Question

1. tell whether each statement about the diagram is true or false.

a. ∠fgd is an exterior angle of △dhg.
b. m∠efd + m∠hgd = m∠fdg
c. m∠dfh + m∠fdh = m∠ghm
d. m∠gdh + m∠dhg = m∠egl
e. m∠dhj + m∠dge + m∠bdh = 360°

1. what is the value of x? show your work.
2. a. what is m∠qtr? show your work.

b. what is m∠ptr? show your work.

Explanation:

4.

a.

An exterior - angle of a triangle is an angle formed by one side of the triangle and the extension of an adjacent side. In \(\triangle DHG\), \(\angle FGD\) is an exterior angle. So, this statement is True.

b.

The exterior - angle of a triangle is equal to the sum of the two non - adjacent interior angles. In a triangle related to these angles, there is no such relationship as \(m\angle EFD + m\angle HGD=m\angle FDG\). So, this statement is False.

c.

There is no triangle relationship that would make \(m\angle DFH + m\angle FDH=m\angle GHM\). So, this statement is False.

d.

There is no triangle relationship that would make \(m\angle GDH + m\angle DHG=m\angle EGL\). So, this statement is False.

e.

\(\angle DHJ\), \(\angle DGE\), and \(\angle BDH\) are angles around a point (the intersection of the lines). The sum of angles around a point is \(360^{\circ}\). So, this statement is True.

5.

The sum of the exterior angles of a triangle is \(360^{\circ}\).

Step1: Set up the equation

We have \(20x + 8x+17x = 360\).

Step2: Combine like terms

\(45x=360\).

Step3: Solve for \(x\)

\(x=\frac{360}{45}=8\).

6.

a.

First, find the third angle of the triangle with angles \(74^{\circ}\) and \(110^{\circ}\)'s non - adjacent interior angle.

Step1: Use the exterior - angle property

Let the third angle of the smaller triangle be \(y\). The exterior angle of \(110^{\circ}\) gives \(y + 74^{\circ}=110^{\circ}\), so \(y = 110 - 74=36^{\circ}\).
In the larger triangle with \(\angle QTR\), using the angle - sum property of a triangle, we know that \(\angle QTR=180-(48 + 36)\).

Step2: Calculate \(\angle QTR\)

\(\angle QTR = 96^{\circ}\).

b.

\(\angle PTR\) and the \(110^{\circ}\) angle are supplementary.

Step1: Use the supplementary - angle property

\(m\angle PTR=180 - 110=70^{\circ}\).

Answer:

4.
a. True
b. False
c. False
d. False
e. True

1. \(x = 8\)

6.
a. \(m\angle QTR = 96^{\circ}\)
b. \(m\angle PTR = 70^{\circ}\)