle), it creates two similar
triangles. Since the sides of similar triangles are in proportion, the third sides of
these triangles are parallel.
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1. In the diagram,
nearest tenth.
is parallel to
, BD = 4, DA = 6 and EC = 8. Find BC to the
Use Parts of Sides
Use Full Sides
2. Find BC to the nearest tenth.
Use Parts of Sides
Use Full Sides
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3. At a certain time of the day, the shadow of a 5' boy is 8' long. The shadow of a tree
at this same time is 28' long. How tall is the tree to the nearest tenth? (assume the
boy and tree are both vertical so they are parallel)
4. In triangle ABC, angle A = 90º and angle B = 35º. In triangle DEF, angle E = 35º and
angle F = 55º. Are the triangles similar?
5. In triangle ABC , point D lies on AB and point E lies on BC such that DE ll AC. If
AD = x, DB = 4, DE =x, and AC = 8, find AB.
6. In triangle PQR, point S lies on QR and point T lies on PR such that ST ll PQ .
If RS = 3, PT = 20, RT = x2, QS = 5, find PR in simplest radical form.
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Day 2 HW Worksheet
Set up proportions for each set of similar triangles and solve for x. Put all answers in
simplest radical form.
1.
2.
3.
4.
5.
6.
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Day 3 Guided Notes: Using Similar Polygons
HW: Finish Day 3 Notes
Warm Up
1. In the diagram,  XYZ ~  MNP.
a) Find the scale factor.
b) Find the lengths of all the sides in both triangles.
c) Find the length of the altitude shown in  XYZ.
d) Find and compare the perimeters of both triangles.
e) Find and compare the areas of both triangles.
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Examples
1. If the corresponding sides are in a ratio of 1/2, list the ratio of the corresponding:
Altitudes: ___________
Medians: ____________
Perimeters: __________
Areas: ______________
2. Triangles RST and WXY are similar. The side lengths of RST are 10 inches, 14 inches
and 20 inches and the length of the altitude is 6.5 inches. The shortest side of
triangle WXY is 15 inches long.
RS
a)

XY

RT
Find the lengths of the other two sides of triangle WXY.
b) Find the length of the corresponding altitude of triangle WXY.
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3. A triangle has sides whose lengths are 5, 12, and 13. A similar triangle could have
sides with lengths of
(1) 3, 4, and 5
(3) 7, 24, and 25
(2) 6, 8, and 10
(4) 10, 24, and 26
4. The accompanying diagram shows two similar triangles. Set up a proportion and solve
for x?
5. In the accompanying diagram,  QRS ~  LMN, RQ = 30, QS = 21, SR = 27
and LN = 7. What is the length of LM?
QR

MN

QS
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6. Two triangles are similar. The lengths of the sides of the smaller triangle are 3, 5,
and 6, and the length of the longest side of the larger triangle is 18. What is the
perimeter of the larger triangle?
7. The base of an isosceles triangle is 5 and its perimeter is 11. The base of a similar
isosceles triangle is 10. What is the perimeter of the larger triangle?
8. Pentagon ABCDE is similar to pentagon FGHIJ. The lengths of the sides of ABCDE
are 8, 9, 10, 11, and 12. If the length of the longest side of pentagon FGHIJ is 18,
what is the perimeter of pentagon FGHIJ?
9. Two triangles are similar, and the ratio of each pair of corresponding sides is 2:1.
Which statement regarding the two triangles is not true?
(1) Their areas have a ratio of 4 : 1.
(2) Their altitudes have a ratio of 2 : 1.
(3) Their perimeters have a ratio of 2 : 1.
(4) Their corresponding angles have a ratio of 2 : 1.
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Day 4 Guided Notes: Proving Triangles Similar By AA  AA
HW: Finish Day 4 Notes
Warm Up
1. Which is not a property of all similar triangles?
(1) The corresponding angles are congruent.
(2) The corresponding sides are congruent.
(3) The perimeters are in the same ratio as the corresponding sides.
(4) The altitudes are in the same ratio as the corresponding sides.
2. a) Graph and label triangle ABC with vertices A(1, 6), B (2, 1), C (7, 2).
b) Show triangle ABC is an isosceles triangle.
c) Find the coordinates of D, the point that bisects AC.
d) Show AC  BD
*In an ISOSCELES TRIANGLE the median and the altitude (as well as the perpendicular
bisector & angle bisector) are the same line segment*
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Proof Reason #____
1.
15
2.
3.
Given: FE || AD
Prove:  AGD ~  EGF
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4. Given: CD is perpendicular to AB
CD bisects ACB
Prove: ∆CDA ~ ∆CDB
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Day 5 Guided Notes: Proving Triangles Similar By AA  AA (including Proportions and
Cross Products)
HW: Finish Day 5 Notes
Warm Up
1.
Solve for the value(s) of x: 2x2 – 50 = x2 + 3x + 4
2. Right triangle ABC is similar to right triangle DEF. Side AB = 8, Side AC = 6,
Side DE = n+ 2 and side DF = 3. Find the value of n. (MUST draw the triangles first!)
Steps to Proofs:
1. What is always true about similar triangles?
- All corresponding angles are congruent.
- All corresponding sides are in proportion.
2. Prove triangles similar by AA, SSS or SAS (AA most common!)
3. The proportion is true because:
Corresponding sides of similar triangles are in proportion.
Proof Reason # ______
4. Cross multiply:
To prove cross products - The product of the means = the product of the extremes.
Proof Reason # ________
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3.
First think about what triangles must be similar to get the correct proportion!
First step: Prove triangle similar by AA…which triangles? _______ ~ ________
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4.
First think about what proportion you need to get those cross products…then think
about which triangles must be similar to get the correct proportion!
Work Backwards!!!
Which triangles are similar? ________ ~ ___________
What is your proportion? _________ = _________
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Day 6 Guided Notes: Prove Triangles Similar by SSS and SAS
HW: Start Unit 8 Test Review Worksheet
Remember what is always true for Similar Triangles:
1. Corresponding angles are ______________________.
2. Corresponding sides are _______________________.
Three ways to prove triangles similar:
1)
AA  AA (Most Common!)
You need to verify two pairs of ____________ angles.
2) SSS ~ Theorem (Proof Reason # ______)
You need to verify _________ proportions.
3) SAS ~ Theorem (Proof Reason # ______)
You need to verify _________ proportion and one pair of ____________ angles.
Practice:
1.
Find the value of x that makes  ABC ~  DEF.
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2. Determine whether the following can be proven similar using AA, SAS or SSS.
a) ___________
b) ___________
c)
___________
d) ___________
e) ___________
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More Practice!
3) Determine whether the following can be proven similar using AA, SAS or SSS.
a) ____________
b) ____________
c) ____________
d) ____________
e) ____________
f) ____________
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g) ____________
h) ____________
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Day 7 Guided Notes: Proportionality Theorems
HW: Finish Unit 8 Test Review Worksheet
Warm Up
1. DE//AC.
2. Which method could you use to prove these triangles similar? Are they similar?
Justify.
a)
b)
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Two Other Similarity Theorems:
If three parallel lines intersect two
transversals, then they divide the transversals
proportionally.
If a ray bisects an angle of a triangle, then it
divides the opposite side into segments whose
lengths are proportional to the lengths of the
other two sides.
If a line parallel to one side of a triangle
intersects the other two sides, then it
divides the two sides proportionally.
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Practice:
1. Given a//b//c, solve for x.
2. Find the length of AB.
3. AD bisects  CAB, find AC.
4. Find the length of AB.
5.
Solve for x.
6.
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7.
8.
Round to the nearest tenth.
9. In the diagram to the right, lines a, b, and c are cut by transversals d and e such that
a//b//c . Solve for x.
10. In the diagram to the right, AD bisects  BAC. Solve for the value of x.
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Day 8 Guided Notes: Common Core Review Questions
1.
2.
30
3.
4.
31
5.
6.
32
7.
8.
9.
33
10.
11.
34
12.
13.
35
14.
Common Core Review Questions Answer Key
1.
2. ANS: 4
3. ANS: 4
4.
5. ANS: 2
6.
7.
36
8.
9.
10.
11.
12.
13.
14.
37