Given: <2 = <4, m<2 = 110
Proof: m<3 = 70
<2 = <4, m<2 = 110 Given
m<2 = m<4 Definition of congruent angles
m<4 = 110 b._______
<3 and <4 are a linear pair Definition of a linear pair(shown in diagram)
<3 and <4 are supplementary Linear Pair postulate
m<3 + m<4 = 180 c.________
m<3 = 70 d._______
I know the answers are a. Transitive
b. Definition of supplementary angles
c. Subtraction property of equality
but why? explain please!!
a. Transitive Property: This property states that if a = b and b = c, then a = c. In this case, <2 = <4 and m<2 = m<4, so <2 = m<4.
b. Definition of Supplementary Angles: Supplementary angles are two angles whose measures add up to 180 degrees. Since <3 and <4 are a linear pair, they are supplementary angles.
c. Subtraction Property of Equality: This property states that if a = b, then a - c = b - c. In this case, m<3 + m<4 = 180, so m<3 + 110 = 180, so m<3 = 70.
a. Transitive: The transitive property states that if two angles are congruent to a third angle, then they are congruent to each other. In this case, we know that <2 is congruent to <4, and since m<2 = 110, it follows that m<4 = 110 as well. Therefore, this is an application of the transitive property.
b. Definition of supplementary angles: Two angles are supplementary if their measures add up to 180 degrees. In this case, <3 and <4 are said to be a linear pair, which means they form a straight line and therefore their measures add up to 180 degrees. This is the definition of supplementary angles.
c. Subtraction property of equality: The subtraction property of equality states that if two quantities are equal, then subtracting or adding the same quantity to both sides will still result in equality. In this case, we know that m<3 + m<4 = 180, and we want to find m<3. Subtracting m<4 from both sides of the equation gives us m<3 = 180 - m<4. And since we know that m<4 = 110, substituting this in yields m<3 = 180 - 110, which simplifies to m<3 = 70. Thus, this is an application of the subtraction property of equality.
a. The reason why statement b is true is because of the transitive property. The transitive property states that if A = B and B = C, then A = C. In this case, since <2 = <4 and m<2 = 110, we can use the transitive property to conclude that m<4 = 110.
b. The reason why statement b is true is because of the definition of supplementary angles. Supplementary angles are two angles that add up to 180 degrees. In this case, since <3 and <4 are a linear pair, they form a straight line and therefore add up to 180 degrees.
c. The reason why statement c is true is because of the addition property of equality. The addition property of equality states that if A = B, then A + C = B + C. In this case, since m<3 + m<4 = 180 (according to the definition of supplementary angles), we can add m<4 to both sides of the equation to get m<3 + m<4 + m<4 = 180 + m<4. Simplifying this gives us m<3 + 2m<4 = 180 + m<4. Since m<4 = 110 (according to statement b), we can substitute this value in to get m<3 + 2(110) = 180 + 110. By solving this equation, we can find the value of m<3.
d. The reason why statement d is true is because of the subtraction property of equality. The subtraction property of equality states that if A = B, then A - C = B - C. In this case, we have already solved the equation m<3 + 2(110) = 180 + 110 and found that m<3 + 220 = 290. By subtracting 220 from both sides of the equation, we get m<3 + 220 - 220 = 290 - 220, which simplifies to m<3 = 70.
a. Transitive: The given statement <2 = <4, along with the fact that m<2 = 110, can be used to establish a relationship between <2 and <4, which is known as congruency. When two angles are congruent, their measures are equal. Therefore, since m<2 = 110 and <2 = <4, it follows that m<4 = 110. Transitivity is applied here because we are establishing the equality of angles by relating them through another equal angle.
b. Definition of supplementary angles: In the given proof, it is stated that <3 and <4 are a linear pair. A linear pair consists of two adjacent angles formed by intersecting lines, and these angles are always supplementary, meaning their measures add up to 180 degrees. So, the fact that <3 and <4 are a linear pair allows us to conclude that they are supplementary.
c. Subtraction property of equality: The proof utilizes the concept that the sum of the measures of two supplementary angles is 180 degrees. By the Linear Pair postulate, we know that m<3 + m<4 = 180. Since we have the value of m<4 (which is 110), we can substitute it into the equation: m<3 + 110 = 180. To solve for m<3, we need to isolate it by subtracting 110 from both sides, resulting in m<3 = 70. This is an application of the subtraction property of equality, which allows us to subtract the same value from both sides of the equation without changing its validity.