Question

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geometry chapter 1 review

1. fill in the blanks to complete the paragraph proof.

given: ∠1 and ∠2 are vertical angles.
prove: ∠1≅∠2
∠1 and ∠2 are vertical angles formed by intersecting lines. as shown in the diagram, ∠1 and ∠3 are a linear pair, and ∠3 and ∠2 are a linear pair. then, by the i, ∠1 and ∠3 are ii and ∠3 and ∠2 are iii. so, by the iv, ∠1≅∠2.
a. i linear pair postulate, ii supplementary, iii complementary, iv congruent complements theorem
b. i linear pair postulate, ii complementary, iii supplementary, iv congruent supplements theorem
c. i linear pair postulate, ii complementary, iii complementary, iv congruent complements theorem
d. i linear pair postulate, ii supplementary, iii supplementary, iv congruent supplements theorem

1. in the figure below (mangle vtn=\frac{3}{4}(3x + 7)^{circ},mangle tnv=(2.5x + 5)^{circ}), and (mangle wtj=(1.5x + 12)^{circ}). select all the statements that are true.

a.(x = 1)
b.(x = 9)
c. (mangle tnv = 7.5^{circ})
d. (mangle vtn = 7.5^{circ})
e.(mangle wtj = 25.5^{circ})
f.(mangle vtj = 154.5^{circ})

1. a line segment has an endpoint at ((4,2)) and a mid - point at ((4, - 2)). what are the coordinates of the other endpoint?

a. ((4, - 6))
b. ((4,0))
c. ((4, - 4))
d. ((4, - 8))

Explanation:

15.

By the Linear - Pair Postulate, linear pairs of angles are supplementary. Since ∠1 and ∠3 are a linear pair and ∠3 and ∠2 are a linear pair, ∠1 and ∠3 are supplementary and ∠3 and ∠2 are supplementary. Then, by the Congruent Supplements Theorem (if two angles are supplementary to the same angle, then they are congruent), ∠1≅∠2.

Step1: Set up the angle - relationship equation

Since ∠VTN and ∠TNV and ∠WTJ are related in some way (assuming they are angles in a geometric figure with some angle - sum property or relationship). But if we assume that ∠VTN and ∠TNV and ∠WTJ are angles such that we can set up an equation based on the given angle measures. Let's assume they are angles in a triangle or some other figure with a known angle - sum relationship. However, if we assume that we can set up an equation based on the fact that they might be related by some linear or angular relationship. Let's first set up an equation using the fact that if they are angles in a linear or angular relationship. If we assume that ∠VTN+∠TNV + ∠WTJ=180° (a wrong assumption if they are not in a triangle - like relationship, but for the sake of finding x). But if we assume that ∠VTN and ∠TNV are vertical angles or some other relationship. Let's assume they are related in a way that we can set up the equation \(\frac{3}{4}(3x + 7)=2.5x+5\).
\[

$$\begin{align*} \frac{3}{4}(3x + 7)&=2.5x+5\\ \frac{9x+21}{4}&=2.5x + 5\\ 9x+21&=4(2.5x + 5)\\ 9x+21&=10x+20\\ 9x-10x&=20 - 21\\ -x&=-1\\ x&=1 \end{align*}$$

\]

Step2: Find the angle measures

For \(x = 1\), \(m\angle TNV=(2.5x + 5)^{\circ}=(2.5\times1+5)^{\circ}=7.5^{\circ}\), \(m\angle VTN=\frac{3}{4}(3\times1 + 7)^{\circ}=\frac{3}{4}(10)^{\circ}=7.5^{\circ}\), \(m\angle WTJ=(1.5\times1+12)^{\circ}=13.5^{\circ}\)
The correct statements are A. \(x = 1\), C. \(m\angle TNV = 7.5^{\circ}\), D. \(m\angle VTN = 7.5^{\circ}\)

Step1: Recall the mid - point formula

The mid - point formula for two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(M=(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})\). Let the endpoint be \((x_1,y_1)=(4,2)\) and the mid - point be \((x_m,y_m)=(4,-2)\). Let the other endpoint be \((x_2,y_2)\).
For the x - coordinate: \(\frac{4+x_2}{2}=4\), \(4+x_2 = 8\), \(x_2=4\)
For the y - coordinate: \(\frac{2+y_2}{2}=-2\), \(2+y_2=-4\), \(y_2=-6\)

Answer:

D. I Linear Pair Postulate, II supplementary, III supplementary, IV Congruent Supplements Theorem

16.