Download Lesson 14: Angle – Angle Similarity

Lesson 14
NYS COMMON CORE MATHEMATICS CURRICULUM
Name:__________________________________
M2
Period: ________ Date: __________
Lesson 14: Angle – Angle Similarity
Learning Targets

I can use the AA criteria to solve for missing angles or sides in triangle problems.

I can prove two triangles to be similar by using Angle - Angle criteria
Opening Exercise: (You will be working with a partner)
A. On a separate sheet of paper, each person will draw a triangle with one angle of 45o and another angle
of 90o. You can trace the angle templates at right for accuracy.
B. One person label the 45o vertex A, the 90o vertex B, and the third vertex C, creating ∆𝐴𝐵𝐶, while your
partner label the 45o vertex A’, the 90o vertex B’, and the third vertex C’, creating ∆𝐴′𝐵′𝐶′.
C. Find the lengths of the sides of your triangle, and compare them to the lengths of the corresponding
sides of a neighbor’s triangle.
Your ∆𝑨𝑩𝑪
Your partner’s ∆𝑨′𝑩′𝑪′
𝐴𝐵 =__________
𝐴′𝐵′ =__________
𝐵𝐶 =__________
𝐵′𝐶′ =__________
𝐴𝐶 =__________
𝐴′𝐶′ =__________
D. Are the triangles of you and your neighbor similar? Explain.
____________________________________________________________________________________
____________________________________________________________________________________
E. Why is it that you only needed to construct triangles where two pairs of angles were equal
and not three?
____________________________________________________________________________________
____________________________________________________________________________________
Lesson 14
NYS COMMON CORE MATHEMATICS CURRICULUM
Name:__________________________________
M2
Period: ________ Date: __________
Notes:
Two triangles, ∆ABC and ∆DEF, are ________________ if there is a similarity transformation that maps
___________ to ____________.
In the figure above, ∆𝐴𝐵𝐶~∆𝐷𝐸𝐹. The similarity transformation takes 𝐴 to 𝐷, 𝐵 to 𝐸, and 𝐶 to 𝐹 such that:

Corresponding angles are _______________ in measurement

Corresponding lengths of sides are _____________________.
Label each pair of corresponding angles in the diagram.
Use the diagram above the complete the following theorem:
Theorem on Similar Triangles:
If ∆𝐴𝐵𝐶~∆𝐷𝐸𝐹, then ____________ , _________, ___________
and
=
=
.
List the conditions that need to exist for two triangles to be similar:
1. _____________________________________
Angle-Angle Similarity – Notation: ____________
If _______ angles of one triangle are _______________
to two angles of another triangle, then the two triangles
are _________________.
2. ______________________________________
Lesson 14
NYS COMMON CORE MATHEMATICS CURRICULUM
Name:__________________________________
M2
Period: ________ Date: __________
Example 1: Determine if the triangles below are similar.
a) Find the degrees of the missing angles. ∠𝐵 =_________ ∠𝐷 =_________
b) Are the triangles similar? If so, how do you know? __________________________________________
____________________________________________________________________________________
c) Write a similarity statement: ________________________________
d) Find the lengths of the missing sides.
𝐵𝐶 =_________
𝐸𝐹 =_________
Example 2. Triangles shown below are similar. Use what you know about similar triangles to find the missing
side lengths 𝒙 and 𝒚.
Lesson 14
NYS COMMON CORE MATHEMATICS CURRICULUM
Name:__________________________________
M2
Period: ________ Date: __________
Lesson 14: Angle – Angle Similarity
Classwork
1. In the diagram below, ∆𝐴𝐵𝐶~∆𝐸𝐹𝐺, 𝑚∠𝐶 = 4𝑥 + 30, and 𝑚∠𝐺 = 5𝑥 + 10.
Determine the value of 𝒙.
2. As shown in the diagram below, ∆𝐴𝐵𝐶~∆𝐷𝐸𝐹, 𝐴𝐵 = 7𝑥, 𝐵𝐶 = 4, 𝐷𝐸 = 7, and 𝐸𝐹 = 𝑥.
What is the length of
a)
28
b)
2
c)
14
d)
4
?
Lesson 14
NYS COMMON CORE MATHEMATICS CURRICULUM
Name:__________________________________
M2
Period: ________ Date: __________
3. Write an explanation that shows that the two
triangles are similar. If so, solve for 𝒙 and 𝒚.
The triangles are / are not similar because:
____________________________________________
____________________________________________
𝒙 =__________
𝒚 =__________
4. Are the triangles shown below similar? Explain. If the triangles are similar, identify any missing angle
and side length measures.
The triangles are / are not similar because:
____________________________________________
____________________________________________
𝑬𝑭 =__________ 𝑨𝑪 =__________
∠𝑫 =__________ ∠𝑪 =__________
5. In the diagram below ∆𝐴𝐵𝐶~∆𝑅𝑆𝑇
Which statement is not true?
a)
b)
c)
d)
∆
Lesson 14
NYS COMMON CORE MATHEMATICS CURRICULUM
Name:__________________________________
M2
Period: ________ Date: __________
Lesson 14: Angle – Angle Similarity
Homework
1. In the figure to the right, ∆𝐿𝑀𝑁~∆𝑀𝑃𝐿.
a) Mark the corresponding angles congruent in the diagram.
b) If 𝑚∠𝑃 = 20°, find the remaining angles in the diagram.
∠𝐿𝑀𝑁 =_______
∠𝑃𝐿𝑀 =_______
∠𝑀𝑁𝐿 =_______
∠𝑁𝐿𝑃 =_______
∠𝑁𝐿𝑀 =_______
∠𝐿𝑁𝑃 =_______
2. In the diagram below, ∆𝐴𝐵𝐶~∆𝐴𝐹𝐷. Determine whether the following statements must be true from the
given information, and explain why.
a) ∆𝐶𝐴𝐵~∆𝐷𝐴𝐹
b) ∆𝐴𝐷𝐹~∆𝐶𝐴𝐵
c) ∆𝐵𝐶𝐴~∆𝐴𝐷𝐹
d) ∆𝐴𝐷𝐹~∆𝐴𝐶𝐵
3. Scalene triangle ABC is similar to triangle DEF. Which statement is false?
a)
b)
c)
d)
4. If ∆𝐴𝐵𝐶~∆𝑍𝑋𝑌, 𝑚∠𝐴 = 50, and 𝑚∠𝐶 = 30, what is 𝑚∠𝑋 ?
a)
30
b)
50
c)
80
d)
100