Select the correct answer from each drop-down menu.
Given:
is complementary to
Prove:
Two lines intersect each other. Another horizontal line bisects to create angles 1,2 and 3, 4
Complete the proof in paragraph form.
Given angle 1 is complementary to angle 2 is pointing to m angle 1 plus m angle 2 equals 90 degrees pointing to the blank box and another blank box. And angle 4 congruent angle 1 also pointing m angle 4 equals m angle 1 pointing down 40 degrees
Complete the proof in paragraph form.
By the definition of complementary angles, since
is complementary to
,
. By the vertical angles theorem,
, and
by the definition of congruence. Combined with the given equation,
, the substitution property of equality means that
. Using the
,
. Finally, using the
,
.
By the definition of complementary angles, since angle 1 is complementary to angle 2, the sum of their measures is equal to 90 degrees. This can be written as m angle 1 + m angle 2 = 90 degrees.
By the vertical angles theorem, angle 1 and angle 3 are vertical angles and are therefore congruent. Similarly, angle 2 and angle 4 are also vertical angles and are congruent.
Given that angle 4 is congruent to angle 1, we can replace m angle 4 with m angle 1 in the equation.
So, m angle 1 = m angle 1, and substituting this into the equation, we have m angle 1 + m angle 1 = 90 degrees.
Using the addition property of equality, we can simplify this to 2m angle 1 = 90 degrees.
Finally, using the division property of equality, we can divide both sides of the equation by 2 to solve for m angle 1.
Therefore, m angle 1 = 45 degrees.
By the definition of complementary angles, since angle 1 is complementary to angle 2, we know that the measure of angle 1 plus the measure of angle 2 equals 90 degrees. This can be represented as m(angle 1) + m(angle 2) = 90 degrees. In addition, we are given that angle 4 is congruent to angle 1, meaning that they have the same measure. Thus, m(angle 4) = m(angle 1) = 40 degrees.
Using the vertical angles theorem, we can conclude that angle 2 and angle 4 are congruent. This can be represented as m(angle 2) = m(angle 4). Furthermore, since angle 4 is congruent to angle 1, we can substitute m(angle 1) in place of m(angle 4) in the equation, resulting in m(angle 2) = m(angle 1).
Combining the given equation m(angle 1) + m(angle 2) = 90 degrees with the substitution property of equality, we find that m(angle 1) + m(angle 1) = 90 degrees. Simplifying this equation, we get 2(m(angle 1)) = 90 degrees.
To solve for m(angle 1), we divide both sides of the equation by 2, giving us m(angle 1) = 45 degrees. Therefore, the measure of angle 1 is 45 degrees.
Finally, using the angle bisector theorem, we can conclude that the angles created by the bisecting horizontal line are congruent. Thus, m(angle 3) = m(angle 1) and m(angle 4) = m(angle 1) = 45 degrees.
To complete the proof in paragraph form, we start by using the given information that angle 1 is complementary to angle 2, which means the sum of their measures is 90 degrees. This can be expressed as m(angle 1) + m(angle 2) = 90.
Next, we can apply the vertical angles theorem, which states that when two lines intersect, the vertical angles formed are congruent. Therefore, angle 3 and angle 1 are congruent, and angle 4 and angle 2 are congruent.
Since we know that angle 4 is congruent to angle 1, we can substitute m(angle 4) = m(angle 1) = 40 degrees.
By the substitution property of equality, we can substitute m(angle 1) + m(angle 2) = 90 into m(angle 4) = m(angle 1), giving us m(angle 4) + m(angle 2) = 90.
Using the transitive property, we can combine the given equation m(angle 4) = 40 with m(angle 4) + m(angle 2) = 90 to get 40 + m(angle 2) = 90.
Finally, by subtracting 40 from both sides of the equation, we find that m(angle 2) = 50 degrees. This completes the proof by showing that angle 2 measures 50 degrees.