Answered


Statements		Reasons

$\begin{array}{l}
m \angle TRV=60^{\circ}

m \angle TRS=(4 x)^{\circ}
\end{array}$		1. given

$\angle TRS$ and $\angle TRV$ are a linear pair		2. definition of linear pair

$m \angle TRS + m \angle TRV = 180^{\circ}$		3. ?

$60+4 x=180$		4. substitution property of equality

$4 x=120$		5. subtraction property of equality

$x=30$		6. division property of equality






What is the missing reason in step 3?


Aaddition property of equality
Bsubtraction property of equality
Cangle addition postulate
Dlinear pair postulate

Verified Answer :

  • The definition of a linear pair refers to adjacent angles that form a straight line.
  • The Linear Pair Postulate states that angles forming a linear pair are supplementary, meaning their sum is 180°.
  • Step 3 of the proof correctly applies this postulate to relate the measures of \(\angle TRS\) and \(\angle TRV\).
  • The correct option is d. \(\boxed{d}\).

Explanation

  1. Understand the Linear Pair Postulate

A linear pair of angles consists of two adjacent angles formed by the intersection of two lines. Their non-common sides form a straight line, which measures 180°. The fundamental geometric rule that states the angles in a linear pair are supplementary (i.e., their measures sum to 180°) is known as the Linear Pair Postulate.

  1. Identify the missing reason

In the provided proof, step 2 identifies that \(\angle TRS\) and \(\angle TRV\) form a linear pair. By applying the Linear Pair Postulate, we directly conclude that the sum of the measures of these two angles must be 180°. Therefore, the reason for the statement \(m\angle TRS + m\angle TRV = 180^{\circ}\) is the Linear Pair Postulate.

  1. Conclusion and final selection

The provided options are:

Aaddition property of equality
Bsubtraction property of equality
Cangle addition postulate
Dlinear pair postulate

Based on our analysis, the correct reason is the linear pair postulate, which corresponds to option d.

Examples

Imagine you are designing a staircase. If you have two adjacent steps that meet at a straight edge, the angle formed by the wood along that edge is 180 degrees. If you measure one angle where the support beam hits the step as 60 degrees, you can use the Linear Pair Postulate to immediately determine the other side of that angle is 120 degrees, ensuring your structural cuts are precise and perfectly aligned.

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