Question

which of the following are true?
select all that apply.
a. \\(x = 6\\)
b. \\(x = 30\\)
c. \\(m\angle rdt = 77^\circ\\)
d. \\(m\angle adr = 29^\circ\\)
e. \\(x = 4\\)

Explanation:

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<reasoning>

Identify the relationship between the angles

Using the Angle Measurement knowledge point
The double arc markings on \(\angle ADR\) and \(\angle RDT\) indicate that these two angles are congruent.
\[
m\angle ADR = m\angle RDT
\]

Set up the algebraic equation

Using the Angle Measurement knowledge point
\[
2x + 17 = 3x - 13
\]

Solve for x

Using the Angle Measurement knowledge point
\[

$$\begin{aligned} 17 + 13 &= 3x - 2x \\ 30 &= x \end{aligned}$$

\]

Calculate the angle measures

Using the Angle Measurement knowledge point
\[

$$\begin{aligned} m\angle ADR &= 2(30) + 17 = 77^\circ \\ m\angle RDT &= 3(30) - 13 = 77^\circ \end{aligned}$$

\]

Evaluate the given options

Using the Angle Measurement knowledge point

• Option A: \(x = 6\) (False)
• Option B: \(x = 30\) (True)
• Option C: \(m\angle RDT = 77^\circ\) (True)
• Option D: \(m\angle ADR = 29^\circ\) (False, it is \(77^\circ\))
• Option E: \(x = 4\) (False)

</reasoning>

<answer>
<mcq-option>(A) \(x = 6\)</mcq-option>
<mcq-correct>(B) \(x = 30\)</mcq-correct>
<mcq-correct>(C) \(m\angle RDT = 77^\circ\)</mcq-correct>
<mcq-option>(D) \(m\angle ADR = 29^\circ\)</mcq-option>
<mcq-option>(E) \(x = 4\)</mcq-option>
</answer>

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Answer:

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<reasoning>

Identify the relationship between the angles

Using the Angle Measurement knowledge point
The double arc markings on \(\angle ADR\) and \(\angle RDT\) indicate that these two angles are congruent.
\[
m\angle ADR = m\angle RDT
\]

Set up the algebraic equation

Using the Angle Measurement knowledge point
\[
2x + 17 = 3x - 13
\]

Solve for x

Using the Angle Measurement knowledge point
\[

$$\begin{aligned} 17 + 13 &= 3x - 2x \\ 30 &= x \end{aligned}$$

\]

Calculate the angle measures

Using the Angle Measurement knowledge point
\[

$$\begin{aligned} m\angle ADR &= 2(30) + 17 = 77^\circ \\ m\angle RDT &= 3(30) - 13 = 77^\circ \end{aligned}$$

\]

Evaluate the given options

Using the Angle Measurement knowledge point

• Option A: \(x = 6\) (False)
• Option B: \(x = 30\) (True)
• Option C: \(m\angle RDT = 77^\circ\) (True)
• Option D: \(m\angle ADR = 29^\circ\) (False, it is \(77^\circ\))
• Option E: \(x = 4\) (False)

</reasoning>

<answer>
<mcq-option>(A) \(x = 6\)</mcq-option>
<mcq-correct>(B) \(x = 30\)</mcq-correct>
<mcq-correct>(C) \(m\angle RDT = 77^\circ\)</mcq-correct>
<mcq-option>(D) \(m\angle ADR = 29^\circ\)</mcq-option>
<mcq-option>(E) \(x = 4\)</mcq-option>
</answer>

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