Question

in the diagram below points g, h, and i are collinear. which conjecture is not necessarily true? ∠ghk≅∠khi m∠ghj + m∠jhi = 180° ∠ghj and ∠jhi form a linear pair ∠ghk and ∠jhi are vertical angles

Explanation:

Step1: Recall linear - pair and vertical - angle properties

A linear pair of angles are adjacent angles that form a straight line and their sum is 180°. Vertical angles are non - adjacent angles formed by two intersecting lines and are congruent.

Step2: Analyze each option

• Option 1: $\angle GHK$ and $\angle KHI$ are not necessarily congruent. There is no information in the problem to suggest they are equal.
• Option 2: Since points $G$, $H$, and $I$ are collinear, $\angle GHJ$ and $\angle JHI$ form a linear pair. By the definition of a linear pair, $m\angle GHJ + m\angle JHI=180^{\circ}$.
• Option 3: $\angle GHJ$ and $\angle JHI$ share a common side and a common vertex, and their non - common sides form a straight line. So, they form a linear pair.
• Option 4: $\angle GHK$ and $\angle JHI$ are non - adjacent angles formed by the intersection of two lines (the lines containing $\overrightarrow{GK}$ and $\overrightarrow{JI}$). So, they are vertical angles.

Answer:

$\angle GHK\cong\angle KHI$ is not necessarily true.