Question

geometry unit 4 test

1. in the diagram below, \\( \overline{rs} \\) and \\( \overline{rt} \\) intersect \\( \overline{de} \\) at points t and s.

which of these statements is true about \\( \triangle rst \\)?
a \\( \triangle rst \\) is an isosceles triangle
b. \\( \triangle rst \\) is a scalene triangle
c. \\( \triangle rst \\) is a right triangle
d. \\( \triangle rst \\) is an equilateral triangle

1. given: \\( \triangle abc \\) with \\( m\angle a=(7x)^{circ} \\), \\( m\angle b=(6x)^{circ} \\), and \\( m\angle c=(2x)^{circ} \\). find the \\( m\angle a \\).

a. \\( m\angle a = 15^{circ} \\)
b. \\( m\angle a = 24^{circ} \\)
c. \\( m\angle a = 72^{circ} \\)
d. \\( m\angle a = 84^{circ} \\)

1. given the following triangle, what is the value of x?

Explanation:

Step1: Find the angles of \(\triangle RST\)

We know that the sum of angles on a straight - line is \(180^{\circ}\).
For \(\angle RTS\), since \(\angle DTR = 130^{\circ}\), then \(\angle RTS=180 - 130=50^{\circ}\) (linear - pair of angles).
For \(\angle RST\), since \(\angle ESR = 130^{\circ}\), then \(\angle RST = 180 - 130=50^{\circ}\) (linear - pair of angles).

Step2: Use the angle - sum property of a triangle

In \(\triangle RST\), by the angle - sum property of a triangle (\(\angle R+\angle RTS+\angle RST = 180^{\circ}\)). Let \(\angle R=x\). Then \(x + 50+50=180\).

$$x=180-(50 + 50)=80^{\circ}$$

Since \(\angle RTS=\angle RST = 50^{\circ}\), two angles of \(\triangle RST\) are equal.

Step3: Solve for \(x\) in \(\triangle ABC\)

By the angle - sum property of a triangle (\(\angle A+\angle B+\angle C=180^{\circ}\)).
Given \(\angle A=(7x)^{\circ}\), \(\angle B=(6x)^{\circ}\), \(\angle C=(2x)^{\circ}\).

$$7x+6x + 2x=180$$

$$15x=180$$

$$x = 12$$

Then \(\angle A=7x\). Substitute \(x = 12\) into \(\angle A\), \(\angle A=7\times12 = 84^{\circ}\)

Step4: Solve for \(x\) in the equilateral triangle

Since the triangle is equilateral (all angles are equal), each angle is \(60^{\circ}\).
If one angle is \((4x + 8)^{\circ}\), then \(4x+8=60\)

$$4x=60 - 8$$

$$4x=52$$

$$x = 13$$

Answer:

1. A. \(\triangle RST\) is an isosceles triangle
2. D. \(m\angle A = 84^{\circ}\)
3. \(x = 13\)