Download Theorems and Angles of a Triangle

6 Oct 2015 9:50 - 11:20
Geometry Agenda
Binder check - triangle / street sheet,+ Unit 2
pages
Angles of a Triangle
Practice problems
Congruent Triangles (SAS Postulate) worksheet
Homework
10/6/2015
Angles of a Triangle
As you probably know, the sum of the interior
angles of a triangle is 180°. We could use a
protractor and look at several triangles to
reinforce that, but several examples do not
(necessarily) make something true. How can
we prove that all triangles have interior angles
that add up to 180°? That is what we will try to
do below.
To the right
we have
triangle
ABC, where
we do not
know the
exact
measure of B
any of the
angles.
A
C
A
B
C
To make our proof easier, extend line segment BC into line
BC. Also create line DE parallel to line BC so that A is in
between D and E and point A lies on line DE.
A
D
B
E
C
To make our proof easier, extend line segment BC into line
BC. Also create line DE parallel to line BC so that A is in
between D and E and point A lies on line DE.
A
D
B
E
C
After having made some changes to our picture, have we
changed any of the properties of the angles in triangle ABC?
A
D
B
E
C
After having made some changes to our picture, have we
changed any of the properties of the angles in triangle ABC?
No.
A
D
B
m angle DAB + m angle BAC + m angle CAE =
angle ABC is congruent to angle ______
angle ACB is congruent to angle______
E
C
A
D
E
B
m angle DAB + m angle BAC + m angle CAE = 180°
angle ABC is congruent to angle DAB
angle ACB is congruent to angle CAE
C
A
D
B
E
C
Use the space below and the information above, to prove the
sum of the interior angles for triangle ABC is 180 degrees. Be
specific in how you know what you do. Another person should
be able to follow your reasoning.
Paragraph Proof
(do on white board)
On the previous page we did a paragraph proof. Some
students and teachers prefer to do two column proofs. Let’s
look at a two column proof of the same idea. Correct diagram.
A
D
B
E
C
Example 2 Column Proof
Given: line DE is parallel to line BC
Prove: m(angleABC) + m(angleBAC) +
m(angle ACB) = 180 degrees
Statement
Reason
1.
1.
line DE is parallel to line BC
Given.
2. m(angle DAB) + m(angleBAC) + m
(angle EAC) = 180 degrees
2. A group of 3 angles adding up to a
line.
3. angle DAB is congruent to angle ABC;
angle EAC is congruent to angle ACB
3. For parallel lines cut by a transversal,
alternate interior angles are congruent
4. m(angleABC) + m(angle BAC) + m
(angle ACB) = 180 degrees
4. By the definition of congruent angles
and line 3. and then substitution of values
from line 3 into line 2.
Let’s try another proof about
triangles.
In this case we are proving a theorem. The
measure of an exterior angle of a triangle
equals the sum of the measure of the two
X
interior angles.
1
2
Y
3
4
Z
X
1
2
Y
Two Column Proof
3
4
Z
Paragraph Proof
Given triangle XYZ we
can say that the
measures of angles 1, 2
and 3 will add up to 180
degrees.
X
1
2
Y
3
4
Z
Paragraph Proof
Given triangle XYZ we can say that the measures
of angles 1, 2 and 3 will add up to 180 degrees.
We also know that angles 3 and 4 are a linear
pair, so the sum of their measures is 180
degrees.
X
1
2
Y
3
4
Z
So we can write
m∠4 + m∠3 = m∠1 + m∠2 + m∠3
and by the subtraction property of equality and
substitution this can be simplified to
m∠4 = m∠1 + m∠2 , which is what we were
asked to prove.
X
1
2
Y
Two Column Proof
Statement
Reason
1
1
2
2
3
3
4
4
5
5
3
4
Z
Paragraph Proof
Given triangle XYZ we
can say that the
measures of angles 1, 2
and 3 will add up to 180
degrees.
X
1
2
3
4
Y
Z
Two Column Proof
Statement
Reason
1 Triangle XYZ
1 Given
2
m∠1 + m∠2 + m∠3 = 180°
3
m∠3 + m∠4 = 180°
4
2 The sum of the internal angles of a triangle
(theorem)
3 Linear angle pair
4
Two Column Proof
Statement
Reason
1 Triangle XYZ
1 Given
2
m∠1 + m∠2 + m∠3 = 180°
3
m∠3 + m∠4 = 180°
m∠1 + m∠2 + m∠3 = m∠3 +
m∠4
4
5
m∠1 + m∠2 = m∠4
2 The sum of the internal angles of a triangle
(theorem)
3 Linear angle pair
4 Substitute line 3 into line 2 since both
equations equal 180 degrees
5 By the subtraction property of equality and
substitution property of equality.
Angles of a Triangle page 3
Practice Problems
Find the values of x and y for the problems
below. Show your work in the space below
each diagram as well as marking up the
diagram as needed.
Congruent Triangles (SAS Postulate)
Work through the steps in the worksheet to see
how the SAS postulate works.
Homework
Angles of a Triangle Homework (Unit
2 Lesson 1)
1 - 4 - all parts.
Draw diagrams out again on separate paper if
not enough room to show what you need to do
to find the values.
Geometry Dictionary Project - due Nov 18 email Mrs. D-B re partner or individual.