Warm-up 1/5/12 1.Find x, m 1, if m 3 = 3x + 40 and m 4 = 5x + 60. 2. Find the value of x that...
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Transcript of Warm-up 1/5/12 1.Find x, m 1, if m 3 = 3x + 40 and m 4 = 5x + 60. 2. Find the value of x that...
![Page 1: Warm-up 1/5/12 1.Find x, m 1, if m 3 = 3x + 40 and m 4 = 5x + 60. 2. Find the value of x that would make lines m and n parallel. 1 4 3 2 x = 10 3x.](https://reader036.fdocuments.us/reader036/viewer/2022062314/56649f045503460f94c18b85/html5/thumbnails/1.jpg)
Warm-up 1/5/12
1.Find x, m1, if m 3 = 3x + 40 and m 4 = 5x + 60.
2. Find the value of x that would make lines m and n parallel.
14
32
x = 10
3x – 4 + x + 10 = 180
4x + 6 = 180
m1 = m3 = 3(10) + 40
m1 = 70
3x – 4
x + 10
m
nt
4x = 174 x = 43.5
3x + 40 + 5x + 60 = 180
8x+ 100 = 180 8x = 80
![Page 2: Warm-up 1/5/12 1.Find x, m 1, if m 3 = 3x + 40 and m 4 = 5x + 60. 2. Find the value of x that would make lines m and n parallel. 1 4 3 2 x = 10 3x.](https://reader036.fdocuments.us/reader036/viewer/2022062314/56649f045503460f94c18b85/html5/thumbnails/2.jpg)
Classifying Triangles
![Page 3: Warm-up 1/5/12 1.Find x, m 1, if m 3 = 3x + 40 and m 4 = 5x + 60. 2. Find the value of x that would make lines m and n parallel. 1 4 3 2 x = 10 3x.](https://reader036.fdocuments.us/reader036/viewer/2022062314/56649f045503460f94c18b85/html5/thumbnails/3.jpg)
Lesson 3.1B
Essential Questions
1.How are triangles classified based on their angle measures and side lengths?
2.Define and draw the median and altitude of a triangle.
3.What are the special characteristics of an isosceles triangle?
4.What is an exterior angle and how do you find the measure of the exterior angle?
![Page 4: Warm-up 1/5/12 1.Find x, m 1, if m 3 = 3x + 40 and m 4 = 5x + 60. 2. Find the value of x that would make lines m and n parallel. 1 4 3 2 x = 10 3x.](https://reader036.fdocuments.us/reader036/viewer/2022062314/56649f045503460f94c18b85/html5/thumbnails/4.jpg)
Standards
MM1G1. Students will investigate properties of geometric figures in the coordinate plane.
e. Use the coordinate plane to investigate properties of and verify conjecture related to triangles and quadrilaterals
![Page 5: Warm-up 1/5/12 1.Find x, m 1, if m 3 = 3x + 40 and m 4 = 5x + 60. 2. Find the value of x that would make lines m and n parallel. 1 4 3 2 x = 10 3x.](https://reader036.fdocuments.us/reader036/viewer/2022062314/56649f045503460f94c18b85/html5/thumbnails/5.jpg)
Essential Question #1
How are triangles classified based on their angle measures and side lengths?
![Page 6: Warm-up 1/5/12 1.Find x, m 1, if m 3 = 3x + 40 and m 4 = 5x + 60. 2. Find the value of x that would make lines m and n parallel. 1 4 3 2 x = 10 3x.](https://reader036.fdocuments.us/reader036/viewer/2022062314/56649f045503460f94c18b85/html5/thumbnails/6.jpg)
Classification of triangles based
on angle measure:
Acute Triangle: a triangle with three acute angles
Right Triangle: a triangle with two acute angles and one right angle
Obtuse Triangle: a triangle with two acute angles and one obtuse angle
![Page 7: Warm-up 1/5/12 1.Find x, m 1, if m 3 = 3x + 40 and m 4 = 5x + 60. 2. Find the value of x that would make lines m and n parallel. 1 4 3 2 x = 10 3x.](https://reader036.fdocuments.us/reader036/viewer/2022062314/56649f045503460f94c18b85/html5/thumbnails/7.jpg)
1. 2. 3.
37o
90o 53o
44o
68o 68o
32o
116o
32o
___________________Triangle _____________________Triangle _________________TriangleRight Acute Obtuse
Sum for ∆1:________ Sum for ∆2:________ Sum for ∆3:________180o 180o 180o
Conjecture: The sum of the interior angles of a triangle is _________180o
![Page 8: Warm-up 1/5/12 1.Find x, m 1, if m 3 = 3x + 40 and m 4 = 5x + 60. 2. Find the value of x that would make lines m and n parallel. 1 4 3 2 x = 10 3x.](https://reader036.fdocuments.us/reader036/viewer/2022062314/56649f045503460f94c18b85/html5/thumbnails/8.jpg)
Using the paper triangle provided by your teacher, tear off the three corners as shown:
Now place the three corners together at their tips (called vertices). Make sure that all three corners or angles are placed so they are adjacent. In the space above make a sketch of what these corners look like when they are arranged as adjacent angles. Note they will form a line. Does this support your conjecture above? _______YES
![Page 9: Warm-up 1/5/12 1.Find x, m 1, if m 3 = 3x + 40 and m 4 = 5x + 60. 2. Find the value of x that would make lines m and n parallel. 1 4 3 2 x = 10 3x.](https://reader036.fdocuments.us/reader036/viewer/2022062314/56649f045503460f94c18b85/html5/thumbnails/9.jpg)
Classifying triangles by the lengths of their edges (sides):
Equilateral Triangle – all three sides of the triangle are equalIsosceles Triangle – at least two sides of the triangle are equal. Scalene Triangle – none of the sides of the triangle are equal.
2.2 cm
2.2 cm2.2 cm 1.6 cm3.4 cm
3.7 cm2.6 cm
3 cm
2.9 cm
_________ triangle _________ triangle _________ triangle _________ triangleEquilateral
6. 7.
3 cm
2.6 cm4.8 cm
NOTE: If a triangle is equilateral then it is also ________________.Equiangular
4. 5.
Scalene Isosceles Scalene
![Page 10: Warm-up 1/5/12 1.Find x, m 1, if m 3 = 3x + 40 and m 4 = 5x + 60. 2. Find the value of x that would make lines m and n parallel. 1 4 3 2 x = 10 3x.](https://reader036.fdocuments.us/reader036/viewer/2022062314/56649f045503460f94c18b85/html5/thumbnails/10.jpg)
. . put our heads together to….
http://wrir4.ucdavis.edu/PHOTOS/CROPS/images/Lettuce%20WideBeds.jpg
Answer Essential Question #1
How are triangles classified based on their angle measures and side lengths?
![Page 11: Warm-up 1/5/12 1.Find x, m 1, if m 3 = 3x + 40 and m 4 = 5x + 60. 2. Find the value of x that would make lines m and n parallel. 1 4 3 2 x = 10 3x.](https://reader036.fdocuments.us/reader036/viewer/2022062314/56649f045503460f94c18b85/html5/thumbnails/11.jpg)
Essential Question #2
Define and draw the median and altitude of a triangle.
![Page 12: Warm-up 1/5/12 1.Find x, m 1, if m 3 = 3x + 40 and m 4 = 5x + 60. 2. Find the value of x that would make lines m and n parallel. 1 4 3 2 x = 10 3x.](https://reader036.fdocuments.us/reader036/viewer/2022062314/56649f045503460f94c18b85/html5/thumbnails/12.jpg)
The ________________________of a triangle is a segment drawn from the vertex perpendicular to the opposite edge. A triangle contains __________________ altitudes. Draw the altitude from point A to the opposite edge. Label the altitude AD.
Altitude
3
A AA
DD D
8. 9. 10.
In the first triangle, the altitude was drawn on the _______________of the triangle.
In the second triangle, the altitude was an _______________of the triangle.
In the third triangle, the opposite edge was extended and the altitude was drawn on the ____________________ of the triangle.
INSIDE
EDGE
OUTSIDE
![Page 13: Warm-up 1/5/12 1.Find x, m 1, if m 3 = 3x + 40 and m 4 = 5x + 60. 2. Find the value of x that would make lines m and n parallel. 1 4 3 2 x = 10 3x.](https://reader036.fdocuments.us/reader036/viewer/2022062314/56649f045503460f94c18b85/html5/thumbnails/13.jpg)
The ___________________of a triangle is a segment drawn from the vertex to the midpoint of the opposite edge. A triangle contains _______ medians. Draw the median from point A to the opposite edge. Label the median AM.
median
3
AA A
MM M
8. 9. 10 .
![Page 14: Warm-up 1/5/12 1.Find x, m 1, if m 3 = 3x + 40 and m 4 = 5x + 60. 2. Find the value of x that would make lines m and n parallel. 1 4 3 2 x = 10 3x.](https://reader036.fdocuments.us/reader036/viewer/2022062314/56649f045503460f94c18b85/html5/thumbnails/14.jpg)
. . put our heads together to….
http://wrir4.ucdavis.edu/PHOTOS/CROPS/images/Lettuce%20WideBeds.jpg
Answer Essential Question #2
Define and draw the median and altitude of a triangle.
![Page 15: Warm-up 1/5/12 1.Find x, m 1, if m 3 = 3x + 40 and m 4 = 5x + 60. 2. Find the value of x that would make lines m and n parallel. 1 4 3 2 x = 10 3x.](https://reader036.fdocuments.us/reader036/viewer/2022062314/56649f045503460f94c18b85/html5/thumbnails/15.jpg)
Essential Question #3
What are the special characteristics of an isosceles triangle?
![Page 16: Warm-up 1/5/12 1.Find x, m 1, if m 3 = 3x + 40 and m 4 = 5x + 60. 2. Find the value of x that would make lines m and n parallel. 1 4 3 2 x = 10 3x.](https://reader036.fdocuments.us/reader036/viewer/2022062314/56649f045503460f94c18b85/html5/thumbnails/16.jpg)
Special features of an Isosceles Triangle:
Legs: the two congruent edges
Base: the edge that is not congruent to the other edges.
Vertex Angle : The angle formed by the two congruent edges (its vertex is the point where the congruent sides intersect)
Base Angle: The angles formed by a leg and the base (their vertices are the points where the legs intersect the base)
Vertex
Vertex
LEG LEG LEG
LEG Base
Base
Base
Base
Base
Base
![Page 17: Warm-up 1/5/12 1.Find x, m 1, if m 3 = 3x + 40 and m 4 = 5x + 60. 2. Find the value of x that would make lines m and n parallel. 1 4 3 2 x = 10 3x.](https://reader036.fdocuments.us/reader036/viewer/2022062314/56649f045503460f94c18b85/html5/thumbnails/17.jpg)
Investigation A:
Step 1: Using a straightedge, draw an isosceles triangle on a sheet of patty paper. Label the vertices
at the base A and B and the remaining vertex C.
Step 2: Fold the triangle so that A and B overlap. Crease the patty paper carefully.
Let’s make the following conjectures based on this fold.
Base angles of an isosceles triangle are ______________________.
The altitude drawn from the vertex angle to the base:
(a). _________________ the base.
(b). is the same segment as the ___________________.
(c). _________________ the vertex angle.
Special features of an Equilateral Triangle:
If a triangle is equilateral, then it is also _________________________.
The measure of each angle in an equilateral triangle is _________o.
All properties of __________________ triangles apply to equilateral triangles.
Congruent
Bisects
Bisects
Median
Equiangular
60
Isosceles
![Page 18: Warm-up 1/5/12 1.Find x, m 1, if m 3 = 3x + 40 and m 4 = 5x + 60. 2. Find the value of x that would make lines m and n parallel. 1 4 3 2 x = 10 3x.](https://reader036.fdocuments.us/reader036/viewer/2022062314/56649f045503460f94c18b85/html5/thumbnails/18.jpg)
. . put our heads together to….
http://wrir4.ucdavis.edu/PHOTOS/CROPS/images/Lettuce%20WideBeds.jpg
Answer Essential Question #3
What are the special characteristics of an isosceles triangle?
![Page 19: Warm-up 1/5/12 1.Find x, m 1, if m 3 = 3x + 40 and m 4 = 5x + 60. 2. Find the value of x that would make lines m and n parallel. 1 4 3 2 x = 10 3x.](https://reader036.fdocuments.us/reader036/viewer/2022062314/56649f045503460f94c18b85/html5/thumbnails/19.jpg)
Essential Question #4
What is an exterior angle and how do you find the measure of the exterior angle?
![Page 20: Warm-up 1/5/12 1.Find x, m 1, if m 3 = 3x + 40 and m 4 = 5x + 60. 2. Find the value of x that would make lines m and n parallel. 1 4 3 2 x = 10 3x.](https://reader036.fdocuments.us/reader036/viewer/2022062314/56649f045503460f94c18b85/html5/thumbnails/20.jpg)
Investigation B:
Step 1: Record the measure of A, D, and C from the overhead.
Step 2: Extend to the left of the triangle. Label a point
on the extension T. ACT is called an exterior angle.
Step 3: Compute: mACT = ___________o
Step 4: Compute: mA + mD = __________o
A and D are called remote interior angles.
Conjecture: The exterior angle of a triangle is equal to the __________ of the measures of the two remote interior angles.
A
C D
90˚
60˚ 30˚T
120˚
120˚
SUM
120˚
![Page 21: Warm-up 1/5/12 1.Find x, m 1, if m 3 = 3x + 40 and m 4 = 5x + 60. 2. Find the value of x that would make lines m and n parallel. 1 4 3 2 x = 10 3x.](https://reader036.fdocuments.us/reader036/viewer/2022062314/56649f045503460f94c18b85/html5/thumbnails/21.jpg)
. . put our heads together to….
http://wrir4.ucdavis.edu/PHOTOS/CROPS/images/Lettuce%20WideBeds.jpg
Answer Essential Question #4
What is an exterior angle and how do you find the measure of the exterior angle?