Download Math 2 – Unit 2: Triangles (Spring 2016) Day 5: Triangle

Math 2 – Unit 2: Triangles (Spring 2016)
Day 5: Triangle Congruence
1.
2.
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Congruent Figures: have the same __________________ & same _______________________.
Each _____________________ (“matching”) side and angle of congruent figures will also be _______.
Example #1:
ABCDE ≅ ____________
Congruent Angles
(Points can be named in any consecutive order, but each
corresponding vertex must be in the same order for each figure).
Congruent Sides
You Try #1: Given: ABCD≅ EFGH. Complete the following
a) Rewrite the congruence statement in at least 2 more ways.
b) Name all congruent angles
c) Name all congruent sides
We will deal mostly with congruent triangles. Two triangles are congruent if and only if their vertices can be matched up
so that the _______________________________ (both angles & sides) are congruent.
Because triangles only have three sides, we can take some shortcuts in proving them congruent…
I. If all three sides are given, we call this ___________.
_______ Postulate: If 3 sides of one triangle are congruent to 3 sides of another triangle, then the triangles are
congruent.
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II. If 2 sides and the angle BETWEEN those sides are given, we call this _________.
_______ Postulate: If 2 sides and the included angle of one triangle are congruent to 2 sides and the included
angle of another triangle, then the triangles are congruent.
Included means _______________
III. If 2 angles and the side BETWEEN those angles are given, we call this _________.
_______ Postulate: If 2 angles and the included side are congruent to 2 angles and the include side of another
triangle, then the triangles are congruent.
IV. If 2 angles and the side NOT BETWEEN those angles are given, we call this _______.
_______ Postulate: If 2 angles and their non-included side are congruent to 2 angles and the non-included side
of another triangle, then the triangles are congruent.
Example #3: Are the triangles congruent? If so, why?
Anytime that 2 triangles share a side, think _________________ property!
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̅ . Are the triangles congruent? If so, why?
̅̅̅̅ ≅ HF
̅̅̅̅ and F is the midpoint of GI
Example #4: EF
There are 3 ways that triangles are not congruent:
1.
2.
3.
You Try #2: State whether each pair of triangles is congruent by SSS, SAS, ASA, or AAS. If none of the method
work, write “Not Congruent”
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1. Indicate which triangles are congruent. Be sure to have the correspondence of the letters correct.
̅̅̅̅
b. E is the midpoint of 𝑇𝑃
Δ𝑆𝑃𝐸 ≅ _______
a. Δ𝐸𝑅𝐶 ≅ _______
̅̅̅̅ ≅ 𝑅𝐶
̅̅̅̅ ?
Why is 𝑅𝐶
E
C
R
c. Δ𝐵𝑂𝑊 ≅ _______
Why is ∠1 ≅ ∠2?
O
S
T
1
2
W
T
R
T
E
P
B
Y
2. The perimeter of ABCD is 85. Find the value of x. Is Δ𝐴𝐵𝐶 congruent to Δ𝐴𝐷𝐶? Explain.
C
4x
6x - 11
B
D
5x - 7
3x + 4
A
Draw and label a diagram. Then solve for the variable and the missing measure or length.
3. If ∆𝐵𝐴𝑇 ≅ ∆𝐷𝑂𝐺, and 𝑚∠𝐵 = 14, 𝑚∠𝐺 = 29, 𝑎𝑛𝑑 𝑚∠𝑂 = 10𝑥 + 7. Find the value of x and 𝑚∠𝑂.
x = ___________
𝑚∠𝑂= _________
4. If ∆𝐶𝑂𝑊 ≅ ∆𝑃𝐼𝐺, and 𝐶𝑂 = 25, 𝐶𝑊 = 18, 𝐼𝐺 = 23, 𝑎𝑛𝑑 𝑃𝐺 = 7𝑥 − 17 . Find the value of x and PG.
x = ___________
PG=___________
5. If ∆𝐷𝐸𝐹 ≅ ∆𝑃𝑄𝑅 and 𝐷𝐸 = 3𝑥 − 10, 𝑄𝑅 = 4𝑥 − 23, 𝑎𝑛𝑑 𝑃𝑄 = 2𝑥 + 7. Find the value of x and EF.
x = ___________
EF = __________
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Complete the congruence statement for each pair of congruent triangles. Then state the reason you are able
to determine the triangles are congruent. If you cannot conclude that triangles are congruent, write “none” in
the blanks.
1.
∆𝐸𝐹𝐷 ≅ ∆___________
∆𝐴𝐵𝐶 ≅ ∆___________
2.
by ________
3. ∆𝐿𝐾𝑀 ≅ ∆___________
by ________
E
by ________
L
A
D
F
B
H
C
K
M
T
G
J
1. ∆𝐴𝐵𝐶 ≅ ∆___________
∆𝐴𝐵𝐶 ≅ ∆___________
5.
by ________
by ________
A
X
A
C
D
C
Y
B
E
B
Use the given information to mark the diagram and any additional congruence you can determine from the
diagram. Then complete the triangle congruence statement and give the reason for triangle congruence .
6.
C
1
7.
A
2
4
D
B
3
B
A
Given: ∠1 ≅ ∠3,
∠2 ≅ ∠4
∆𝐴𝐵𝐶 ≅ ∆__________ by __________
8.
C
D
Given: ∠𝐴𝐵𝐷 ≅ ∠𝐶𝐵𝐷,
F
∆𝐴𝐵𝐷 ≅ ∆__________ by __________
A
9.
C
1
A
2
G
E
B
̅̅̅̅
̅̅̅̅ 𝑎𝑛𝑑 𝐸𝐴
Given: 𝐺 𝑖𝑠 𝑡ℎ𝑒 𝑚𝑖𝑑𝑝𝑜𝑖𝑛𝑡 𝑜𝑓 𝐹𝐵
∠𝐴𝐷𝐵 ≅ ∠𝐶𝐷𝐵
D
Given: ∠1 ≅ ∠3,
4
3
B
̅̅̅̅ ≅ 𝐴𝐵
̅̅̅̅
𝐶𝐷
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Use the given information and triangle congruence statement to complete the following.
̅̅̅̅? How do you know?
10. ∆𝐴𝐵𝐶 ≅ ∆𝐺𝐸𝑂, AB = 4, BC = 6, and AC = 8.
What is the length of 𝐺𝑂
11. ∆𝐵𝐴𝐷 ≅ ∆𝐿𝑈𝐾, 𝑚∠𝐷 = 52°, 𝑚∠𝐵 = 48°, 𝑎𝑛𝑑 𝑚∠𝐴 = 80°.
a. What is the largest angle of ∆𝐿𝑈𝐾?
b. What is the smallest angle of ∆𝐿𝑈𝐾?
12. ∆𝑆𝑈𝑁 ≅ ∆𝐻𝑂𝑇. ∆𝑆𝑈𝑁 is isosceles. Is there enough information to determine if ∆𝐻𝑂𝑇 is isosceles?
Explain why or why not.
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