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Chapter 5 TEST
Study Guide
Scheduled for ​Thursday, March 9, 2017
- Lines (Lesson 1)
- Angles of Triangles (Lesson 3)
-Pythagorean Theorem(Lesson 5)
-Use the Pythagorean Theorem (Lesson 6)
-Distance on the Coordinate Plane (Lesson 7)
Lesson 1 - Lines
In the figure above, lines m and n are parallel to one another and are intersecting
by a line called a “Transversal.” When this happens, special angles are made.
Adjacent Angles​-Angles that are ​NEXT TO​ each other. They share a
common side. These angles are S
​ UPPLEMENTARY​ which means the sum of
these two angles is equal to 180 DEGREES.
Adjacent angles:​ < 1 & < 2
<3 &<4
<5 &<6 <7 &<8
<1 &<3
<2 &<4
<5 &<7 <6 &<8
Vertical Angles​- Angles that are ​OPPOSITE​ from each other. They share a
common vertex. These angles are C
​ ONGRUENT​ which means they are equal in
measure.
Vertical angles:​ < 1 & < 4
<2 &<3
<5 &<8 <6 &<7
Corresponding Angles​-When two parallel lines are crossed by another
line (which is called the Transversal), the angles in matching corners are called
corresponding angles. These angles are C
​ ONGRUENT​. (“same spot different
group”)
Corresponding Angles:​ < 1 & < 5
<2 &<6
<3 &<7 <4 &<8
Alternate Interior Angles​-​When two parallel lines are crossed by
another line (which is called the Transversal), the pairs of angles on opposite
sides of the transversal but inside the two lines are called Alternate Interior
Angles. These angles are ​CONGRUENT​. (“inside the walls of the parallel lines”)
Alternate Interior Angles: ​< 3 & < 6
<4 &<5
Alternate Exterior Angles​-​When two parallel lines are crossed by
another line (which is called the Transversal), the pairs of angles on opposite
sides of the transversal but outside the two lines are called Alternate Exterior
Angles. These angles are ​CONGRUENT​. (“outside the walls of the parallel
lines”)
Alternate Exterior Angles: ​< 1 & < 8
<2 &<7
Finding Missing Angles
Suppose you know that m<1 = 50 ° You can use that information to figure out the
measures of angles 2, 3, and 4.
YOU MUST JUSTIFY USING VOCABULARY TERMS!!
(vertical, supplementary, corresponding, alt. interior, alt. exterior)
m < 2 = 130 because < 1 and < 2 are supplementary angles (add up to 180)
m < 3 = 50 because < 1 and < 3 are vertical angles
m < 4 = 130 < 1 and < 4 are supplementary or < 4 and < 2 are vertical angles.
Lesson 3 ​–​ Angles of Triangles
All ​INTERIOR​ angles in a triangles add to give you 180°
Finding Interior Angles of Triangles
Find the value of c in the given triangle
38 + 85 + c = 180
123 + c = 180
C = 57°
The missing angle is 57°.
Find the value of x in the given triangle
45°, 65°
1​ → Add the two angles given 45 + 65 = 110
nd​
2​ → Subtract the sum from 180 (180° in total triangle) 180-110 =70
The missing angle is 70°.
st​
Finding Exterior Angles of Triangles
interior angle + interior angle = exterior angle
The two “interior angles” are x° and 22°. The “exterior angle” is 134°.
x + 22 = 134 (interior + interior = exterior)
x = 112°
The two “interior angles” are x° and 58°. The “exterior angle” is 120°.
x + 58 = 120 (interior + interior = exterior)
x = 62°
Lesson 5 ​–​ Pythagorean Theorem
The formula is ​a2​ ​+ b​2​ = c​2
Pythagorean Theorem​- In a right triangle, the square of the measure of
the hypotenuse is equal to the sum of the measure of the squares of the
legs.
Example- “c” is missing
5​2​ + 12​2​ = c​2
25 + 144 = c​2
169 = c​2
c​2​ = 169
c = ​√​169
c = 13
Example-- “a” or “b” is missing
Example:
Is this triangle a right triangle? Prove using the Pythagorean Theorem
Does a​2​ + b​2​ = c​2​ ?
*​The longest side is always the hypotenuse (c).
●
●
●
●
a​2​ + b​2​ = c​2
10​2​ + 24​2​ = 26​2
100 + 576 = 676
​676 =​ ​676
They are equal, so ...
Yes, it is a Right Triangle!
Example:​ ​Is a triangle with lengths 8 in,15 in.,and 16 in. a right triangle?
Does 8​2​ + ​15​2​ = ​16​2 ​?
8​2​ + 15​2​ = 16​2
●
●
64 + 225 = 256,
●
289​
​
​256
They are NOT equal so, NO, it is not a right triangle!
Example: ​Using the Pythagorean theorem twice
-For triangle FBD: We know that leg FB= 6 cm. We have to find leg BD. Then
hypotenuse FD
Triangle: ABD- leg AD= 4 cm, leg AB= 3 cm. BD is hypotenuse of this triangle.
Use the Pythagorean Theorem to find BD (the hypotenuse of triangle ABC)
4​2​+3​2​=c​2
16 + 9 = c​2
25 = c​2
√25 = c
5=c (BD = 5 cm)
Use the Pythagorean theorem to find the hypotenuse FD. (legs are BD & FB)
5​2​+6​2​=c​2
25 + 36 = c​2
61 = c​2
√61 = c
7.8 = c
Answer: FD = 7.8 cm
Lesson 6 ​–​ Use the Pythagorean Theorem
Example:
The formula is ​a2​ ​+ b​2​ = c​2
A 13 feet ladder is placed 5 feet away from a wall. The distance from the ground straight up to
the top of the wall is 13 feet Will the ladder the top of the wall?
Let the length of the ladder represents the length of the
hypotenuse or c = 13 and a = 5 the distance from the ladder
to the wall.
c​2​ ​= a​2​ ​+ b​2
13​2​ ​= 5​2​ ​+ b​2
169 = 25 + b​2
169 - 25 = 25 - 25 + b​2​ ​(minus 25 from both sides to isolate
b​2​ ​)
144 = 0 + b​2
144 = b​2
b = ​√​144 = 12
The ladder will never reach the top since it will only reach 12 feet high from the ground
yet the top is 14 feet high.​
Example:
c​2​ ​= a​2​ ​+ b​2
c​2​ ​= 16​2​ ​+ 12​2
c​2​ ​= 256​ ​+ 144
c​2​ ​= 400
c​ ​= 20 ft
Lesson 7​–​Distance on the Coordinate Plane
You can use the ​Pythagorean Theorem​ to find the distance between two
points on the coordinate plane.
Steps:
1. Graph the two points
2. Draw a right triangle
3. Count the boxes to find the length of the two legs (a and b)
4. Find the hypotenuse using the Pythagorean Theorem
Example: ​Graph the ordered pairs (2, –3) and (5, 4). Then find the distance ​e between the two points.
Step 1: Graph the two points
Step 2: Draw a right triangle
Step 3: Count the boxes → a = 3, b = 7
Step 4: Use the pythagorean theorem
a2​ + ​b2​ = ​c2
3​2​ + 7​2​ = ​c2
58 = ​c2
=
7.6 ≈ ​c
​The
Pythagorean Theorem
Replace ​a with 3 and ​b with 7.
2​
​3​
​
​
+ ​72​​ = ​9 ​+ ​49, or 58.
​Definition of square root
​Use a calculator.
The points are about 7.6 units apart.