warm-up since they are polygons, what two things must be true about triangles if they are similar?
TRANSCRIPT
![Page 1: Warm-Up Since they are polygons, what two things must be true about triangles if they are similar?](https://reader034.vdocuments.us/reader034/viewer/2022052701/56649f3e5503460f94c5e0f5/html5/thumbnails/1.jpg)
Warm-UpWarm-Up
Since they are polygons, what two things must be true about triangles if they are similar?
![Page 2: Warm-Up Since they are polygons, what two things must be true about triangles if they are similar?](https://reader034.vdocuments.us/reader034/viewer/2022052701/56649f3e5503460f94c5e0f5/html5/thumbnails/2.jpg)
Similar PolygonsSimilar Polygons
Two polygons are similar polygonssimilar polygons iff the corresponding angles are congruent and the corresponding sides are proportional.
MAIZCORN ~
ZMNC
IZRN
AIOR
MACO
ZNIR
AOMC
C
OR
N
C
OR
NM
A
I
Z
Similarity Statement:Similarity Statement:
Corresponding Angles:Corresponding Angles:
Statement of Proportionality:Statement of Proportionality:
![Page 3: Warm-Up Since they are polygons, what two things must be true about triangles if they are similar?](https://reader034.vdocuments.us/reader034/viewer/2022052701/56649f3e5503460f94c5e0f5/html5/thumbnails/3.jpg)
Example 1Example 1
Triangles ABC and ADE are similar. Find the value of x.
6 cm
8 cm9 cm
xE
D
A
B
C
![Page 4: Warm-Up Since they are polygons, what two things must be true about triangles if they are similar?](https://reader034.vdocuments.us/reader034/viewer/2022052701/56649f3e5503460f94c5e0f5/html5/thumbnails/4.jpg)
Example 2Example 2
Are the triangles below similar?
3
5
4 6
8
1053
37
Do you really have to check all the sides and angles?
![Page 5: Warm-Up Since they are polygons, what two things must be true about triangles if they are similar?](https://reader034.vdocuments.us/reader034/viewer/2022052701/56649f3e5503460f94c5e0f5/html5/thumbnails/5.jpg)
6.4-6.5: Similarity Shortcuts6.4-6.5: Similarity Shortcuts
Objectives:
1. To find missing measures in similar polygons
2. To discover shortcuts for determining that two triangles are similar
![Page 6: Warm-Up Since they are polygons, what two things must be true about triangles if they are similar?](https://reader034.vdocuments.us/reader034/viewer/2022052701/56649f3e5503460f94c5e0f5/html5/thumbnails/6.jpg)
Investigation 1Investigation 1
In this Investigation we will check the first similarity shortcut. If the angles in two triangles are congruent, are the triangles necessarily similar?
4050
C
A B 50 40
F
ED
![Page 7: Warm-Up Since they are polygons, what two things must be true about triangles if they are similar?](https://reader034.vdocuments.us/reader034/viewer/2022052701/56649f3e5503460f94c5e0f5/html5/thumbnails/7.jpg)
Investigation 1Investigation 1
Step 1: Draw ΔABC where m<A and m<B equal sensible values of your choosing.
4050
C
A B
![Page 8: Warm-Up Since they are polygons, what two things must be true about triangles if they are similar?](https://reader034.vdocuments.us/reader034/viewer/2022052701/56649f3e5503460f94c5e0f5/html5/thumbnails/8.jpg)
Investigation 1Investigation 1
Step 1: Draw ΔABC where m<A and m<B equal sensible values of your choosing.
Step 2: Draw ΔDEF where m<D = m<A and m<E = m<B and AB ≠ DE.
4050
C
A B 50 40
F
ED
![Page 9: Warm-Up Since they are polygons, what two things must be true about triangles if they are similar?](https://reader034.vdocuments.us/reader034/viewer/2022052701/56649f3e5503460f94c5e0f5/html5/thumbnails/9.jpg)
Investigation 1Investigation 1
Now, are your triangles similar? What would you have to check to determine if they are similar?
4050
C
A B 50 40
F
ED
![Page 10: Warm-Up Since they are polygons, what two things must be true about triangles if they are similar?](https://reader034.vdocuments.us/reader034/viewer/2022052701/56649f3e5503460f94c5e0f5/html5/thumbnails/10.jpg)
Angle-Angle SimilarityAngle-Angle Similarity
AA Similarity AA Similarity PostulatePostulate
If two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar.
![Page 11: Warm-Up Since they are polygons, what two things must be true about triangles if they are similar?](https://reader034.vdocuments.us/reader034/viewer/2022052701/56649f3e5503460f94c5e0f5/html5/thumbnails/11.jpg)
Example 3Example 3
Determine whether the triangles are similar. Write a similarity statement for each set of similar figures.
![Page 12: Warm-Up Since they are polygons, what two things must be true about triangles if they are similar?](https://reader034.vdocuments.us/reader034/viewer/2022052701/56649f3e5503460f94c5e0f5/html5/thumbnails/12.jpg)
ThalesThales
The Greek mathematician Thales was the first to measure the height of a pyramid by using geometry. He showed that the ratio of a pyramid to a staff was equal to the ratio of one shadow to another.
![Page 13: Warm-Up Since they are polygons, what two things must be true about triangles if they are similar?](https://reader034.vdocuments.us/reader034/viewer/2022052701/56649f3e5503460f94c5e0f5/html5/thumbnails/13.jpg)
Example 4Example 4
If the shadow of the pyramid is 576 feet, the shadow of the staff is 6 feet, and the height of the staff is 5 feet, find the height of the pyramid.
![Page 14: Warm-Up Since they are polygons, what two things must be true about triangles if they are similar?](https://reader034.vdocuments.us/reader034/viewer/2022052701/56649f3e5503460f94c5e0f5/html5/thumbnails/14.jpg)
Example 5Example 5
Explain why Thales’ method worked to find the height of the pyramid?
![Page 15: Warm-Up Since they are polygons, what two things must be true about triangles if they are similar?](https://reader034.vdocuments.us/reader034/viewer/2022052701/56649f3e5503460f94c5e0f5/html5/thumbnails/15.jpg)
Example 6Example 6
If a person 5 feet tall casts a 6-foot shadow at the same time that a lamppost casts an 18-foot shadow, what is the height of the lamppost?
![Page 16: Warm-Up Since they are polygons, what two things must be true about triangles if they are similar?](https://reader034.vdocuments.us/reader034/viewer/2022052701/56649f3e5503460f94c5e0f5/html5/thumbnails/16.jpg)
Investigation 2Investigation 2
What if you decide to indirectly measure a height on a day when there are no shadows? The following GSP Animation will help you discover an alternate method of indirect measurement using a mirror.
![Page 17: Warm-Up Since they are polygons, what two things must be true about triangles if they are similar?](https://reader034.vdocuments.us/reader034/viewer/2022052701/56649f3e5503460f94c5e0f5/html5/thumbnails/17.jpg)
Example 7Example 7
Your eye is 168 centimeters from the ground and you are 114 centimeters from the mirror. The mirror is 570 centimeters from the flagpole. How tall is the flagpole?
![Page 18: Warm-Up Since they are polygons, what two things must be true about triangles if they are similar?](https://reader034.vdocuments.us/reader034/viewer/2022052701/56649f3e5503460f94c5e0f5/html5/thumbnails/18.jpg)
Investigation 3Investigation 3
Each group will be given one of the three candidates for similarity shortcuts. Each group member should start with a different triangle and complete the steps outlined for the investigation. Share your results and make a conjecture based on your findings.
![Page 19: Warm-Up Since they are polygons, what two things must be true about triangles if they are similar?](https://reader034.vdocuments.us/reader034/viewer/2022052701/56649f3e5503460f94c5e0f5/html5/thumbnails/19.jpg)
Side-Side-Side SimilaritySide-Side-Side Similarity
SSS Similarity Theorem:SSS Similarity Theorem:
If the corresponding side lengths of two triangles are proportional, then the two triangles are similar.
![Page 20: Warm-Up Since they are polygons, what two things must be true about triangles if they are similar?](https://reader034.vdocuments.us/reader034/viewer/2022052701/56649f3e5503460f94c5e0f5/html5/thumbnails/20.jpg)
Side-Angle-Side SimilaritySide-Angle-Side Similarity
SAS Similarity Theorem:SAS Similarity Theorem:
If two sides of one triangle are proportional to two sides of another triangle and the included angles are congruent, then the two triangles are similar.
![Page 21: Warm-Up Since they are polygons, what two things must be true about triangles if they are similar?](https://reader034.vdocuments.us/reader034/viewer/2022052701/56649f3e5503460f94c5e0f5/html5/thumbnails/21.jpg)
Example 8Example 8
Are the triangles below similar? Why or why not?
![Page 22: Warm-Up Since they are polygons, what two things must be true about triangles if they are similar?](https://reader034.vdocuments.us/reader034/viewer/2022052701/56649f3e5503460f94c5e0f5/html5/thumbnails/22.jpg)
Example 9Example 9
Use your new conjectures to find the missing measure.
18
24
x
24
28
y
![Page 23: Warm-Up Since they are polygons, what two things must be true about triangles if they are similar?](https://reader034.vdocuments.us/reader034/viewer/2022052701/56649f3e5503460f94c5e0f5/html5/thumbnails/23.jpg)
Example 10Example 10
Find the value of x that makes ΔABC ~ ΔDEF.
![Page 24: Warm-Up Since they are polygons, what two things must be true about triangles if they are similar?](https://reader034.vdocuments.us/reader034/viewer/2022052701/56649f3e5503460f94c5e0f5/html5/thumbnails/24.jpg)
AssignmentAssignment
• P. 384-387: 1-4, 7, 8, 10, 12, 14-17, 20, 30, 31, 32, 36, 41, 42
• P. 391-395: 4, 6-8, 10-14, 33, 39, 40
• Challenge Problems