Question

1. what property of equality is illustrated by the statement? (g1g) if ab = cd, then cd = ab. 18) using the transitive property of congruence, what is the completion of the statement? (g1g) if x + 2 = 7 and 7 = y + 1, then _. 19) which of the following represents a linear - pair? (g1d) a. diagram of angles b. diagram of angles c. diagram of angles d. diagram of angles in problems 20 and 21, use the following diagram. diagram of intersecting lines 20) what is the relationship between ∠1 and ∠3? (g6a) a. they are vertical angles. b. they are a linear pair. c. they are supplementary angles. 21) if m∠2 = 128°, what is m∠1? (g6a) 22) what is the value of x? (g6a) (4x - 13)° and (6x + 5)° shown in angles formed by intersecting lines 23) what is the value of x? (g6a) (9x + 1)° and (11x - 21)° shown in angles formed by intersecting lines 24) given a(10, - 4) and b(- 3, 7), find the distance and mid - point of segment ab. (g2b) distance formula: (d=sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}) midpoint formula: ((\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})) a. (d = sqrt{200}), (m=(\frac{7}{2},\frac{3}{2})) c. (d = 58), (m=(\frac{13}{2},\frac{11}{2})) b. (d = 290), (m=(7,3)) d. (d=sqrt{58}), (m=(13, - 11)) ***last question is on the back!***

Explanation:

Step1: For 17

The symmetric property of equality states that if \(a = b\), then \(b = a\). Here \(AB = CD\) implies \(CD = AB\).

Step2: For 18

The transitive property of congruence states that if \(a=b\) and \(b = c\), then \(a = c\).

Step3: For 19

Linear - pair angles are adjacent and their sum is \(180^{\circ}\). In option A, \(\angle1\) and \(\angle2\) are adjacent and form a straight - line.

Step4: For 20

Vertical angles are formed by two intersecting lines and are opposite each other. \(\angle1\) and \(\angle3\) are opposite each other when two lines intersect.

Step5: For 21

Vertical angles are equal. Since \(\angle2\) and \(\angle1\) are vertical angles, \(\angle1=\angle2\).

Step6: For 22

Set the two angle expressions equal since they are corresponding angles (assuming parallel lines). Then solve the resulting linear equation for \(x\).

Step7: For 23

Set the two angle expressions equal since they are corresponding angles (assuming parallel lines). Then solve the resulting linear equation for \(x\).

Step8: For 24

Use the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\) and mid - point formula \(M=(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})\) to find the distance and mid - point respectively.

Answer:

1. Symmetric Property of Equality
2. If \(x + 2=7\) and \(7=y + 1\), then \(x+2=y + 1\)
3. A. (Linear - pair angles are adjacent and supplementary, and in option A, \(\angle1\) and \(\angle2\) are adjacent and form a straight - line)
4. A. They are vertical angles.
5. Since \(\angle2\) and \(\angle1\) are vertical angles, \(\angle1=\angle2 = 128^{\circ}\)
6. If the two lines are parallel, then \((14x-13)=(6x + 53)\)

• First, subtract \(6x\) from both sides: \(14x-6x-13=6x-6x + 53\), which gives \(8x-13 = 53\)
• Then, add 13 to both sides: \(8x-13 + 13=53 + 13\), so \(8x=66\)
• Divide both sides by 8: \(x=\frac{66}{8}=\frac{33}{4}=8.25\)

1. If the two lines are parallel, then \((9x + 1)=(11x-21)\)

• Subtract \(9x\) from both sides: \(9x-9x + 1=11x-9x-21\), so \(1 = 2x-21\)
• Add 21 to both sides: \(1+21=2x-21 + 21\), then \(22 = 2x\)
• Divide both sides by 2: \(x = 11\)

1. Distance formula: \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), where \(x_1 = 10,y_1=-4,x_2=-3,y_2 = 7\)

• \(d=\sqrt{(-3 - 10)^2+(7+4)^2}=\sqrt{(-13)^2+11^2}=\sqrt{169 + 121}=\sqrt{290}\)
• Mid - point formula: \(M=(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})=(\frac{10-3}{2},\frac{-4 + 7}{2})=(\frac{7}{2},\frac{3}{2})\)