Is it a Triangle?
Can three sides with lengths 2, 3, and 4 make a triangle?
Yes!
Adding any two of those sides gives us a number bigger than the third side, so those sides can make a triangle (Triangle Inequality Theorem).
Angle 1 and Angle 2 are vertical angles. The measure of angle 1 is 40 degrees. What is the angle measure of angle 2?
40 degrees. Vertical angles are angles that are opposite to each other. They have the same angle measure.
Angles A, B and C are the interior angles of a triangle. Their measures are 30 degrees, 27 degrees, and x degrees. What is the value of x?
x = 123
The interior angles of a triangle add up to 180 degrees. x = 180 - 30 - 27 = 123.
What is the sum of the interior angles of a pentagon?
540 degrees.
We use (n-2)*180, where n is the number of sides. In this case, we use (5-2)*180 = 3*180 = 540 degrees.
A right triangle has legs with lengths 4 units and 3 units. What is the length of the hypotenuse?
5 units.
Use Pythagorean Theorem to find the length of the hypotenuse. a^2 + b^2 = c^2.
4^2 + 3^2 = c^2
16 + 9 = c^2
25 = c^2
5 = c
Can three sides with lengths 1, 9, and 11 make a right triangle?
Nope.
1 + 9 = 10, which is not greater than 11. The Triangle Inequality Theorem does not hold for these three sides.
Angle 3 and Angle 4 are complementary angles. The measure of angle 4 is 53 degrees. What is the measure of angle 3?
37 degrees.
Complementary angles are angles that add up to 90 degrees. If the measure of angle 4 is 53 degrees, then the measure of angle 3 is 90 - 53 = 37 degrees.
68 degrees.
The Exterior Angle Theorem says that the sum of the remote interior angles equals the measure of the exterior angle. So 23 + 45 = Z. 68 = Z.
What is the sum of the interior angles of a decagon?
1440 degrees.
(n-2)*180 ---> (10-2)*180 = 8*180 = 1440.
A triangle has sides of lengths 6, 3, and square root of 45. Is it a right triangle?
Yes.
a^2 + b^2 = c^2.
Make 3 and 6 a and b. Make square root of 45 the value of c. Then verify that both sides are equal.
Can three sides with lengths 4.3, 1.7, and 6 make a triangle?
NAH.
4.3+1.7=6. Since the left side is equal to the right side instead of greater than it, this is not a triangle.
Angles 1, 2, and 3 are supplementary angles. The measure of angle 1 is 20 degrees. The measure of angle 2 is 36 degrees. What is the measure of angle 3?
124 degrees.
Supplementary angles are angles that add up to 180 degrees. 180 - 36 - 20 = 124.
Angles Q and R are the remote interior angles of a triangle. Angle S is the exterior angle of a triangle. Angle Q is 33 degrees and Angle S is 103 degrees. What is the measure of Angle R?
70 degrees.
We know that the sum of the remote interior angles equals the measure of the exterior angle.
So 33 + R = 103. This means that R = 103-33 = 70.
What is the value of each angle in a hexagon?
120 degrees.
(n-2)*180 ---> (6-2)*180 = 4*180 = 720 degrees. Then divide by 6 to get the measure of one angle. 720 divided by 6 is 120.
Can a shelf with side lengths of 6, 8, and 10 fit in the corner of a rectangular room? Why/why not?
Yes, because the sides of the shelf form a right triangle.
Which of the following are the possible lengths to complete a triangle with side lengths of 10 feet and 7 feet?
The third side can be between 3 feet and 17 feet.
To solve this, set up the triangle inequality theorem. We know that the sum of any two sides must be bigger than the third side.
So 10 + 7 > x. That means x is less than 17.
Also 7 + x > 10, so x is greater than 3.
21 degrees.
Complementary angles add up to 90 degrees, so 3x + 27 = 90. Solve the equation by subtracting 27 on both sides to get 3x = 63, and then divide by 3 on both sides to get x = 21.
Angles X, Y, and Z are the interior angles of a triangle. Angle X is 5x degrees, Angle Y is 44 degrees, and Angle Z is 3x degrees. What is the value of x?
x = 17.
We know that the sum of the interior angles is 180 degrees. So 5x + 3x + 44 = 180. So 8x + 44 = 180. So 8x = 136, so x = 138/8 = 17.
1080 degrees.
(8-2)*180 = 6*180 = 1080
The top of a ladder rests at a height of 12 feet against the side of a house. If the base of the ladder is 5 feet from the house, what is the length of the ladder?
13 feet.
The length of the ladder is the hypotenuse of a right triangle. Treat 12 and 5 as the legs and then solve for the hypotenuse using Pythagorean Theorem.
Which of the following are the possible lengths to complete a triangle with side lengths of 11 feet and 20 feet?
9 < x < 31
We know that the sum of any two sides must be greater than the length of the third side.
11 + 20 > x, which means 31 > x (x is less than 31).
Also, 11 + x > 20, so x > 9.
Angle X and Angle Y are vertical angles. The measure of angle x is 3x + 40 degrees. The measure of angle y is 13x degrees. What is the value of x?
x = 4.
X and Y are vertical angles, so their measures are the same. We can set up an equation to represent this. 3x + 40 = 13x is the equation. Then subtract 3x on each side to get 40 = 10x. Divide by 10 on both sides to get x = 4.
Angles E and F are remote interior angles of a triangle. Angle G is the exterior angle of a triangle. The measure of Angle E is 2x degrees and the measure of Angle F is 6x degrees. The measure of Angle G is 64 degrees. What is the value of x?
x = 8.
2x + 6x = 64. So 8x = 64. So x = 64/8 = 8.
What is the measure of each angle in a 14-sided shape?
154.38 degrees.
(n-2)*180 --> (14-2)*180 = 12*180 = 2160. Then divide 2160 by 14 to get 154.28 degrees.
The middle school is located at (0, 0) on a coordinate plane. Town Hall is located 3 miles directly east of the middle school. The fire station is located 5 miles directly north of Town Hall. How many miles long is a straight line between the school and the fire station? Round to the nearest tenth.
Square root of 34.
Treat 3 and 5 as the legs of the triangle, and then solve for the length of the hypotenuse.