Mathematics, 05.10.2019 02:30 susannaking5742

*50 pointsurgentplz **
in the figure shown, the angles formed when two parallel lines are cut by a transversal are labeled 1 through 8.

question:

a. prove that any two vertical angles are congruent.

b. one of the two angles from part a has a corresponding alternate interior angle. use the corresponding angles postulate, which says that corresponding angles are congruent, to prove that the angle from part a and its alternate interior angle are congruent.

c. how can you use the proof from part a and the proof from part b to prove that alternate exterior angles are congruent?

*50 points**urgent**plz ** in the figure shown, the angles formed when two parallel lines are

Answers: 3

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Answer from: serenityarts123

See explanation

Step-by-step explanation:

A. Prove that angles 1 and 3 are vertical.

1. $\angle 1, \angle 2$ - linear pair of angles

2. $m\angle 1+m\angle 2 =180^{\circ}$ - linear pair of angles are supplementary

3. $\angle 2, \angle 3$ - linear pair of angles

4. $m\angle 2+m\angle 3 =180^{\circ}$ - linear pair of angles are supplementary

5. $m\angle 1+m\angle 3 =m\angle 2+m\angle 3$ - substitution property of equality

6. $m\angle 1 =m\angle 3$ - subtraction property of equality

7. $\angle 1\cong \angle 3$ - definition of congruent angles.

So, any two vertical angles are congruent.

B. Prove that angle 3 and angle 5 are congruent.

1. $\angle 1\cong \angle 5$ - Corresponding Angles Postulate

2. $m\angle 1=m\angle 5$ - definition of congruent angles

3. $\angle 1\cong \angle 3$ - vertical angles

4. $m\angle 1=m\angle 3$ - definition of congruent angles

5. $m\angle 3=m\angle 5$ - substitution property of equality

6. $\angle 3\cong \angle 5$ - definition of congruent angles

So, alternate interior angles are congruent

C. Prove that angle 1 and angle 7 are congruent.

1. $\angle 1\cong \angle 5$ - Corresponding Angles Postulate

2. $m\angle 1=m\angle 5$ - definition of congruent angles

3. $\angle 5\cong \angle 7$ - vertical angles

4. $m\angle 5=m\angle 7$ - definition of congruent angles

5. $m\angle 1=m\angle 7$ - substitution property of equality

6. $\angle 1\cong \angle 7$ - definition of congruent angles

So, alternate exterior angles are congruent

Answer from: Quest

5. y=6; de=7; fg=8

6. x=4.2; jl=14.2; mn=13

step-by-step explanation:

when chords cross inside a circle, the product of the segment lengths is a constant.

5. dh×eh = fh×gh

y = gh = dh×eh/fh = 3×4/2 . . . divide by the coefficient of gh

y = 6

of course, the total chord length is the sum of the lengths of its segments.

de = dh +eh = 3+4 = 7

fg = fh +gh = 2+6 = 8

6. this problem works the same way as the previous one.

x = lk = mk×nk/jk = 6×7/10 = 4.2

jl = jk +lk = 10 +4.2 = 14.2

mn = mk +nk = 6 +7 = 13

comment on chords and secants

you can think of the points h (problem 5) and k (problem 6) as points where the chords meet. a chord is part of a secant line, a line that intersects the circle in two points. one can also consider these points (h or k) to be the points where the respective secant lines meet.

then, the product that is a constant is the product of the distance from the meeting point (h or k) to one circle intersection with the distance from that meeting point to the other circle intersection. for problem 5, the constant is the product hd×he = hf×hg.

it turns out that this rule regarding the products of lengths to points of intersection is also true when the secants intersect outside the circle.

that is, you only need to remember one rule to work both kinds of problems.

Find the value of the variable and the length of each chord for #5 and #6.

Answer from: Quest

A: 18/15b : 7 - -8 = 15 , 12 - -6 = 18 the y variable always goes over the x variables, c : same thing a

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in the figure shown, the angles formed when two parallel lines are...

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