Question

identify the pair(s) of congruent angles in the figure. explain how you know they are congruent.
figure: lines with points g, l, m, h, j, k (k is perpendicular to gj?)
options:

• ☐ ∠gml ≅ ∠hmj by the vertical angles congruence theorem
• ☐ ∠jmk ≅ ∠gml because they form a linear pair and are supplementary to congruent angles
• ☐ ∠gmh ≅ ∠lmj by the vertical angles congruence theorem
• ☐ ∠hmj ≅ ∠lmk by the vertical angles congruence theorem
• ☐ ∠gmk ≅ ∠jmk by the right angles congruence theorem - they form a linear pair, which means they are supplementary by the linear pair postulate, and because one is a right angle, so is the other by the subtraction property of equality.

Explanation:

To solve this, we analyze each option using angle properties (Vertical Angles Theorem, Right Angles Congruence, Linear Pair, Subtraction Property of Equality):

Option 1: $\boldsymbol{\angle GML \cong \angle HMJ}$ by Vertical Angles Theorem

Vertical Angles Theorem states that vertical angles (opposite angles formed by intersecting lines) are congruent. $\angle GML$ and $\angle HMJ$ are vertical angles (formed by intersecting lines $GLH$ and $GJH$ at $M$), so they are congruent by this theorem. This is valid.

Option 2: $\boldsymbol{\angle JMK \cong \angle GML}$ (linear pair + supplementary to congruent angles)

A linear pair of angles sums to $180^\circ$ (supplementary). However, $\angle JMK$ and $\angle GML$ are not necessarily supplementary to congruent angles in a way that implies congruence. This reasoning is flawed (supplementary to congruent angles does not guarantee congruence unless the angles are right angles, which is not established here).

Option 3: $\boldsymbol{\angle GMH \cong \angle LMJ}$ by Vertical Angles Theorem

$\angle GMH$ and $\angle LMJ$ are vertical angles (formed by intersecting lines), so they should be congruent. Wait—did we mislabel? Wait, the diagram: $G$ and $J$ are on one line, $L$ and $H$ on the other, with $MK \perp GJ$. Wait, maybe a typo, but the key is vertical angles. However, the first option ($\angle GML \cong \angle HMJ$) is clearly vertical angles.

Option 4: $\boldsymbol{\angle HMJ \cong \angle LMK}$ by Vertical Angles Theorem

$\angle HMJ$ and $\angle LMK$: Are these vertical angles? No—vertical angles are opposite each other. $\angle HMJ$ and $\angle GML$ are vertical, $\angle LMK$ and $\angle GMH$ would be vertical. So this is incorrect.

Option 5: $\boldsymbol{\angle GMK \cong \angle JMK}$ by Right Angles Congruence + Subtraction

$MK \perp GJ$ (right angle symbol), so $\angle GMK$ and $\angle JMK$ are right angles? Wait, $GJ$ is a straight line, and $MK$ is perpendicular, so $\angle GMK = \angle JMK = 90^\circ$. By Right Angles Congruence Theorem (all right angles are congruent), they are congruent. The explanation adds linear pair (supplementary) and subtraction, but the core is right angles congruence. However, the first option ($\angle GML \cong \angle HMJ$) is a direct application of Vertical Angles Theorem, which is more straightforward.

Final Analysis:

The most correct and straightforward is Option 1 ( $\angle GML \cong \angle HMJ$ by Vertical Angles Theorem ), as vertical angles are always congruent, and this pair is clearly vertical.

Answer:

$\boldsymbol{\angle GML \cong \angle HMJ}$ by the Vertical Angles Congruence Theorem (because vertical angles formed by intersecting lines are congruent).