Download ∆аAngle Sum Thm. ∆аExt. Angle Thm.

∆ Angle Sum Thm.
∆ Ext. Angle Thm.
47o
∆ Angle Sum Thm.
Aim #3 Continued...
5.
m≮g = _______
6.
m≮h = _______
7.
m≮i = _______
1250
1150
i
8. Draw a diagram for each problem. Find the answer.
a) Find the measure of the vertex angle of an isosceles triangle if an exterior
angle formed by extending the base measures 1120.
112 = 68 + x
x
x = 44
68
68 112
b) If the vertex angle of an isosceles triangle measures 400, find the measure of
either of the exterior angles formed by extending the base.
40 + 2x = 180
40
x = 70
x
x
180 ­ 70 = 1100
9.
0
50
m≮j = _______
5x ­ 3 + 4x ­ 2 = 10x ­ 20
∆ Ext. Angle Thm.
x = 15
10(15) ­ 20 = 1300
180 ­ 130 = 500
A linear pair of angles sums to 1800
0
10.
36
x = _______
d0
0
(2x)
0
41
d = _______
(2x ­ 5)0
113
113 + 2x ­ 5 = 180
x = 36
­ When || lines are cut by a transv., corr. angles are =.
­Linear pairs of angles sum to 1800.
1130
2(36) ­ 5 + 2(36) + d = 180
∆ Angle Sum Thm.
d = 41
Let's sum it up!!!
180
• The sum of the angle measures in a triangle is ____________°.
remote
• An exterior angle of a triangle equals the sum of the two _________________
__________________
angles.
interior
Aim #4: How do we use deductive reasoning to write angle proofs?
CC Geometry H
Do Now: One of the main goals in studying geometry is to develop your ability to
reason critically, to draw valid conclusions based upon observations and proven facts.
• A proof is a detailed explanation of how a statement follows logically from
other statements already accepted as true.
• Deductive reasoning is reasoning based on using observations and accepted
facts, which lead to specific conclusions.
Determine if each is an example of correct DEDUCTIVE REASONING. If not,
explain why it is not:
1. If you live in Manhasset, then you live on Long Island.
You live in Manhasset.
∴ You live on Long Island.
∴
"therefore" symbol
2. If you live in Manhasset, then you live on Long Island.
You live on Long Island.
You live in Manhasset.
3. If you live in Manhasset, then you live on Long Island.
You do not live in Manhasset
You do not live on Long Island.
4. If you live in Manhasset, then you live on Long Island.
You do not live on Long Island.
∴ You do not live in Manhasset.
In geometry, we follow a similar deductive thought process to provegeometric claims. Let's
revisit solving for unknown angles.
You needed to figure out the measure of a, and used the
"fact" that an exterior angle of a triangle equals the sum of
the measures of the opposite interior angles. The measure
0
of ≮a must therefore be ____ .
0
Recall that the angles of a triangle sum to 180 .
Given the labeled diagram at the right, we can prove
that m≮x + m≮y = m≮z.
Theorem to prove:
An exterior angle of a triangle equals the sum of the two opposite interior angles.
A two-column proof:
Statements
Reasons
1. m x + m y + m w = 1800
1.
2. m w + m z = 1800
2.
3. m≮x + m≮y + m≮w = m≮w + m≮z
3. Angles that form a linear pair are supplementary.
4.
4. m≮w = m≮w
5.
5. m≮x + m≮y = m≮z
Notice that each step in the proof was justified by a previously known or
demonstrated fact. We end upwith a newly proven fact (that an exterior angle of
any triangle is the sum of the measures of the opposite
interior angles of the triangle).
This ability to identify the steps used to reach a conclusion based on knownfacts is
deductive reasoning.
1. Prove: Vertical angles are equal.
w and
Plan: What do you know about
x?
y and
x?
What conclusions can you draw based on both pieces of knowledge?
Statements
1.
Reasons
1.
0
Angles that form a linear pair sum to 180 .
0
2.
2. Angles that form a linear pair sum to 180 .
3.
3. Substitution Property of Equality
4.
4. Reflexive Property
5.
5. Subtraction Property of Equality
2. Prove: m≮w + m≮x + m≮z = 180
Reasons
Statements
1.
1.
2.
2.
3.
3.
3. Prove: m≮w = m≮y + m≮z
Reasons
Statements
1.
1.
2.
2.
3.
3.
4. Given: AB ll CD and BC ll DE
Prove: m≮ABC = m≮CDE
Reasons
Statements
1.
1.
AB ll CD and BC ll DE
2.
2.
3.
3.
4.
4.
x
5. Prove: m y + m z = m w + m x
y
a
Statements
Reasons
1. m≮y + m≮z + m≮a = 180
1.
2. m≮w + m≮x + m≮a = 180
2.
3.
3. Substitution Property of Equality
4.
4.
5.
5.
0
6. Prove: The sum of the angles marked by arrows is 900 .
Reasons
Statements
1. a + b + c + d = 360
1.
2. e + f + g + h = 360
2.
3. i + j + k + m = 360
3.
4. a+b+c+d+e+f+g+h+i+j+k+m = 10804.
5. d + h + i = 180
5.
6. a+b+c+e+f+g+j+k+m = 900
6.
Let's Sum it Up!!
A proof is a detailed explanation of how a statement follows logically from other
statements already accepted as true.
Deductive reasoning is reasoning based on using observations and accepted
facts, which lead to specific conclusions.
Name ______________________
Date ________________
CC Geometry H
HW #4
1. In the figure at the right, explain whym ll n.
e
b
2. In the diagram at the right, prove that the sum
a
0
of the angles marked by arrows is 360 .
d
Statements
c
f
Reasons
1.
1. m≮a + m≮d = 180
2. m≮b + m≮e = 180
2. same as (1)
3. m≮c + m≮f = 180
3. same as (1)
4. m≮a + m≮d + m≮b + m≮e + m≮c + m≮f = 540
5. m≮a + m≮b + m≮c = 180
4.
5.
6.
6. m≮d + m≮e + m≮f = 360
3. In the diagram at the right, prove
that m≮a - m≮b + m≮d = 180
Statements
x
y
Reasons
z
1. x ll y ll z 1. Given
2. m≮c + m≮d = 180
2. If parallel lines are cut by a transversal, then
0
interior angles on the same side sum to 180 .
3. m≮c + m≮b = m≮a
3.
4. m≮b = m≮b
4.
5. m≮c = m≮a - m≮b
5.
6. m≮a - m≮b + m≮d = 180
6.
Mixed Review:
1) Complete the table. AB || CD
C
A
88o
42o
xo
w
o
yo
zo
D
B
Angle
Angle
Measure
Reason
≮y
≮z
≮w
≮x
o
S
2) In the diagram, m≮SRA = 37 .
A
a. Name an angle complementary to ≮SRA.
R
C
I
b. Name an angle supplementary to ≮MRI.
c. Name an angle that is a supplement to ≮TRI.
d. Find m≮ARI.
o
e. Why is the m≮CRM = 53 ? Explain.
f. Find the m≮CRA.
M
T