Vinny's Practice Problems:
Question 1: Three roads make a triangle. Two of the roads are exactly the length of two sides of the triangle, but the third road extends further, past one of the vertices. For a school project, Timmy needs to find out the measures of all the angles the roads make. He is given this information: the exterior angle of the triangle is 141 degrees, and the vertex of the angle that is further north is 36 degrees. What are the measures of every single angle the roads make?
Question 1: Three roads make a triangle. Two of the roads are exactly the length of two sides of the triangle, but the third road extends further, past one of the vertices. For a school project, Timmy needs to find out the measures of all the angles the roads make. He is given this information: the exterior angle of the triangle is 141 degrees, and the vertex of the angle that is further north is 36 degrees. What are the measures of every single angle the roads make?
Answer: Exterior angle: 141 degrees
"Top" angle of the triangle: 36 degrees
"Bottom left" angle of the triangle: 105 degrees
"Bottom right" angle of the triangle: 39 degrees
Explanation: First of all, half of the answer is given in the problem. So, you already know that the exterior angle equals 141 degrees, and the "top" angle of the triangle is 36 degrees. Now, using the Exterior Angle Theorem, you can find out the measures of the other two angles. The Exterior Angle Theorem states that an exterior angle of a triangle is equal to the sum of the two interior angles not adjacent to it. So, using x as the measure of the unknown "bottom left" angle, the new equation would be 36+x = 141. Simplifying, with the Subtraction POE, x would be equal to 105 degrees. The final angle left is the third angle of the triangle. To find this angle, you could use the equation 105+36+x = 180 degrees, as the interior angles of a triangle always add up to 180 degrees. That would lead to 141+x = 80 degrees, and x would equal 39 degrees.
Question 2: There is an equilateral triangle. One of its midsegments cuts one of its sides into two segments. One of these segments equals x+3. If the triangle's perimeter is 139, what is the length of the midsegment?
Answer: 23 1/6 units
Explanation: Since it is a midsegment, the other segment it cuts the side of the triangle into is also x+3. This means that the entire side of the triangle, adding x+3+x+3, equals 2x+6. Since the triangle is equilateral, all sides equal 2x+6. So, 3(2x+6) = 139. Using the Distributive Property, 6x+18 = 139. From this, 6x = 121, and x = 20 1/6. Now, the side length of the triangle is 2x+6, but since the midsegment is exactly half of that, you can just plug x in to the equation x+3 to get the length of the midsegment. So, 20 1/6+3 equals 23 1/6, and that is how long the midsegment of the triangle is.
"Top" angle of the triangle: 36 degrees
"Bottom left" angle of the triangle: 105 degrees
"Bottom right" angle of the triangle: 39 degrees
Explanation: First of all, half of the answer is given in the problem. So, you already know that the exterior angle equals 141 degrees, and the "top" angle of the triangle is 36 degrees. Now, using the Exterior Angle Theorem, you can find out the measures of the other two angles. The Exterior Angle Theorem states that an exterior angle of a triangle is equal to the sum of the two interior angles not adjacent to it. So, using x as the measure of the unknown "bottom left" angle, the new equation would be 36+x = 141. Simplifying, with the Subtraction POE, x would be equal to 105 degrees. The final angle left is the third angle of the triangle. To find this angle, you could use the equation 105+36+x = 180 degrees, as the interior angles of a triangle always add up to 180 degrees. That would lead to 141+x = 80 degrees, and x would equal 39 degrees.
Question 2: There is an equilateral triangle. One of its midsegments cuts one of its sides into two segments. One of these segments equals x+3. If the triangle's perimeter is 139, what is the length of the midsegment?
Answer: 23 1/6 units
Explanation: Since it is a midsegment, the other segment it cuts the side of the triangle into is also x+3. This means that the entire side of the triangle, adding x+3+x+3, equals 2x+6. Since the triangle is equilateral, all sides equal 2x+6. So, 3(2x+6) = 139. Using the Distributive Property, 6x+18 = 139. From this, 6x = 121, and x = 20 1/6. Now, the side length of the triangle is 2x+6, but since the midsegment is exactly half of that, you can just plug x in to the equation x+3 to get the length of the midsegment. So, 20 1/6+3 equals 23 1/6, and that is how long the midsegment of the triangle is.
Aniket's Problems
https://docs.google.com/a/k12.friscoisd.org/presentation/d/1LdRsFk-5NMt2JtM9LlITUj-LQRmwX0uYSGL-_DYnEOQ/edit#slide=id.p
The link above is the powerpoint presentation of my practice problems.
The link above is the powerpoint presentation of my practice problems.
Brice's problem
Triangle ABC has the measure of the angles as A=6x+2, B=12x-13, C=11x+17
Solve for X
Triangle sum theorem
measure of angle A+ measure of angle B+ measure of angle C=180 degrees
6x+2+12x-13+11x+17=180
29x+6=180
29x=174
x=6
Solve for X
Triangle sum theorem
measure of angle A+ measure of angle B+ measure of angle C=180 degrees
6x+2+12x-13+11x+17=180
29x+6=180
29x=174
x=6
Dhruv's Problems
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