Integrated Math 2 – WS 4-1 - Classifying Triangles 10/26 Name ______
1. What are three ways to classify triangles according to the sides?
· An equilateral triangle has three congruent sides.
· An isosceles triangle has two congruent sides.
· A scalene triangle has as no congruent sides.
2. What are three ways to classify triangles according to the angles
· A right triangle is defined as exactly one 90 degree angle.
· An obtuse triangle is defined as exactly one angle between 90 and 180 degrees.
· An acute triangle is defined as three angles less than 90 degrees
Define the following.
3. Scalene Triangle
A scalene triangle has as no congruent sides.
4. Isosceles Triangle
An isosceles triangle has two congruent sides.
5. Equilateral Triangle
An equilateral triangle has three congruent sides.
For each triangle below, determine whether it is scalene, isosceles, or equilateral.
6.
An equilateral triangle
7.
An isosceles triangle
8.
A scalene triangle
9.
An isosceles triangle
Find the value of the variable in each figure below.
10. Equilateral triangle
b – 8 = 12 and 2a – 2 = 12
+ 8 + 8 + 2 + 2
b = 20 and 2a = 14
2 2
a = 7
11.
4x – 2 = 3x + 3
– 3x + 2 – 3x + 2_
x = 5
12.
AB = 3g + 3 BC = 5h – 2 AC = f – 3
9 = 3g + 3 3 = 5h – 2 7 = f – 3
– 3 – 3 + 2 + 2 + 4 + 4
6 = 3g 5 = 5h 11 = f
3 3 5 5
g = 2 h = 1 f = 11
Use the information below about ∆ABC to identify whether the triangle is acute, right, or obtuse. Then determine which side will be the longest side of the triangle side a, b, or c.
13. ÐA = 32°, ÐB = 90°, ÐC = 58°
A right triangle. – Side b is longest side
14. ÐA = 52°, ÐB = 41°, ÐC = 87
An acute triangle. – Side c is longest side
15. ÐA = 104°, ÐB = 22°, ÐC = 54°
An obtuse triangle. – Side a is longest side
16. ÐA = 70°, ÐB = 80°, ÐC = 30°
An acute triangle. – Side b is longest side
17. ÐA = 90°, ÐB = 27°, ÐC = 63°
A right triangle. – Side a is longest side
18. ÐA = 60°, ÐB = 58°, ÐC = 62°
An acute triangle. – Side b is longest side
19. ÐA = 95°, ÐB = 33°, ÐC = 52°
An obtuse triangle. – Side a is longest side
20. ÐA = 85°, ÐB = 5°, ÐC = 90°
A right triangle. – Side a is longest side
21. How are the angles of a triangle related to the sides of the triangle?
The largest side is opposite the largest angle and the smallest side is opposite the smallest angle
Integrated Math 2 – WS 4-2 - Triangle Congruence Day 1 10/28 Name ______
Determine whether the following triangles are congruent. If they are congruent, write the reason
(SSS, SAS, AAS, or ASA) vertically, justifying each part by identifying the parts of the triangle that make the statement true.
1.
B E
A C D F
2.
M A
P O R
3.
A O
J N D
4. Y O
R T E G
5. B D
A C E
6.
7. P N
A
M T
Integrated Math 2 – WS 4-3 – Reflexive Property 10/31/11 Name ______
1. The reflexive property can be used to show a side that is shared between two shapes is congruent to itself. Draw a picture that represents an example of the reflexive property.
2. What are the four ways we have discussed to prove triangles congruent? SSS SAS ASA AAS
Define the following:
3. Vertical Angles Angles on the opposite side of an “X” and are therefore congruent
4. Congruent Segments or angles that have the same measure.
5. Alternate Interior Angles Angles which are on the interior (between to parallel lines) and are on opposite sides of the transversal. These angles are congruent
Each of the following shows two congruent triangles. Write the reason they are congruent (SSS, SAS, AAS, or ASA) on the line for justification. Identify which parts are congruent and give a reason that you know that. IF it is shown in the picture, write given. If not, list the reason you know it to be true.
6. ΔABD @ ΔCDB
A → L BAD @ L DCB Why Given
A → L BAD @ L DCB Why Given
S → BD @ BD Why Reflexive
Justification: AAS (Angle, Angle Side)
------
7. ΔBIK @ ΔKEB
S → BI @ KE Why Given
A → L IBK @ L EKB Why Alt. Int L Thm.
S → BK @ KB Why Reflexive
Justification: SAS (Side, Angle Side)
------
8 . ΔEHM @ ΔHEG
A → L HME @ L EGH Why Given
A → L MEH @ L GHE Why Alt. Int L Thm.
S → EH @ EH Why Reflexive
Justification: AAS (Angle, Angle Side)
Integrated Math 2 – WS 4-4 – Bisectors 11/1//11 Name ______
1. What does a bisector do? A bisector cuts something (angle or segment) into two congruent sections.
Segment Bisector: A segment (ray or line) that intersects a segment at its midpoint – makes two congruent segments.
Angle Bisector: A ray that divides the angle into two congruent angles.
2. What is a midpoint? Midpoint: The point that divides the segment into two congruent segments.
3. What is the difference between an angle bisector and a segment bisector?
An angle bisector is a ray that cuts an angle into two congruent angles whereas a segment bisector can be a segment, ray or line and cuts a segment into two congruent segment lengths.
In each figure below, find the value of the variable and the measure of each unknown segment or angle.
4. Ray BC bisects L ABD 5. Z is the midpoint of line XY 6. Ray KH bisects LGKJ
m L ABD @ m L CBD = 37o XZ = ZY = 15 units m L GKH @ m L HKJ
4x – 9 = 37 3x – 3 = 15 4x – 8 = 3x + 2
+ 9 + 9 + 3 + 3 – 3x + 8 – 3x + 8
4x = 46 3x = 18 x = 10
4 4 3 3 m L GKH m L HKJ
x = 11 1/2 x = 6 3(10) + 2 4(10) – 8
32 o 32 o
Draw a picture of each of the following and then find the value of the variable and each unknown segment or angle.
7. N is the midpoint of LM, LN = 8x – 3 and MN = 7x
LN = NM LN = NM LN + NM = LM
8x – 3 = 7x 8(3) – 3 = 7(3) 21 + 21 = LM
- 7x + 3 - 7x + 3 21 = 21 42 = LM
x = 3
8. Ray UT is the angle bisector of LVUW. The measure of LVUT = 84o .
The measure of LTUW = 9x – 3; and the measure of LVUT = 7y
7y = 84/2 9x – 3 = 84/2 LVUT = 42o
7y = 42 9x – 3 = 42 LTUW = 42o
7 7 + 3 = + 3
y = 6 9x = 45
9 9
x = 5
9. R is the midpoint of ST, SR = 5x + 4 and RT = 3y – 1
SR = 28/2 RT = 28/2 SR = 14
5x + 4 = 14 3y – 1 = 14 RT = 14
– 4 – 4 + 1 + 1
5x = 10 3y = 15
5 5 3 3
x = 2 y = 5
The following shows two congruent triangles. Justify the congruency using SSS, SAS, AAS, or ASA, then identify which parts are congruent and give a reason that you know that.
10. bisects
ΔASM @ ΔASP
S → AS @ AS Why Reflexive
A → L ASM @ L ASP Why Given
S → SM @ SP Why Def. Seg. Bisector
Justification: SAS
Simplify each radical as far as you can. Show work where you “took it down”.
1.
2.
3.
4.
5.
= 12
6.
= 2
7.
8.
9.
10.
11.
12.
13.
Solve for x in the following problems
14. x2 = 100
x =
15. x2 + 2 = 20
– 2 – 2
x2 = 18
x =
x =
x = 3
16. x2 – 20 = 28
+ 20 + 20
x2 = 48
x =
x = 4
x = 4
17. x2 + 5 = 130
– 5 – 5
x2 = 125
x =
x =
x = 5
18. x2 – 10 = 62
+ 10 + 10
x2 = 72
x =
x =
x = 6