Download 4.1 Triangle Sum.notebook

4.1 Triangle Sum.notebook
November 04, 2014
Agenda for Geometry:
• Must complete all missing assessments by the end of
the day today.
• Pick up a scratch piece of paper and the definition
sheet for Chapter 4.
• Correct last nights assignment, I will be checking
that it is done soon after the bell rings so have it
visible.
• Triangle Sum Conjecture investigation
• First paragraph proof and examples.
• Return tools and then work on assignment.
• Assignment is 4.1 worksheet not the book problems
Nov 6­8:27 AM
Key for Slopes of Parallel and Perpendicular lines Worksheet
19. No, not parallel
20. Yes, parallel
21. Yes, opp reciprocal slopes
22. No, slopes are Nov 6­8:14 AM
For Exercises 1‐4, determine whether each pair of lines through the points given below is parallel, perpendicular, or neither. A(1, 2) B(3, 4) C(5, 2) D(8, 3) E(3, 8) F( −6, 5) 1. AB and BC 2. AB and CD 3. AB and DE 4. CD perpendicular
perpendicular
neither
parallel
and EF
For Exercises 7‐9, find the slope of each side, and then determine whether each figure is a trapezoid, a parallelogram, a rectangle, or just an ordinary quadrilateral. Explain how you know.
5. Given A(0, − 3), B(5, 3), and Q( −3, −1), find two possible locations for a point P such that PQ is parallel to AB.
mAB =6/5
(7,11)
Ordinary Quadrilateral
(2,5)
(­8,­7)
Ordinary Quadrilateral
6. Given C( −2, −1), D(5, −4), and (4, 2), find two possible locations for a point P such that PQ is perpendicular to CD .
mCD= ­3/7
(7,9)
Trapezoid:
(1,­5)
(­2,­12)
Nov 9­8:47 AM
Nov 9­8:45 AM
10. Quadrilateral HAND has vertices H(−5, −1), A(7, 1), N(6, 7), and D(−6,5). a. Is quadrilateral HAND a parallelogram? A rectangle? Neither? Explain how you know. b. Find the midpoint of each diagonal. What can you conjecture?
a. mHA =mND =1/6; m HD = mNA = ­6.
So, Quad HAND is a rectangle because opposite sides are parallel and adjacent sides are perpendicular.
b. Midpoint HN = Minpoint AD = (1/2, 3). The diagonals of the rectangle bisect each other.
not
13. Given A( −3, 2), B(1, 5), and C(7, −3), find point D such that quadri
ABCD is a rectangle.
(3,-6)
not
Nov 9­8:47 AM
Nov 9­8:48 AM
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4.1 Triangle Sum.notebook
November 04, 2014
Lesson 4.1 Triangle Sum Conjecture
Nov 4­10:17 AM
Nov 3­1:20 PM
We will do and investigation in Pairs.
You will need a piece of paper, ruler, and protractor for this investigation.
1)
Start by having each group member draw 2 large different triangles on their paper using a ruler. Make sure you have one acute and one obtuse triangle.
2) Next, pass your triangles to another person in your group.
3) Now select a triangle and use a protractor to find the We will need to share scissors, one per every 2­3 students.
measurements of each angle in that triangle.
4) Add these together and share this sum of the three Once all students are seated with above materials in front of them we will get started.
Nov 3­12:56 PM
5)
angles with your group.
Nov 3­12:58 PM
Now use the other triangle from the sheet and label each corner(vertice) with s, t, and u.
s
u
Triangle Sum Conjecture: t
The sum of the measures of the angles in every triangle is ________?
6) Cut out this triangle and wait.
7)
Now tear each corner off so you have three good size corners.
s
u
t
8) Line up the corners so the vertices meet at one point.
9)
What is the measure of the angle that is created by these three angles
Nov 3­1:21 PM
Nov 3­1:19 PM
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4.1 Triangle Sum.notebook
November 04, 2014
Let's write a paragraph proof to show that the Triangle Sum
Conjecture is always true for any triangle.
0
We want to show m<2 + m<4 +m<5 = 180
E
C
We will first sketch a triangle and use an auxiliary line and
answer the questions:
• What are we trying to prove?
• Why might we draw an
auxiliary line to be parallel to
one of the sides?
• What is the relationship among
1 2 3
If EC is parallel to side AB
we know that and
2
4
A
5
5
4
B
angles
• What congruences can be
determined from this diagram?
Nov 6­6:01 AM
Sep 28­4:31 PM
Paragraph Proof for Triangle Sum Conjecture:
Consider
and
together
as a single angle that forms a
linear pair with
.
E
E
C
C
1 2 3
1 2 3
and
Since
we will use the substitution
property to attain
Thus, by the linear pairs conjecture,
A
4
5
B
A
4
5
B
m<2 + m<4 +m<5 = 1800 , as desired.
Nov 3­3:05 PM
Sep 28­4:31 PM
Find the measure of each angle indicated
EX 1:
EX 2:
?
?
Nov 5­3:40 PM
Nov 6­6:24 AM
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4.1 Triangle Sum.notebook
November 04, 2014
EX 4:
EX 3:
?
?
Nov 6­6:29 AM
EX 5:
Nov 6­6:38 AM
Solve for x.
EX 6:
Nov 6­6:40 AM
Solve for x.
Nov 6­6:44 AM
Let's do this Mini-investigation together
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
0°
0
1
2
Nov 2­2:41 PM
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4
5
Nov 3­3:34 PM
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4.1 Triangle Sum.notebook
November 04, 2014
Write this onto your definition sheet.
Third Angle Conjecture ­
If two angles of one triangle are equal in measure to two angles of another triangle, then the third angles of the triangles are equal in measure. Nov 2­3:23 PM
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