Practice Problems
Prashanth Gowda
Question- What is the measure of <ACB? (Figure not drawn to scale)
Prashanth Gowda
Question- What is the measure of <ACB? (Figure not drawn to scale)
Answer- 132º
Work- <BAC + <ABC = <ACD
x + 5 + 12x + 17 = 18x + 12
13x + 22 = 18x + 12
5x = 10
x = 2
<BAC + <ABC + <ACB= 180º
x + 5 + 12x + 17 + < ACB = 180º
13x + 22 + <ACB = 180º
13(2) + 22 + <ACB = 180º
26 +22 + <ACB = 180º
48 + <ACB = 180º
<ACB = 132º
Explanation- The first thing I realized was that <BAC and <ABC add up to equal <ACD. I knew this because of the Exterior Angles Theorem. This theorem states the measure of an exterior angle of a triangle is equal to the sum of the measures of its remote interior angles. Next I plugged the expressions into equations and started solving for x. I got x to be equal to 2. Next I made the equation <BAC + <ABC + ACB = 180º. This is true because of the Triangle Sum Theorem. This theorem states that the angles of a triangle add up to 180º. After that, I plugged in the expressions into the equation and solved for m<ACB. Finally after simplifying the equation, I got <ACB to be equal to 132º.
Question- What is the length of Segments DE?
Work- <BAC + <ABC = <ACD
x + 5 + 12x + 17 = 18x + 12
13x + 22 = 18x + 12
5x = 10
x = 2
<BAC + <ABC + <ACB= 180º
x + 5 + 12x + 17 + < ACB = 180º
13x + 22 + <ACB = 180º
13(2) + 22 + <ACB = 180º
26 +22 + <ACB = 180º
48 + <ACB = 180º
<ACB = 132º
Explanation- The first thing I realized was that <BAC and <ABC add up to equal <ACD. I knew this because of the Exterior Angles Theorem. This theorem states the measure of an exterior angle of a triangle is equal to the sum of the measures of its remote interior angles. Next I plugged the expressions into equations and started solving for x. I got x to be equal to 2. Next I made the equation <BAC + <ABC + ACB = 180º. This is true because of the Triangle Sum Theorem. This theorem states that the angles of a triangle add up to 180º. After that, I plugged in the expressions into the equation and solved for m<ACB. Finally after simplifying the equation, I got <ACB to be equal to 132º.
Question- What is the length of Segments DE?
Answer- 74
Work- 2(DE) = BC
2(13x + 35)= 35x + 43
26x + 70 = 35x + 43
9x = 27
x = 3
Segment DE = 13(3) + 35
Segment DE = 36 + 35
Segment DE = 74
Explanation- First I knew that the length of Segment DE is half the length of Segment BC because of the Midsegment Theorem. This theorem states that a midsegment of a triangle is parallel to a side of the triangle, and its length is half the length of that side. That's why I set up the equation 2(DE) = BC. After setting up the equation, I solved for x. I got x to be equal to 3. Then I solved for the length of segment DE. Finally I got Segment DE to be equal to 74.
Work- 2(DE) = BC
2(13x + 35)= 35x + 43
26x + 70 = 35x + 43
9x = 27
x = 3
Segment DE = 13(3) + 35
Segment DE = 36 + 35
Segment DE = 74
Explanation- First I knew that the length of Segment DE is half the length of Segment BC because of the Midsegment Theorem. This theorem states that a midsegment of a triangle is parallel to a side of the triangle, and its length is half the length of that side. That's why I set up the equation 2(DE) = BC. After setting up the equation, I solved for x. I got x to be equal to 3. Then I solved for the length of segment DE. Finally I got Segment DE to be equal to 74.
Patrick's Practice Problems
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Pavan's Practice Problems
Problem 1
Question: In the diagram below, <W is the exterior angle of Triangle ZXY. If the m<Z=20 degrees and the m<X=60 degrees, what is the m<W? Keep in mind that the diagram is not draw to scale.
Answer and Explanation: To solve this problem, I first made a plan. I would find the m<Y, and then I would find the sum of the m<X and the m<Y to find the m<W. After I came up with this plan, I executed it and took the first step. So, the question tells us that the m<Z=20 degrees and the m<X=60. With this information given to us, we can find the m<Y using the Triangle Angle Sum Theorem, and formulating that m<Y=180-(m<Z+m<X). With the Substitution Property, we can further break down this equation into: m<Y=180-(20+60). Now, from this point, we would simply simplify the equation by solving. In the end, I got the m<Y as 100 degrees. Next, after completing the 1st step in our plan, we move on the 2nd and final step: finding the m<W. As we have learned in Unit 3, the Exterior Angle Theorem states that the measure of an exterior angle of a triangle is equal to the sum of the measures of its remote interior angles. The problem tells us that <W is the exterior angle, and based on the diagram, we can state that <X and <Y are the interior angles of the triangle shown above. So now, we can conclude that the m<W=m<X+m<Y. And once again, through substitution, we can further break down this equation to get: m<W=60+100. Now, from this point, we can proceed to find our answer by simplifying or basically solving this equation. Your final answer, after all calculations should be: m<W=160 degrees.
Problem 2
Question: Answer the following question based on the diagram below. Given that the m<A is 60 degrees, the m<B is 60 degrees, and the length of Segment AB is 10 cm, what is the length of Line Segment CB?
Answer and Explanation: To solve this problem, I first made a plan. I would find the m<C, and then use any postulates or theorems that I have learned to find determine the length of Line Segment BC. After I came up with this plan, I executed it and took the first step. So, the question tells us that the m<A=60 and the m<B=60. With this information given to us, we can find the m<C using the Triangle Angle Sum Theorem, and formulate that the m<C=180-(m<A+m<B). With the Substitution Property, we can further break down this equation into: m<C=180-(60+60). Now, from this point, we would simply simplify the equation by solving. In the end, I got the m<C as 60 degrees. Next, after completing the first step in our plan, we move on to the 2nd and final step: using any postulates or theorems that I have learned to find determine the length of Line Segment BC. As we have learned in Unit 3, a triangle is considered equiangular if all 3 of its interior angles are congruent. So with the information we gathered so far in this problem, we can state that Triangle ABC is an equiangular triangle. Then, using the Equiangular Triangle Theorem, we can also assume that Triangle ABC is considered equilateral. If a triangle is equilateral, that means that all 3 of its sides are congruent. The problem gives us the length of 1 side, Line Segment AB, as 10 units, so that means that all 3 of Triangle ABC's sides are 10 units long, which leads us to our answer. The length of Line Segment CB is 10 units.
Meredith Derucki's Practice Problems
Question: Looking at the triangle below, find the perimeter.
Explanation:
First you can see that the triangle has all angles congruent, therefore making it an equilateral triangle. Equilateral triangles have congruent sides. After that you need to find the value of X. You can do 2x+6=5x. After that you can either move 2x or 5x. In this case, move 2x. Your equation is now 3x=6. Then you divide everything by 3. You end up with x=2. After this you plug X back into any side of the triangle, because they are all congruent. Each side ends up being 10. If a triangle has 3 sides, then the perimeter is 30.
Answer: Perimeter is 30
First you can see that the triangle has all angles congruent, therefore making it an equilateral triangle. Equilateral triangles have congruent sides. After that you need to find the value of X. You can do 2x+6=5x. After that you can either move 2x or 5x. In this case, move 2x. Your equation is now 3x=6. Then you divide everything by 3. You end up with x=2. After this you plug X back into any side of the triangle, because they are all congruent. Each side ends up being 10. If a triangle has 3 sides, then the perimeter is 30.
Answer: Perimeter is 30
Question 2: Looking at the triangles below, find the length of the hypotenuse of the triangles. Then find the total perimeter. If necessary round the hypotenuse and perimeter to the nearest whole number.
Explanation:
You can conclude that these triangles are congruent due to SSS congruence. Therefore the side lengths are 12 and 32. You can also tell that the triangle is a right triangle due to the angle shown. After this you need to find the length of the hypotenuse which would be 12^2+32^2=c^2. When you find this it is 144+1024= sq. root of C. That ends up to be 34.17601496127012. Rounding this to the nearest whole number. The length of the hypotenuse is 34. To find the perimeter, add 34+12+32. The perimeter ends up being 76.
Answer: Hypotenuse- 34. Perimeter- 76.
You can conclude that these triangles are congruent due to SSS congruence. Therefore the side lengths are 12 and 32. You can also tell that the triangle is a right triangle due to the angle shown. After this you need to find the length of the hypotenuse which would be 12^2+32^2=c^2. When you find this it is 144+1024= sq. root of C. That ends up to be 34.17601496127012. Rounding this to the nearest whole number. The length of the hypotenuse is 34. To find the perimeter, add 34+12+32. The perimeter ends up being 76.
Answer: Hypotenuse- 34. Perimeter- 76.