Module 3
Study Guide
3.01 Pythagorean Theorem
a2 + b2 = c2
Steps to find the unknown length of a right triangle
Step 1: Identify the legs, a and b, and the hypotenuse, c.
Step 2: Substitute the known values into the Pythagorean Theorem formula.
Step 3: Solve the equation.
Is it a Triangle?
Triangle Inequality Theorem: The sum of the two sides of a triangle must be greater than the length of the third side.
Step 1: Locate the longest side.
Step 2: Add together the other two sides.
Step 3: Is the sum > the longer side?
If yes - it is a triangle. If no, it is not.
Is it a right triangle?
Step 1: Identify the shortest sides as a and b and the longest side as c.
Step 2: Substitute the known values into the Pythagorean Theorem formula.
Step 3: Simplify: If you get a true statement (like 25=25) then yes- it is a right triangle. If you get a false statement (like 20=25), then no - it is not a right triangle.
Step 1: Legs: a = ? b = 8 Hypotenuse: c = 22�Step 2: a2 +82 = 222
Step 3: a2 + 64 = 484
Subtract 64 from both sides a2 = 420
Square root on both sides a = 20.5
3.02 Pythagorean Theorem and the Coordinate Plane
Steps for finding the distance between two points using the Pythagorean Theorem without a coordinate plane
Step 1: Find the distance between the x-values of the two points given. Find the absolute value of the difference of the numbers. The distance should always be positive!
Step 2: Find the distance between the y-values of the two points given.
Step 3: Find the length of the hypotenuse, c, using the Pythagorean Theorem formula.
Determine the distance between the points (-2, -3) and (-10, -9)
Step 1: l-2 - -10l = 8
Step 2: l-3 - -9l = 6
Step 3: a2 + b2 = c2
82 + 62 = c2 64+36 = c2
100 = c2 10 = c
Steps for finding the distance between two points using the Pythagorean Theorem
Step 1: Plot the points on a coordinate plane and connect them with a line segment.
Step 2: Create a right triangle using the line segment as the hypotenuse.
Step 3: Determine the length of each leg, a and b, of the right triangle by counting each side.
Step 4: Find the length of the hypotenuse, c, using the Pythagorean Theorem formula.
a2 + b2 = c2
Vertical Angles: �
Adjacent Angles:�
3.03 Angle Relationships
Complementary Angles:�
Supplementary Angles:�
Naming Angles:
Angles FGH and JKL are supplementary and m∠JKL = 101.8°. Find m∠FGH.
Step 1: The angles are supplementary, so add to
180°
Step 2: FGH + JKL = 180°
FGH + 101.8° = 180°
Subtract 101.8 from both sides of the equation
Step 3: FGH = 78.2°
3.03 Angle Relationships continued
Steps to solve for unknown angles
Step 1: Identify the angle relationships.
Step 2: Write an equation that can be used to solve for the unknown value.
Step 3: Solve the equation.
Step 4: Substitute the value of the variable to determine the angle measures.
Angles A and B are complementary to each other. If m∠A = 48° and
m∠B = (2x + 18)°, find the value of x.
Step 1: The angles are complementary, so add to
90°
Step 2: 48 + 2x + 18 = 90
Step 3: Combine like terms (add 48 and 18)
2x + 66 = 90°
Subtract 66 from both sides of the equation 2x = 24
Divide each side of the equation by 2.
x = 12
3.04 Interior and Exterior Angles
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Determine the Interior Angle of the Triangle
For triangle XYZ, m∠X = 47°, m∠Y = (2x − 30)°, and m∠Z = (6x − 21)°. Find m∠Y.
Write an equation:
X + Y + Z = 180°
Substitute what you know from the problem:
47 + 2x - 30 + 6x -21 = 180°
Combine like terms:
47 - 30 -21 = -4
2x + 6x = 8x
8x - 4 = 180
Add 4 to both sides of the equation:
8x = 184
Divide both sides of the equation by 8:
x = 23°
Substitute 23° for x to solve for the m∠Y:
2(23) - 30
46 - 30 = 16 m∠Y = 16°
Pentagon EFGHI is shown with specific angle measures. Find the measure of angle F��
Step 1: Determine the number of sides of the polygon: 5 sides
Step 2: Substitute the value for n into the Polygon Angle Sum formula and simplify: (5-2) x 180 . . . 3 x 180 = 540 degrees
Step 3: Use the interior angle sum to solve the problem: Since the angles add up to 540, you can write an equation to solve for the measure of angle F
70 + 121 + 89 + 121 + F = 540
Combine Like Terms:
401 + F = 540
Isolate the Variable F:
F = 139 degrees
3.05 Polygon Angle Sums
Polygon Angle Sum Formula
(n - 2) • 180°, where n is the number of sides of the polygon
Regular Polygon Angle Formula
(n - 2) • 180° , where n is the number of sides of the polygon
n
Use this formula only on a regular polygon - a polygon that is both equilateral (all sides congruent) and equiangular (all angles congruent)
Matching
If a triangle is a right triangle, then a^2 + b^2 = c^2, where a and b are legs and c is the hypotenuse.
The sum of the interior angles of a triangle is 180 degrees
Two angles that add up to 90 degrees.
The sum of the lengths of any two sides of a triangle must be greater than the third
Having the same shape and size
Two angles that add up to 180 degrees.
The opposite angles formed when two lines intersect.
A polygon that is both equilateral and equiangular
Complementary angles
Supplementary angles
Vertical Angles
Pythagorean Theorem
Regular polygon
Triangle Sum Theorem
Triangle Inequality Theorem
Congruent
A right triangle has a leg length of
and a hypotenuse length of 5. Determine the length of the other leg of the right triangle.
Practice Problems
Practice Problems
Can a triangle be formed with side lengths 8, 14, 5? Explain.
Determine which set of side measurements could be used to form a right triangle.
B. 13, 8, 6
C. 7, 11, 17
D. 16, 12, 20
Determine the perimeter of the right triangle shown. Round to the nearest tenth.
Angles W and X are complementary.
Determine the degree measure of <W if m<X = 59.3 degrees.
Angles A and B are supplementary angles.
Angle A measures 82 degrees and angle B measures (9x + 17) degrees. Find x.
One interior angle of a triangle is 52°, and the other two angles are congruent. Choose the equation that could be used to determine the degree measure of one of the congruent angles.
In triangle XYZ, m∠Y = 71.09° and m∠Z = 66.2°.
Determine the measure of the exterior angle to ∠X.
In triangle XYZ, m∠Z = (6m − 2)° and the exterior angle to ∠Z measures (11m + 12)°. Determine the value of m.
What is the sum of all interior angles of a 26-sided regular polygon?
A regular polygon is shown with one of its angle measures labeled as a.
If m<a = (5z + 30) degrees, find the value of z.
A pentagon has 5 sides. One angle of a regular pentagon measures
(6w + 13)°.
Determine the value of w.
Round to the nearest whole number.
Part D: Autumn traveled from the Pigpen to the Petting Zoo and then to the Horse Barn. Billy traveled from the Pigpen to the Horse Barn along a straight path. Who went the shortest distance? Explain.
Part C: Find the shortest distance, in miles, from the Horse Barn to the Pigpen. Show every step of your work.
Essay #1
A map of 3 animal exhibit locations at the fair was created using a coordinate plane where the origin represents the entrance. The Petting Zoo is graphed at (-3, 4). The Horse Barn is graphed at (4, 4), and the Pigpen is graphed at (-3, -5). Each unit on the graph represents 1 meter.
Part A: Find the shortest distance, in miles, from the Petting Zoo to the Horse Barn. Show every step of your work.
Part B: Find the shortest distance, in miles, from the Petting Zoo to the Pigpen. Show every step of your work.
Essay #2
Angles DEF and GEH have the following measures:
m∠DEF = (x − 8)°, m∠GEH = (2x + 29)°.
Part A: If angle DEF and angle GEH are complementary angles, find the value of x. Show every step of your work.
Part B: Use the value of x from Part A to find the measures of angles DEF and GEH. Show every step of your work.
Part C: Could the angles also be vertical angles? Explain.